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COMPLETELY SHAMEFUL PLUG: FALL 2024

I will be teaching a dyadic data analysis course in Fall 2024 (HDFS 6583: Dyadic Data Analysis) will focus on dyadic data and finding + obtaining dyadic data from existing sources

Software free (mplus and lavaan will be used for demonstrations)

Topics

How to work with non-independent/clustered data

Clustered adjusted SE, fixed effect approaches

Greater focus on measurement / levels of measurement

Generate preliminary ideas of scale development / “develop” scale that is consistent with theory

Determining whether dyadic analyses are needed

Is the dyadic nature of data a nuisance or is it a part of the research question?

SEM vs MLM
More depth on models discussed in this workshop (APIM / CFM)

APIM (mediation/moderation)
CFM (mediation/moderation/individual/dyadic CFM

Discuss other dyadic models with a focus on longitudinal models

Multivariate APIM
Moderated Mediation within the APIM (Maybe common fate?)
Common Fate Growth Modeling
Hybrid models (APIM-CFM)
Modeling differences among dyads (Latent Congruence Model / Latent Score model)

Possible integration of missing data

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ASK QUESTIONS AT ANY TIME

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TENTATIVE SCHEDULE

DAY 1

Morning Session 1 (9-1030)

Introduction to Structural Equation Modeling
Theoretical Underpinnings and Need for Dyadic Data
Basic Principles of Dyadic Data

Morning Break (1030-1045)
Morning Session 2 (1045-12)

Actor Partner Interdependence Model

Basic APIM

Lunch 12-1
Afternoon Session 1 (1-230)

APIM with Mediation

Afternoon Bread (2-245)
Afternoon Session 2 (245-4)

APIM with Moderation

Individual Consultation (4-5)

DAY 2

Morning Session 1 (9-1030)

APIM

APIM with Within-Dyad Effects

Morning Break (1030-1045)
Morning Session 2 (1045-12)

Basic CFM

Lunch 12-1
Afternoon Session 1 1-230)

Common Fate Model

CFM with Mediation
CFM with Moderation

Afternoon Bread (2-245)
Afternoon Session 2 (245-4)

CFM with Individual and Dyadic Effects

Individual Consultation (4-5)

Tentative: Based on Time

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LOCATION OF FILES USED IN WORKSHOP

Access to files will be available through my website:
https://sites.google.com/view/michaelfitzgeraldphd/home
There is a page called dyadic data workshop 2024

Linked to a google drive will be the powerpoint as well as data files + input files

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OBJECTIVES OF TWO DAY WORKSHOP

Understand the need for dyadic data / consequences of not addressing dyadic / nested data
Describe basic dyadic principles and how they map onto analytics
Connect level of measurement to choice of dyadic model
Describe the Actor-Partner Interdependence Model and articulate what types of research questions the model can answer
Describe the Common Fate Model and articulate what types of research questions the model can answer
Compare and contrast the APIM with the CFM

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OBJECTIVES OF TWO DAY WORKSHOP

We will focus more on the conceptual aspects of dyadic data
This workshop will not be an equation / matrix heavy workshop

There will be some equations but will only briefly discussed

There will also be a focus on coding the analyses in Mplus
Our analyses will focus on distinguishable dyads (e.g., cisgender, heterosexual couples), but time-permitting we will allow talk about and analyze indistinguishable dyads (e.g., monozygotic twins)

Rationale: most dyads in the social and medical sciences will be distinguishable

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SLIDES

At the time of writing this slide, there are over 500 slides – we will absolutely and unequivocally not get through all them
I will intentionally be skipping over dozens of slides but I present them so you can go back later and deeper your knowledge
Slide sets most likely to be skipped:

Advanced APIM
Moderated mediation in the APIM
Latent variable moderation
Many of the random-intercept cross lagged panel model slides

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STATISTICS 101, LINEAR REGRESSION, AND STRUCTURAL EQUATION MODELING

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BACK TO STATISTICS BASICS

A + B + C + D = 100% variance
A = unique variance of IV1 on DV

Semipartial correlation (square this and you get the unique variance in DV accounted for by IV1)

B = shared variance in DV accounted for by IV1 and IV2
C = unique variance of IV2 on DV

Semipartial correlation

D = error variance (unexplained variance)
A + B = Partial correlation = total influence of IV1 on DV
B + C = Partial correlation = total influence of IV2 on DV
a is the TRUE contribution of IV1 and c is the TRUE contribution of IV2

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APPLICATION TO DYADS

Now imagine that IV1 is partner 1 and IV2 is partner 2 (same variable measured in both partners) predicting partner 1’s outcome
In research using only individuals, you are capturing a and b (partial correlation)

But b isn’t unique to IV1 and shared with IV2
This is an overestimation!!!

In dyadic research we are capturing a + b + c (and d, I guess)!

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LINEAR REGRESSION: YOUR FIRST CAR

Traditional methods of estimation (e.g., OLS) have wonderful properties

You only need one more person/observation than number of variables in order to run the model
Closed form estimation
Very simple and easy to understand (relatively)

Assumptions of linear regression

The distribution of IVs is fixed and known, or there is no measurement error
Linear in the parameters (intercept, slope)

You can have age + age² in a regression model without violating assumptions

Multicollinearity is not a major issue (it almost never is)

Not a reason your model didn’t find significant results

Residuals are normally distributed
Residuals are independent of one another

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STRUCTURAL EQUATION MODELING: LAMBORGHINI

Structural equation modeling is a highly flexible set of statistical methods
Most of the analyses you learned in first year grad stats are embedded within SEM framework

Linear regression
ANCOVA / ANOVA / MANCOVA / MANOVA
T-test

But adds so much more!

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PATH MODELS OF THE GENERAL LINEAR MODEL WITHIN SEM

MULTIPLE REGRESSION

INDEPENDENT SAMPLES T-TEST

Group

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PATH MODELS OF THE GENERAL LINEAR MODEL WITHIN SEM

ANCOVA

REPEATED MEASURES ANOVA

Group

Covariate

Group

Y1 T2

Y1 T1

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SEM

Define Your Terms: What is the “best” model?
In OLS regression, the “best” estimates are a function of minimizing the sum of the square residuals
In SEM we are trying to maximize the likelihood that our model represents population values

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SEM

OLS regression DOES NOT test hypotheses and is not theoretically driven– I will die on this hill

It only rescales means, variances, and covariances
You can have a highly significant predictor, but your overall model is GARBAGE and you’d never know it
You are not “testing” your model

In linear regression we think of model fit as an R-square (that’s a variance, not the extent to which we are effective in reproducing the characteristics of our data or means, variances, and covariances)

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SEM

We have a theoretical model and we test how well our model fits the data
Our theoretical model is a testable hypothesis in SEM
We have an observed variance-covariance matrix and we have an estimated variance-covariance

Difference between them is our model fit (largely based on different forms of the non-centrality parameter (x2-df) –CFI, TLI, RMSEA
More on these statistics in a few slides

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SO WE CAN HAVE COMPLEX MODELS LIKE THIS

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SEM

Because our models are testable hypotheses, we can make modifications to our model

This is an exceptionally slippery slope and misguided and well-intention researchers alike can misuse this feature
Simulation research shows that if you the follow the modifications provided by your SEM package, you almost never get back to the correct model
This is a data driven approach and you just end up back in the linear regression / EFA world

We can identify potentially problematic areas of our model and make appropriate and theoretically justified adjustments

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BENEFITS OF SEM: CONTROL

EFA -> Throw your items in the model and start praying
Multiple Regression -> Constrain one path to be 0? Not a chance.
Multiple group analyses don’t exist – testing moderation relies on the product method
Repeated measures ANOVA, ANCOVA, T-Test can only use measure variables (measurement error)

Repeated measures ANOVA: Using an indicator with alpha of .70 at time 1 and time 2 (good-bye power)

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BENEFITS OF SEM: ACTUALLY TESTING HYPOTHESES

Related to having more control, SEM allows our theories, hypothesis, and models to be falsifiable

With regression, you get what you get -> not an appropriate test of theory

As researchers, we should strive to put our theories and models in mortal danger of being wrong

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BENEFITS OF SEM: LATENT VARIABLES

Bollen (2002) defines a latent variable as a variable “for which there is no sample realization for at least some observation in a given sample”

Definition differentiates the values a variable could assume and the value it does assume when measured
Only assumes that the latent construct has not been measured for individuals within the sample
Extremely broad and general

Most common operationalizations of latent variables are within a confirmatory factor analysis context (effect indicators)
Other forms of latent variable: residuals, formative latent constructs, phantom variables

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BENEFITS OF SEM: LATENT VARIABLES

Modeling things at the construct level – we talk about constructs in our intro + discussed but don’t measure constructs in our methods and analyses

Depressive symptoms vs major depression

Attenuates measurement error
Examine how well our proposed model fits the data (in most latent variable models)

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LATENT VARIABLES

Conceptual Definition is the verbal/written idea – describes how the concept is believed to exist
Focal concept is the concept in how it exists in the world, but it is itself unmeasured. Definition of focal concept provides a variety of ways in which the concept manifests

Nomological network

Proxies are quantitative representations of concepts
Proxies are not equivalent to focal concepts

Conceptual Definition

Focal Concepts

Proxy

Indicator 1

Indicator 2

Indicator 3

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SIMPLEST (RELATIVELY SPEAKING) LATENT VARIABLE MODEL: CONFIRMATORY FACTOR ANALYSIS

Accounts for measurement error
Measures variables at the level in which you theoretically discuss them

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CONFIRMATORY FACTOR ANALYSIS MODEL

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BENEFITS OF SEM: MULTIVARIATE OUTCOMES AND FORMAL TESTS OF MEDIATION

In GLM, multilevel modeling (in general) you are limited to univariate outcomes that are observed

Don’t get me started on MANOVA or MANCOVA

In SEM, you can model 1 outcome or 10 outcomes!
In SEM, we have formal tests of mediation

Bootstrapping or Robust Standard errors
Avoids problems with Sobel test (non-normal distribution of indirect effects) and non-normal residuals (Robust Maximum Likelihood)

We can use observed variables (path analysis), latent variables (latent variable structural equation models), or both!

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BENEFITS OF SEM: FIT STATISTICS (KIND OF)

In SEM, we aim to match our theoretical model to the empirical data and determine to what extent there is “misfit”
Good News: Numerous Common Fit Statistics

Chi-square test
CFI
TLI
RMSEA
SRMR

Bad News: Fit statistics in SEM are a MESS
Local fit indices are helpful in identifying what aspects of the model you are estimating well and what aspects are being over or under estimated

Use gestalt approach integrating both global and local fit

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BENEFITS OF SEM: FIT STATISTICS (KIND OF)

Do NOT say you have good model-data fit because your CFI or TLI are above .95 or RMSEA below .05

CFI at .90 could be a wonderful fit (e.g., large sample, many latent variables) or .95 could be a poor fit (e.g., simple model with small sample)

If you cite Hu and Bentler (1999), you’re probably citing it incorrectly

They only tested a CFA and they said that models below .90 could be improve upon (not the same as good fit)
I’ve cited Hu and Bentler – this is not a holier than thou, it is a be careful

Model fit statistics provide an OVERALL evaluation of fit, but may not identify a mispecified model

Monte Carlo research suggests that many fit statistics would not effectively identify a misspecification

Use local-misfit (residual matrix) in conjunction with traditional fit statistics

Observed – expected covariance matrix

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STEPS IN SEM

Specification: Outline your theoretical model (White Board)
Identification: Ensure your model can be estimated
Estimation: Running your model with the appropriate method of estimation
Evaluation: To what extent does your model reproduce the characteristics of your data – notice how this isn’t relevant to significance?
Modification: Refine your model (e.g., mod indices, addressing fit issues)
Interpretation: Estimates in the final model are interpreted

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RETURNING TO OUR LAMBORGHINI

Lamborghinis are phenomenally sexy
Turns out they can’t handle speed bumps, trips to the grocery store, kids, or other basic things are expeptionally challenging, and the maintenance is STUPID expensive

Small sample sizes, intensive longitudinal data, model-data fit is a mess, can be overly complex, and generally requires a large sample size

But for many dyadic analyses, it remains a Lamborghini

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HYPOTHETICAL EXAMPLE OF A BETWEEN DYADIC DATA FILE IN SEM

Dyad ID	Male_Satisfaction	Female_Satisfaction	Male_DyadCope	Female_DyadCope	Relationship Length	Have_Kids
1	5	2	19	16	10	0
2	4	5	22	22	8	1
3	6	4	23	20	2	1
4	4	5	16	24	4	0
5	3	3	14	10	9	1
6	6	5	25	23	1	1

Continuous Between dyad moderating variable

Binary Between dyad moderating variable

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FOUNDATIONAL PRINCIPLES IN DYADIC DATA

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EXAMPLE OF DYADIC DATA

Some Obvious Interdependent Examples

Parent-Child
Married Couples
Siblings

Perhaps Less Obvious Examples

Teachers and children / children within same classroom
Roommates
Employer / Employee
Clients seeing the same therapist in a clinical trial (multiple clients nested within therapist)

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THEORETICAL UNDERPINNINGS

Interdependence Theory (Kelley & Thibault, 1978)

Between person relationships are just as important as the individual
Actor Control: the impact of each person’s actions on his or her own outcomes
Partner Control: the impact of each person’s actions on the partner’s outcomes
Joint Control: the impact of the partners’ joint actions on each person’s outcomes

Family systems theory

There are sequences of interaction where people are acting and reacting to others
The whole is greater than the sum of its parts
Shared characteristics and values among family members

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THEORETICAL UNDERPINNINGS

Most social and family science researchers presume that an individual’s outcomes are a function of their own thoughts, emotions, behavior, perception, attitudes as well as people they are interdependent on/with
Degree of interdependence varies across people and over time

Levels of interdependence between friends and romantic partners
Levels of interdependence of twins starting school vs after getting married vs after parents passing away

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THE NEED FOR DYADIC DATA

In dyadic data, individuals are nested within dyad
Their residuals are correlated with one another -> violates the assumption of independence of observations

Downwardly biases your standard errors, upwardly biases your test statistics -> Type 1 errors

You do not have uniquely independent pieces of information: if I know something about dyad member 1 then I also know a little something about dyad member 2

Dyad member 2’s information is not unique

There is/are a common reason(s) outside of your model for why relationship quality scores, for example, between marital partners would be correlated

Violates the independence of observations assumptions

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THE NEED FOR DYADIC DATA

A second reason for needing dyadic data is that it allows us to investigate research questions related to 1) individuals affecting one another
Cannot get at this with a single reporter

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THE NEED FOR DYADIC DATA

Couples therapist track sequences

When Betty makes an accusation, Chris gets defensive
When Chris gets defensive, Betty becomes more critical
When Betty gets more critical, Chris then becomes critical
When Chris becomes critical, Betty shuts down
Quick assessment of sequences of interaction – likely reflects are larger pattern within the relationship

Individuals affecting one another or my behavior affects your behavior

According to theory, these behaviors become unconscious and patterned over time

Called the Actor-Partner Interdependence Model

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THE NEED FOR DYADIC DATA

Mutual influence is inherently interpersonal in nature, but is it dyadic?

Dyadic in data collection and methods, but is it measuring dyadic processes??
Isn’t everything measured at the individual level?

We can also test hypothesis about what goes on at the dyad level
Think of a car engine

It has hundreds if not thousands of parts
The parts on their own are just the parts but the engine does not run even though you have all the parts
But when they are assembled in a specific way and work together, the engine runs

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THE NEED FOR DYADIC DATA

I’m trained as a couple’s therapist and have worked with individuals, couples, and families for nearly a decade
After seeing someone for a while, they develop trust in what I say, do, and offer

This means I can take more “risks” as a therapist and, for example, give them a hard truth that the client needs to hear and clients may have a reaction to that

My behavior (give hard truth) -> their behavior (get upset or anxious)

If giving the hard truth doesn’t go well, the client doesn’t automatically disengage from therapy because there is an underlying relationship (attachment, respect, appreciation, understanding) between us

I rarely give hard truths in the first session -> clients wouldn’t ever come back!!!

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THE NEED FOR DYADIC DATA

Partner 1, Partner 2, and what occurs between the partners OR is shared by partners

Think about your partner and bring to mind that you do both do that you established as a couple (e.g., how to communicate, how to manage stress, daily routines)

When you are with your partner, over time you have established ways of communicating, develop shared beliefs, and forge shared meaning

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THE NEED FOR DYADIC DATA

What is shared between partners is analogous to the whole being greater than the sum of it’s parts

1 + 1 = 3

Key here is that the processes being modeled are at the relationship level (level 2 for you multilevel modelers) and NOT at the individual levels

Common Fate Model

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THE NEED FOR DYADIC DATA

The APIM accounts for interdependence through the partner effects and covariances between each dyad members reports
The CFM accounts for covariation by identifying shared variation between partners that causes each partner’s reports

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THE NEED FOR DYADIC DATA: CORRALATENT

Corralatent is a completely made up term to remember how non-independence is handled in dyadic analyses
Corralatent

Correlation: correlating report of dyad members
Latent: a latent variable of shared processes causes each dyad member’s responses

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TYPES OF DYADS

There are dyads that can be theoretically differentiated based on a measured characteristic (e.g., gender)

Refers to having a unique and specific role within the dyad -> how you assign people to either be dyad member number 1 and 2

Two flavors of dyad types

Distinguishable
Indistinguishable

There are implications for theory, statistical power, and model estimation

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DISTINGUISHABLE DYADS

There are two levels of distinguishability

Theoretical – you have a theoretically derived reason to systematically create dyad assignments (e.g., gender, parent/caregiver, parent/child)
Empirical – you can have a theoretically derived rationale for, but there may or may not be statistical evidence to support the theoretical propositions

Obviously, empirical distinguishability is a testable hypothesis

I-SAT model: constraining, means, variances, covariances, actor and partner effects to be equal

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DISTINGUISHABLE DYADS

Distinguishable dyads

A measured variable that places both partners into mutually exclusive groups.

Variable MUST be theoretically relevant to the research question(s)

Avoid arbitrary designations such as first person who enrolls in study is person 1 and 2nd person is person two – assumes distinguishability when there isn’t

Can provide biased estimates and standard errors

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DISTINGUISHABLE DYADS

Dyads can be distinguished in multiple ways within a single data set
Studying heterosexual couples where one partner has cancer and the other is an informal caregiver.

Cancer status (patient / caregiver)
Gender (male / female)
Age (older vs younger partner)

Example: if studying emotional support and mental health in heterosexual couples where one person has cancer -> distinguish across role (patient/caregiver) and not gender (male female)

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INDISTINGUISHABLE DYADS

Indistinguishable dyads

Assignment into groups is random / arbitrary
Same sex twins (unlikely to have variable based on who was born 1^st and is this variable conceptually meaningful?)
Distinguishable indicators should relate to research question

If your theory says there are notable differences then you should measure that difference to make them distinguishable

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BEFORE COLLECTING DYADIC DATA

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PLEASE DON’T COLLECT DATA AND THEN TRY AND MAKE IT DYADIC ON THE BACK END

THERE ARE DESIGN AND MEASUREMENT FEATURES THAT WE NEED TO CONSIDER ON THE FRONT END

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BIG IDEAS IN MEASUREMENT

Measurement is the principled assignment of numerals to observations according to rules (Stevens, 1946)
Measurement is also an attempt to understand the nature of a variable
Common Measurement strategies

Mixture Modeling
Classical Test Theory
Item Response Theory
Moderated Non-Linear Factor Analysis

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LEVEL OF MEASUREMENT IN DYADIC RESEARCH

Individual level: questions that are self-referential Examples: Personality, mental health
Partner level: questions that references an individual’s perception about their partner
Dyadic / relationship level: question reference the relationship

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THE NEED FOR DYADIC DATA

Level of measurement can help clarify the “abstractness” of relationship level processes

Individual Level
Partner Level
Relationship Level

Individual Level: “I could not get going” – self referential
Partner Level: “My partner is warm and supportive” – referential to the partner’s behavior
Dyad Level: “All things considered, I’m happy in my relationship.” – referential to relationship

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THE NEED FOR DYADIC DATA

Level of measurement can help clarify the “abstractness” of relationship level processes

Individual Level
Partner Level
Relationship Level

Individual Level: “I could not get going” – self referential
Partner Level: “My partner is warm and supportive – referential to the partner’s behavior
Dyad Level: “All things considered, I’m happy in my relationship.” – referential to relationship

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CONSIDERATIONS WHEN COLLECTING DYADIC DATA: MEASUREMENT

Assess both partner using the same variables
You can assess both individual, partner, and dyadic processes
”Target” of assessment can be the individual, partner, or relationship

Self Referential: When arguing, I can be defensive (1-5 Likert) – self perception
Partner Referential: When arguing, my partner is defensive (1-5 Likert) – perception of partner
Relationship Referential: When arguing, both my partner and I can both be defensive (1-5 Likert) – dyad level

Framing of the questions you ask, changes in the interpretation of the effects AND the analyses you can run

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COPING SCALE

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DYADIC COPING INVENTORY

37 item scale ranging from very rarely to very often
Assesses stress communication processes in couples
Numerous Subscales

Perceptions of self and partner stress communication processes
Shared stress communication processes
Satisfaction with stress communication processes

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DYADIC COPING INVENTORY

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DYADIC COPING INVENTORY PARTNER STRESS COMMUNICATION SUBSCALE: WHAT IS THE UNIT OF ANALYSIS?

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DYADIC COPING INVENTORY – COMMON DYADIC COPING SUBSCALE: WHAT IS THE UNIT OF ANALYSIS

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DYADIC COPING INVENTORY – SATISFACTION SUBSCALE: WHAT IS THE UNIT OF ANALYSIS?

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THE VERDICT

I love the scale, but…. there are problems when creating on overall composite
1) violates assumptions of Cronbach’s alpha (unidimensionality, tau-equivalence)

Use Omega Hierarchical

2) there are multiple levels of measurement

Self referential items
Partner referential items
Couple referential items

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THE VERDICT

Combining the individual and dyadic levels of measurement can potentially be problematic

All items are an individual’s perception, but there are problems with different units of analyses

Conduct analyses using the subscales and avoid using the total score

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THE VERDICT: THIS IS NOT A HOLIER THAN THOU

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ANOTHER EXAMPLE: WE DISEASE

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CONSIDERATIONS WHEN COLLECTING DYADIC DATA: MEASUREMENT

Measurement invariance is a critical, yet rarely tested feature of dyadic data

Not a holier than thou….I’m guilty of such sins!

When you have two reporters, measurement equivalence is critical

Otherwise, you are modeling measurement error and biasing parameter estimates and standard errors -> Type I and Type II errors

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CONSIDERATIONS WHEN COLLECTING DYADIC DATA: MEASUREMENT

Even if using a measured variable path model, invariance tests are still needed.

More important in models that estimate means (growth curve, latent congruence model) – presumes strong tests of invariance

Without testing invariance, we are a hopin and a wishin that measurement isn’t a problem

Like eating hot dogs and sausage – just let me enjoy it and don’t ever tell me how it’s made

A sum score is imposing an unseen latent variable measurement model

All factor loadings are equivalent and identical residual variances

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PARALLEL MEASUREMENT MODEL WITH BOTH MEMBERS OF THE DYAD

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TAU EQUIVALENT MEASUREMENT MODEL WITH BOTH MEMBERS OF THE DYAD��FREE THE RESIDUALS

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CONSIDERATIONS WHEN COLLECTING DYADIC DATA: MEASUREMENT

Differences in scores across the distinguishing factor (e.g., gender) are hiding within the sum score – if items operate differently across gender (non-invariant) that is not detectible using sum scores
Non-invariance may result in biased parameter estimates that pops up other place in the model

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CONSIDERATIONS WHEN COLLECTING DYADIC DATA: MEASUREMENT

That being said….The mean is always wrong, but we know how it’s wrong…except when we don’t – statistics is actually an emotional roller coaster
Sometimes you make a sacrifice and use a mean so that you can do other things in your model

Knowing to what extent measurement is a problem is critical (knowledge is power)

Lose the battle to win the war

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NOW, FOR WHY YOU ARE HERE: DYADIC MODELS

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NUMEROUS TYPES OF DYADIC ANALYSES

Actor Partner Interdependence Model

Mediation (APIMeM)
Moderation (APIMoM)

Common Fate Model

Mediation Model
Moderation Model

Latent Congruence Model

Growth Modeling

Dyadic Growth Curve
Dyadic Growth Curve with Structured Residuals
Common Fate Growth Model

Social Relations Model (Triadic)
Hybrid APIM-CFM Models

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ACTOR PARTNER INTERDEPENDENCE MODEL

Partner 1

Partner 2

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ACTOR PARTNER INTERDEPENDENCE MEDIATIONAL MODEL

Partner 1

Partner 2

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ACTOR PARTNER INTERDEPENDENCE MODERATION MODEL

Partner 1

Partner 2

X1 x X2 (INT)

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BASELINE COMMON FATE MODEL

Partner 1

Partner 2

Partner 1

Partner 2

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COMMON FATE MODEL WITH INDIVIDUAL AND DYADIC EFFECTS

Mother

Father

Mother

Father

Phantom Variables

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COMMON FATE GROWTH MODEL

Time 1

Time 2

Time 3

Mother

Father

APIM-CFM HYBRID MODEL: CROSS LEVEL MEDIATION

Y4A

Y4B

Y3A

Y3B

X1A

X2B

Y1A

Y2B

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WHAT MODEL DO I USE?!?!?

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WHAT MODEL DO I USE?!?!?

The default is pretty much the APIM – it is the most widely used (and it’s not all that close)

This is highly problematic -> creates chasm between theory and research
Many research questions are implemented using the APIM, but are better fit for the CFM (Galovan et al., 2017)

The APIM is easy to understand and estimate (can be done in MLM and SEM)

It’s a dizygotic twin of the cross-lagged panel model (path model the same, degrees of freedom, only difference is rather than 2 constructs, it’s two people)

Relationship and family levels of analysis are seldom taught in graduate school

Even in family systems and attachment theories, teaching is focused on the interaction nature and not what is common/shared between partners

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WHAT MODEL DO I USE?!?!?

The statistical model you choose is based on your THEORY and RESEARCH QUESTION

What does your theory say?
What is the unit of analysis? Individual or dyad?

Unit of analysis is at the family level -> SRM / CFM
Unit of analysis at the dyadic level -> CFM
Unit of analysis at the individual level / mutual influence -> APIM or DGC-SR
Unit of analysis is at multiple levels -> CFM or Hybrid models
Unit of analyses is similarities and differences in dyad -> LCM

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WHAT WE WILL COVER

Actor Partner Interdependence Model

Base APIM with Distinguishable and Indistinguishable Dyads
APIM with Mediation
APIM with Moderation
Random-Intercept Cross Lagged Actor Partner Interdependence Model (RI-CL-APIM)

Common Fate Model

Base Common Fate Model
Common Fate Model with Individual and Dyad Level Effects
Common Fate Model with Mediation
Common Fate Model with Moderation

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ACTOR PARTNER INTERDEPENDENCE MODEL

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APIM

Consistent with theory (e.g., FST, interdependence theory), family scientists widely recognize that we affect other people and other people affect us

APIM is a statistical method to examine the reciprocal effects

Visually and statistically identical to a 2 wave cross-lagged panel model

Interpretations of coefficients change

Describes patterns of mutual influence between dyad members

Actor effects: how an individual affects their own outcome
Partner effects: how an individual affect the other dyad member’s outcomes

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APIM

Designed to estimate intra individual and interindividual effects

Many variables may not have a relational impact on those we are interdependent on

Example: My educational achievement impacting my brother’s level of depression
But there are common causes of both (e.g., family of origin)

My neuroticism, however, is likely to have an impact on a partner or friend

Estimates the extent to which I influence my own outcomes BEYOND how much my partner influences my outcomes and how much my partner influences my outcomes BEYOND my own influence

INTERDEPENDENCE WITHIN THE APIM

Within the APIM, interdependence is accounted for in several statistical way

Covariances between each partner’s reports on IV and DV
Partner effects where each dyad member affects the other
This is the “corra” in corralatent

Theoretical ways

The dyad members were related to each other before formally becoming a dyad

Example: Assortive mating – couples are similar to each other on values, beliefs, interests and behavior and those factors are what attracted them to each other
Homogamy: tendency to date/marry someone from a similar background

Dyad members can come from the same origin (e.g., born to the same parents)

Shared environment and genetics influences

Share a current environment (e.g., roommates)

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ACTOR PARTNER INTERDEPENDENCE MODEL

Physical Health

Happiness

Sibling 1

Sibling 2

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RELATIONSHIP QUALITY AND PSYCHOPATHOLOGY EXAMPLE

Consider the association between relationship quality and depressive symptoms
We report on our own perceptions of relationship quality and our own depressive symptoms (traditional linear regression)

But can we fully understand the proposed relationship without considering our spouse?

How our spouse evaluates the relationship is likely to ALSO impact our levels of depression

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ACTOR PARTNER INTERDEPENDENCE MODEL WITH MEAN STRUCTURE DISPLAYED

Husband

Wife

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ACTOR PARTNER INTERDEPENDENCE MODEL: MARITAL QUALITY AND DEPRESSION

Depression

Husband

Wife

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WHAT ARE OTHER RESEARCH QUESTIONS FOCUS ON MUTUAL INFLUENCE?

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RESEARCH QUESTIONS THAT CAN BE ANSWERED BY AN APIM

Mutual Influence

To what extent do husband’s unhappiness in his marriage influence his own level of depressive symptoms and his wife’s level of depressive symptoms?

Individual Outcomes Within Context

Does a higher quality parent-child relationship with the mother compared to their fathers strengthen peer relationships among adolescents?

Statistically control for father effects to discern the effects of mothers

Compare Effects Between Dyad Members

Do husbands’ marital happiness influence wives’ depression in a stronger way that wives’ marital happiness influencing their husband’s

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RESEARCH QUESTIONS THAT CANNOT BE ANSWERED BY AN APIM

Does marital satisfaction decrease over time for husbands and wives among newlywed couples?

Assess individual constructs within a dyad – good!
Dyadic Growth Curve (yes the APIM can be testing longitudinally but people start throwing things when you call two timepoint methods “change;” trying to avoid a riot here)
More accurately, “rates of change”

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RESEARCH QUESTIONS THAT CANNOT BE ANSWERED BY AN APIM

Does marital satisfaction among heterosexual couples decrease after 1 year of marriage?

Common Fate model – unit of analysis is the relationship which the APIM does NOT estimate

Do differences in marital satisfaction predict parenting stress

Latent congruence model – focuses on the differences between partners

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BREAKING DOWN THE APIM

2 Actor effects: within person effects

Example: My own reports of marital satisfaction predict my own levels of depression
One actor effect per person

2 Partner effects: between person effects

Example: My own reports of marital satisfaction predict my partner’s levels of depression
One partner effect per person

2 Covariances / Correlations

1: Predictor variables for each person are assumed to be related to each other (husbands reports of satisfaction are correlated with the wife’s)
2: Residual variance of Y’s error terms: represents the non-independence not accounted for by the model

4 Variances

Variances for IV and DV for partner 1 and partner 2

4 Means / Intercepts

We generally don’t care about mean structures in SEM – saturated portion of the model but mean will come into play in the LCM, CFM, CFGM, DGC, and DGC-SR)
Means for IV and DV for partner 1 and partner 2

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APIM PARAMETERS

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EQUATIONS IN THE APIM

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MODEL-DATA FIT WITHIN SEM

The APIM with observed variables is saturated (degrees of freedom)

We have 14 pieces of information that can estimate and we are want information about 14 things -> we’re broke
This is only true for the APIM for distinguishable dyads

Things you can do to estimate model-data fit:

Constrain the actor effects to be equal (Likelihood ratio test; D or χ²)
Constrain the partner effects to be equal (Likelihood ratio test; D or χ²)
Use item level data to estimate a latent variable APIM

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RETURNING TO OUR MARITAL QUALITY AND DEPRESSION EXAMPLE

Depression

Husband

Wife

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ACTOR PARTNER INTERDEPENDENCE MODEL: RESULTS FROM EXAMPLE

Depression

Husband

Wife

β = -.20**

β = -.28**

β = -.10**

β = -.12**

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INTERPRETING ACTOR AND PARTNER EFFECTS

Interpretation when the measurement of the IVs and DVs are self referential (e.g., I am happy in my relationship, I feel depressed) for both dyad members

Use relationship quality and depression example

Actor effects:

Husbands who reporter higher levels of relationship quality tended to also report lower levels of depression (β = -.20, p < .001)
Wives who reported greater levels of relationship quality also tended to report lower levels of depression (β = -.28, p < .001)

Partner Effects

As levels of the husbands’ relationship quality increased, the wives’ levels of depressive symptoms tended to decrease (β = -.10, p < .01)
As levels of the wives’ relationship quality increased, the husbands’ levels of depressive symptoms tended to decrease (β = -.12, p < .01)

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ACTOR PARTNER INTERDEPENDENCE MODEL IN MPLUS

Provide a brief overview of Mplus
In the workshop, you will be provided all the syntax except for the coding of the model command section

I will guide you through the syntax
Learn by doing
Full syntax will be provided after the workshop

You can follow along using other programs (e.g., PROC CALIS or lavaan), but I may not be able to help troublehoot problems

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RUNNING THE APIM

Open Up MPlus
The basic APIM will be run using the following syntax file and data file

APIM.inp
Couples.dat

Create a folder on your desktop / dropbox / somewhere of convenience

Ensure your data file and syntax file are in the same folder

For in person folks, when leaving for lunch, I would move your files onto a jump drive / cloud / one drive to be safe

The files SHOULD be saved, but technology issues may occur

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ISAT

ISAT model (saturated model) constrains all means, variances, and covariances to be equal across dyad members

Constrain the means of the IVs + DVs to be equal across partners
Constrain actor effects (covariances) to be equal across partners
Constrain partner effects (covariances) to be equal across partners
Constrain the variances of each variable to be equal across partners

The actor and partner covariances are not constrained to be equal -> actor with actor and partner with partner
Provides a test of distinguishability

Goal is to confirm indistinguishability for indistinguishable dyads (test is non sig) and confirm distinguishability for distinguishable dyads (test is sig)

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ISAT FOR DISTINGUISHABLE DYADS

If iSAT is significant -> freely estimate parameters across dyad members

This is most typical in APIM research
Indicates that variances, covariances, or means are different across groups

If the iSAT is non-significant, then the dyads are theoretically indistinguishable, but not empirically distinguishable

Indicates that variances, covariances, or means are not different across groups -> group members can randomly assigned

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OUTPUT OF ISAT MODEL

Baseline Model has 0 df, CFI + TLI =1, RMSEA = 0
iSAT model isn’t horrible, but does significantly decrease model-data fit

Chisquare statistics
Difference in CFI

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PARAMETER ESTIMATES OF ISAT MODEL

Unstandardized results showing the results of the iSAT model
Textboxes with common color are the paths constrained to be equal

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ISAT

If iSAT is not significant AND there are no partner effects, then you do not need to use dyadic data – ML with cluster correct standard errors

Type = Complex in Mplus

Fixed effect approaches are less effective when there are numerous (dozens) of clustering variables

In the case of dyads, you would dummy code dyad ID
Not advisable most of the time in dyadic research (VERY effective when you have unbalanced designs such as students within teachers)

Study of 100 people

49 dummy variables for students 100 people (50 dyads) – unmanageable
10 teachers each with 10 kids then you need only 9 dummy variables – manageable

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MARITAL QUALITY AND PSYCHOPATHOLOGY EXAMPLE

Lets presume that we need to use an APIM -> we have a dyadic research question
We believe that husbands and wives are theoretically different across gender, but it remains unclear whether that is empirically supported in our data
We can empirically test whether male and female partners are empirically distinguishable ->ISAT model

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ACTOR PARTNER INTERDEPENDENCE MODEL: MARITAL QUALITY AND DEPRESSION

We run the iSAT model

Tests whether our theoretical distinguishability is empirically supported by conducting a likelihood ratio / chi-square test
We want the test to be significant! Indicates that the highly constrained model is worse than the freely estimated model

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RUNNING THE APIM WITH ISAT MODEL

Data File: couples.data
Syntax File: APIM ISAT

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DISTINGUISHABILITY EXAMPLE

Say you are interested in couple processes and how they affect health behavior

Communication and support predicting substance use and exercise

You recruit 500 couples

380 heterosexual
60 gay
60 lesbian

Who is your analytic sample?

Do you only test 380 dyads?
Do you run separate models for gay and lesbian couples?
Do you throw them in together and hope that gender is a distinguishable variable?

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DISTINGUISHABILITY EXAMPLE

Before making any analytic decisions, run an ISAT test on the distinguishable dyads
If the ISAT test is not-significant, then you can add in the gay and lesbian couples because gender does not influence means, variances, or covariances among partners
Greater statistical power, greater external validity
If they are distinguishable, then separate models for heterosexual and gay/lesbian would be appropriate

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APIM WITH INDISTINGUISHABLE DYADS

Previous discussion has been using dyads that are distinguishable (assuming both theoretical and empirical distinguishability), but we do things differently for indistinguishable dyads
Run ISAT Model and we want the test to be non-significant – tells us that there are no empirical differences between dyad members

If the ISAT model is significant then there are differences and determining what those differences are is needed

When indistinguishable dyads are used, a modified ISAT model is implemented

Utilize the full iSAT model to “confirm” indistinguishability (e.g., non-significant difference tests)
Constrain the actor and partner effects to equality in final model

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MPLUS FILES (NOT PERFORMING THIS ANALYSIS)

Input File: APIM Indistinguishable Dyads
Data File: twins.dat
This is for you to experiment with on your own. Same syntax is used as in the distinguishable dyad example

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TYPES VARIABLES WITHIN THE APIM

Within Dyad Variables:
Between Dyad Variables:
Mixed Variables

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WITHIN DYAD VARIABLES

Variables that do not vary across dyads but vary within dyads
Dyad members can have different scores (there is variation within the dyad), but each dyad will have the same score
Example: percentage of household chores or parenting

Some dyads will be 50/50 others 60/40 (within)
All percentages will add upto 100% (between)

Example: couples in therapy for conflict

Some dyads will have high levels of conflict and others with low level conflict
All dyads experiencing clinical levels of conflict

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BETWEEN DYAD VARIABLES

Between Dyad Variables: scores on a specific variable are the same for both members of a dyad, but varies between dyads.
Example: Household income – male and females (should) have the same household income (within dyad), but different couples have different levels of household income.
Relationship length – partners have been together for the same amount of time, but this will vary across dyads
Number of children

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TYPES VARIABLES WITHIN THE APIM

Mixed variables: varies both within and between dyads

Partner 1 and 2 in the same dyad have different scores
All partner 1s and all partner 2s in the sample can have different scores

Most variables within the social sciences will be mixed

Mental Health
Relationship Quality
Substance Use
Personality

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SEPARATION AND DIVORCE VARIABLES EXAMPLES

Coparenting Relationship Quality= Mixed Variable

Varies for both members of a dyad and across all analyzed dyads

Percentage of time spent with child = Within Dyad Variable

Time spent with child adds up to 100% of the time so it does not vary across dyads– varies within dyad

Time Since Separation = Between Dyad Variables

Time since separation is the same for each dyad member but varies across dyads

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WITHIN AND BETWEEN DYAD VARIABLES

What is a within or between dyad variable in one set of dyads will not be the same type of variable in other sets of dyads

Gender can be a within dyad or between dyad variable depending on research question, sample ect.

Example: Household income

For couples, this would be a between dyad variable
For adult twins, this would be a mixed variable

Gender

In heterosexual couples, gender is a within dyad variable, but in same sex relationships is a between dyad variable (gay + lesbian couples)
Although gender occurs on a continuum, and it could be treated as a “mixed” variable -> assumes a hierarchy of gender which isn’t theoretically or ethnically tenable

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DYADIC PATTERNS

There are patterns of associations that can be broken down into categories: actor oriented, partner oriented, couple oriented, social comparison, and domination

Computed ratios of the actor effects to the partner effects (k ratio)

Essentially, dyadic patterns are ratios of actor effects to partner effects
Provide a general description of the nature of the actor and partner effects
Coefficient is referred to a k, for Larry Kurdek, an early pioneer in dyadic data

Other coefficients have been developed

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DYADIC PATTERNS

Actor oriented patterns: actor effects are significantly stronger than partner – within person effect only

a is not = 0, p = 0

Partner oriented patterns: partner effects are significantly greater than actor effects – between person effect only

p is not = 0, a = 0

Couple oriented pattern: actor and partner effects are non-zero and not different from each other

a = p

Contrast / Social Comparison pattern: actor + partner effects = 0

Equal in magnitude, but have opposing directionality (signs)

Domination pattern: actor only effect for partner 1, and partner only effect for partner 2

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DYADIC PATTERNS

Mathematical(ish) Representations of Dyadic Patterns

Actor Oriented: actor effects < 0, partner effects = 0
Partner Oriented: partner effects < 0, actor effects = 0
Couple Oriented: actor = partner effects
Contrast Oriented: actor + partner = 0
Domination Pattern: simultaneous, significant actor and partner patterns

When examining dyadic patterns, measures of perception (e.g., my partner is warm towards me) tend to demonstrate actor oriented patterns
Measures of behavior (e.g., frequency of chores) will likely lead to partner or couple level patterns

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SPECIFYING DYADIC PATTERNS

Test distinguishability
Estimate the APIM model in the usual manner to obtain estimates of actor + partner effects
Create phantom variables

Variance of 1 + factor loading of 1

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EXAMPLES OF DYADIC PATTERNS

Actor oriented patterns (sig actor, non-sig partner)

Interpretation: The husbands reports of job satisfaction is associated with his own reports of negative affect, controlling for his wife’s level of job satisfaction, but not his wife’s.

Job Satisfaction

Negative Affect

Husband

Wife

sig

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EXAMPLES OF DYADIC PATTERNS

Partner Oriented Patterns (sig partner effect, non-sig actor effect)

Interpretation: My own appraisals of my physical attractiveness likely won’t predict how sexually satisfied I am, but my partner’s level of attractiveness will influence how sexually satisfied I am

Attractiveness

Sexual Satisfaction

Husband

Wife

sig

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EXAMPLES OF DYADIC PATTERNS

Couple oriented pattern (actor and partner effects = and are sig)

Example and Interpretation: one partner’s ability to joke around will predict their own levels of positive affect as well as their partner’s level of positive affect

More general example: If one is concerned with their own outcomes just as much as their partner’s outcomes

Joking

Positive Affect

Husband

Wife

sig

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EXAMPLES OF DYADIC PATTERNS

Contrast / Social Comparison (actor + partner effects = 0; both are sig)

Example and Interpretation: the more each dyad member spends more time with friends would increase their own happiness, but as time spends with friend increases, it would be inversely related to their partner’s happiness

Time with Friends

Happiness

Husband

Wife

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EXAMPLES OF DYADIC PATTERNS

Domination pattern (actor + partner effects but for only one dyad member)
Interpretation: therapist level of empathy (i.e. behavioral measure) predicts their own evaluations of relationship satisfaction. Clients own empathy is unlikely to be related to their own evaluations of the therapeutic relationship. Unlikely (or shouldn’t be) that a client’s empathy increases the therapists’ view of the relationship

Empathy

Therapeutic Alliance

Therapist

Client

sig

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VISUALIZING DYADIC PATTERNS

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DYADIC PATTERNS IN MPLUS

I’m not a huge fan of dyadic patterns

Nothing wrong with running and reporting dyadic patterns
They don’t give me new information -> categorical in nature
I can constrain actor and partner effects to equality or to a fixed value (e.g., 0) if I have a theoretical rationale from which to compare effects

We will not test dyadic patterns

Implement phantom variables – not indicated by any variables in the model
Other, more modern methods for dyadic patterns have been developed and phantom variables are not needed

I will post a video to YouTube on how to estimate dyadic patterns and syntax/code will be provided

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DYADIC PATTERNS

Fitzpatrick, J., Gareau, A., Lafontaine, M. F., & Gaudreau, P. (2016). How to use the actor-partner interdependence model (APIM) to estimate different dyadic patterns in Mplus: A step-by-step tutorial. The Quantitative Methods for Psychology, 12(1), 74-86.
Kenny, D. A., & Ledermann, T. (2010). Detecting, measuring, and testing dyadic patterns in the actor–partner interdependence model. Journal of Family Psychology, 24(3), 359–366. https://doi.org/10.1037/a0019651
Yang, J., Kim, J., & Kim, M. (2023). A comparison of the methods for detecting dyadic patterns in the actor-partner interdependence model. Behavior Research Methods, 1-12.

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EXTENSIONS OF THE APIM

Actor Partner Interdependence Mediational Model (APIMeM)

Ledermann, T., Macho, S., & Kenny, D. A. (2011). Assessing mediation in dyadic data using the actor-partner interdependence model. Structural Equation Modeling: A Multidisciplinary Journal, 18(4), 595-612.

Actor Partner Interdependence Moderation Model

Garcia, R. L., Kenny, D. A., & Ledermann, T. (2015). Moderation in the actor–partner interdependence model. Personal Relationships, 22(1), 8-29.

Cross Lagged Actor Partner Interdependence Model

Fitzgerald, M., & Ledermann, T. (2020). Longitudinal Effects of Adolescent Abuse on Relationship Quality and Posttraumatic Stress Symptoms in Mother–Adolescent Dyads. Journal of Marital and Family Therapy, 46(2), 352-365.

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ACTOR PARTNER INTERDEPENDENCE MEDIATIONAL MODEL (APIMEM)

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APIMEM

We can extend the basic APIM to test mediation

IV of each dyad member is associated with the DV of each dyad member through both individual’s mediators

We assess the IV, mediator, and DV for each dyad member
Estimate actor and partner effects from IV to mediator and from mediator to DV
Answers research questions related to individual pathways influencing one another among dyad members

Own outcomes and partner’s outcomes

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APIMEM

What applies in the APIM, also applies in the APIMeM

ISAT
Type of variables (within, between, mixed)

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ACTOR-PARTNER INTERDEPENDENCE MEDIATION MODEL

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APIMEM

The standard APIMeM for distinguishable dyad members is a saturated model that has 27 free parameters

Six actor effects
Six partner effects
One mean and one variance for each IV
One intercept for each mediator and outcome, one variance for each error term
One covariance between the IV variable
One covariance between the mediators’ error terms
One between the outcomes’ error terms.

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APIMEM

Number of parameters: (p+p(p+1)/2)) = 6+((6*7)/2) = 6 +42/2 = 6+21

P = number of variables

(p+p(p+1)/2))

Mean structure

Variance/ Covariance structure

Variance/covariance matrix is symmetrical what is below the diagonal is a mirror image of what is above the diagonal -> no new information so we divide by 2 (above / below)

In SEM, it is rarer that we care about the mean structure (it is saturated in most model) so you will see just the: p(p+1)/2

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Stress

Coparenting

Stress

Coparenting

Male

Female

Maltreatment

Actor Partner Interdependence Mediational Model Examining Perceived Stress Linking Childhood Maltreatment to Coparenting Following a Separation or Divorce

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Stress

Coparenting

Maltreatment

Stress

Coparenting

Mother

Father

.22

-.36

-.26

-.40

Maltreatment

.20

Actor Partner Interdependence Model Examining Perceived Stress Linking Childhood Maltreatment to Coparenting Following a Separation or Divorce

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APIMEM

We can test the ISAT model with the APIMeM

Constrain each of the actor effects to equality
Constrain each of the partner effects to equality
Constrain the means to equality
Constrain the variances

We won’t be testing the ISAT within the APIMeM…expanded syntax

The 6 variables are considered independent predictors
You can use on statements transforming the variances of the Mediator and DV to residual variances

Coparenting

Mother

Father

.22

-.36

-.26

-.40

Maltreatment

.20

Actor Partner Interdependence Model Examining Perceived Stress Linking Childhood Maltreatment to Coparenting Following a Separation or Divorce

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INDIRECT EFFECTS IN THE APIMEM

Each dyad member will have 4 indirect effects for a total of 8 indirect effects

Y1 = aA1*bA1
Y1 = aP2*bP1
Y1 = aP1 *bA1
Y1= aA2*bP1
Y2 = aA2*bA2
Y2 = aP1*bP2
Y2 = aP2* bA2
Y2 = aA1*bP2

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CONSTRAINING ACTOR AND PARTNER EFFECTS

Testing whether effects are significantly different from one another within the APIM is very rarely done in practice but I strongly recommend this be a common practice
Constrain just actor effects to be equal -> LRT or diff test
Constrain just partner effects to be equal -> LRT or diff test
Constrain actor effects to be equal to partner effects -> LRT or diff test
Actually allows you to say “stronger than” and make firmer conclusions
You can also compare the strength of the INDIRECT effects across partners

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WAYS TO TESTING MEDIATION: BOOTSTRAPPING

Resampling technique whereby individuals are randomly taken from your sample (with replacement) to create a different sample from which your model is tested
Indirect effects are not normally distributed (major problem with the sobel test), so a sampling distribution of the effects are made
Heavily reliant on having a high quality and representative sample
Some recent research (Amanda Montoya) has found that bootstrapping is prone to type 1 errors

Likely due to, in part, non normality of the residuals which downwardly biases SE and upwardly biases test statistics

Does not address possible problems with non-normality in the residuals, which is almost always a problem (ranging from minor to substantial)

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BOOTSTRAPPED SAMPLING DISTRIBUTION OF INDIRECT EFFECTS

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WAYS TO TESTING MEDIATION: BOOTSTRAPPING

Sampstat = mean, variance, range, skew kurtosis ect
Residual = observed – expected covariance matrix
Tech1 = number of each parameter (helps when you have error messages)
Tech4 = latent variable information
Stdyx = standardize x and y
Modindices = modification indices
Cinterval = 90, 95, 99% confidence intervals
BCBootstrap = bias corrected bootstrapping

5000 bootstrapped sample is most common
Recent simulation research (Amanda Montoya) has suggested there is bias (type 1 errors) in Bcbootstrap

My suspicion (contributing factor not causal statement) is that our standard errors are too small in ML when normality is violated -> test stats are too high leading to type 1 errors

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WAYS TO TEST MEDIATION: ROBUST MAXIMUM LIKELIHOOD

Uses Maximum likelihood estimation (MLR) with robust standard errors (address non-normality in the residuals) – Yuan Bentler corrections

Does not require complete datas

Uses Maximum likelihood estimation (MLM) with robust standard errors (address non-normality in the residuals) – Satorra Bentler Correction

Requires complete data

Provides a scaling correction factor that adjusts the chi-square statistic and corrects the standard errors
Cannot use bootstrapping with MLR
Potentially problematic because a sampling distribution of indirect effects cannot be created -> use sobel test
This is my preferential method of testing indirect effects

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MPLUS

Lets move over to Mplus and estimate the APIMeM
Use the following input and data files

Input: APIMeM.inp
Data: couples.dat

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APIMEM WITH INDISTINGUISHIBLE DYADS

We might have theoretically distinguishable dyad members, but the actor and partner effects do not vary by the distinguishing variable and one can create the following set of constraints:

a_A1 = a_A2, b_A1 = b_A2, c’_A1 = c’_A2, a_P1 = a_P2, b_P1= b_P2, and c’_P1 = c’_P2.
One could test these six constraints individually or by an omnibus test. If the constraints hold, we would say the direct effects are empirically indistinguishable.
Testing the six constraints individually, “partial indistinguishability” can occur (i.e., the equality constraints hold for some out of the six effects).

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APIMEM WITH INDISTINGUISHIBLE DYADS

The APIMeM for indistinguishable dyad members within SEM requires 12 equality constraints:

Six for the actor and partner effects

IV to mediator (2)
IV to DV (2)
Mediator to DV (2)

One for mean (IVs)
Two for intercepts (mediator and DV)
1 for variances (IV)
2 for residual variances (mediator, DV)

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ACTOR-PARTNER INTERDEPENDENCE MEDIATION MODEL FOR INDISTINGUISABLE DYADS

Same color = constrained to equality
Can test individually or in an omnibus fashion

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Male

Female

Actor-Partner Interdependence Mediation Model for Indistinguishable Dyads

Same subscripts indicates being constrained to equality�Tau = intercept, psi = variance

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MPLUS

Data File: couples.dat
Syntax File: APIMeM ISAT
Running this analysis will depending on time

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POWER IN ACTOR-PARTNER INTERDEPENDENCE MODEL

Sample size requirements

Sample sizes vary across distinguishability (Monte Carlo Simulation)
Distinguishable dyads require about 250 couples to detect partner effects and 91 for actor effects
Indistinguishable dyads require 121 for partner effects and 45 for actor effects

When there is high skewness and kurtosis of the outcome (values above 3, and 21 respectively), you need to roughly double your sample size
To detect simple indirect effects (mediation) using the APIM

120 dyads recommended when there is only mediation or partner mediation to outcome (M1 to Y1 or Y2 or M2 to Y1 and Y2)
Over 800 dyads are need when there are partner effects from X to M (X1 to M2 or X2 to M1)

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POWER IN ACTOR-PARTNER INTERDEPENDENCE MODEL

Note: sample size recommendations are based on characteristics of the parameters (variances and covariances)
It would not be best practice to draw a sample of dyads and cited the article as “sufficient” evidence that your study is appropriately powered

You would have to replicate the simulation conditions exactly for this to be a justifiable position

Use a Monte Carlo Simulation study

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ACTOR PARTNER INTERDEPENDENCE MODERATION MODEL (APIMOM)

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APIMOM

The demo version of Mplus will not allow us to use the APIMoM – only allows us to use 2 IVs

I will walk you through an example / we’ll add nonsense parameters to estimate the model
You can create and save code/syntax

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MODERATION

Rarely are the social sciences interested in main effects (there seems to always been another possible pathway linking our IVs to DVs,

Yet, we seem to live an die by main effects

Mediation is very much a main effect model – we just have a lot of main effects

Our discussion section frequently discusses other main effects that mediate the relationship between our mediator and our outcome variable

Mediation helps us answer questions to why or how two variables are connected
Moderation answers questions related for whom and under what circumstances are there associations

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VISUAL REPRESENTATION OF MODERATION

Depression

Sexual Satisfaction

Attachment Avoidance

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MODERATION

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MODERATION

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GRAPHICAL REPRESENTATION OF MODERATION

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Graphical representation of moderation (3 way interaction)

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AN EXCERPT FROM ONE OF MY MODERATION STUDIES

A power analysis using G*Power 3.1 was conducted to determine the required sample size needed and using the following assumptions a sample size of 395 participants was needed: power = .80, f² = .02 (small effect size), α = .05, 14 predictors, and 1 interaction. Power was estimated for only one interaction (the three-way interaction term).
This is assuming that every assumption is met within your model – otherwise power isn’t actually the observed value

Hope your predictors are perfectly reliable (I may or may not have, but definitely did use a change score -> implications for reliability)
Hope your residuals are homoscedastic and normally distributed (This one was actually met)

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MODERATION WITHIN THE APIM: APIMOM

Within the APIMoM, we can test under what circumstances do actor and partner effects vary
Moderation can be tested within dyads and between dyads or a combination of both

Within: We can identify differences in parameter estimates based on individual characteristics of dyad members
Between: We can identify differences in parameter estimates based on how couples are different

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APIMOM

A moderator can be a within-dyad variable, between-dyad variable, or a mixed variable

Within-Dyad: only varies within dyads (e.g., every dyad has same average score, but different dyad members have different scores)
Between-Dyad: Both members of the dyad have the same score, but the score varies across dyads in the sample
Mixed: varies within and between dyads: variation within dyads and between dyads

We’ll talk about a variety of different types of moderation that can occur in the APIMoM

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APIMOM

When testing within dyad moderation, we only do this with distinguishable dyads
Recall that indistinguishable dyads have the same means, variances, and covariances across dyad members -> nothing varies within a dyad

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GENERAL QUESTIONS USING THE APIMOM

Do characteristics of the relationship impact how individuals influence their own and their partner’s outcomes
Do the roles that each person plays in the relationship affect the strength of associations
Do characteristics of partners influence each dyad member’s outcomes
Is there a synergistic or buffering influence of each dyad member’s characteristics that influences each partner’s outcomes

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APIMOM: DICHOTOMOUS, WITHIN DYAD MODERATOR

Dichotomous within dyads moderator occurs when there is distinguishability

Recall that to avoid arbitrary assignment to P1 and P2, we need to establish differences between dyad members – this is moderation!!
Example: gender in heterosexual couples

The within dyad variable is dichotomous because it is comparing the effects of one partner relative to another
You can have a trichotomous variable if you add another person

Three-person APIM

A continuous within-dyads moderator (e.g., percent of time talking) could be tested using the between dyads analysis method (discussed later on), but I’m not aware of any studies using a continuous within-day variable

Generally, within dyad moderation is tested with the distinguishing variable (e.g., gender)

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TRIADIC APIM

Parent-Child Relationship - Child

Parent-Child Relationship - Father

Parent-Child Relationship - Mother

Depression - Child

Depression - Father

Depression - Mother

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EFFORTFUL CONTROL AND OUTCOMES IN MOTHER + TWINS

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APIMOM: DICHOTOMOUS, WITHIN DYAD MODERATOR

In MLM, within dyad moderation is tested with an interaction term
In SEM, we constrain the actor paths to be equal and/or the partner paths to be equal

Remember the ISAT tests means, variances and covariances to be equal – actor and partner effects may not be equal or they may
Omnibus tests provide NO diagnostic information -> little use in locating specific effects
Also allows for model-data fit to be examined

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APIMOM: DICHOTOMOUS, WITHIN DYAD MODERATOR

Researchers often make claims that actor effects are stronger than partner effects, but do not constrain the effects to be equal to create a statistical test

You need a statistical result to claim moderation
If effects are not different then you have more statistical power to detect the effects

Think about statistical power here as well

If you have 80 dyads and a standardized parameter difference of .05 between the two groups -> good luck having the power to detect that
If you have 800 dyads and a parameter difference of .10 -> you’re probably ok!

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APIM MODERATION: DICHOTOMOUS, WITHIN DYAD MODERATOR

Female

Male

Effects are constrained to be equal

Actor effect for male is constrained to be equal to female

Partner effect from female is established to be equal to the partner effect from male

This these sequentially (multiple, separate, analyses) – not in the same model

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THINK BACK TO THE APIMOM

Think back to when I advocated for constraining paths within the APIM – this is a way to testing MODERATED mediation or have a within dyad variable moderating the indirect effect!

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APIMOM: WITHIN DYAD MODERATOR

Within the APIMoM, you can go nuts and do all sorts of constraints (theoretically drive, of course)
You can constrain the actor and partner effects to be equal
You can constrain the actor effect of dyad member 1 to be equal to the partner from dyad member 2 to dyad member 1

Essentially the same as the ISAT model but is less restrictive

Each constraint will give 1 degree of freedom – the paths are still freely estimated but they are constrained to equality

To get 2 df, the paths would have to be fixed to a certain value

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APIMOM

Example research questions for within dyad moderation

Do husbands and wives levels of depression equally impact each member’s reports of relationship quality?
Does male’s own neurotic behavior impact their own reports of relationship satisfaction more than female’s reports of neurotic behavior influencing female reports of relationship quality?

Female

Male

-.25***

-.11*

-.21***

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EXAMPLE: DEPRESSIVE SYMPTOMS IMPACTING RELATIONSHIP QUALITY: ACTOR AND PARTNER EFFECTS CONSTRAINED TO EQUALITY

Relationship Quality

Depression

Female

Male

-.23***

-.10*

-.23***

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FINDINGS

Constraining the actor and partner effects to equality did not significantly decrease the fit the model -> this is the most parsimonious model
Gender did not moderate any of the structural paths in the model

Turns out, there are gender differences in the variance in CSI (much more variability in male reports of relationship quality -> significant ISAT test
R2 for males = .116 and females = .194

Structural paths in the model are indistinguishable between dyad members.

Male depression has as much of an impact on their female partners relationship quality as female depression influences male relationship quality
Female depression influences their own perceptions of relationship quality to the same magnitude as male depression influencing their own reports of relationship quality

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HAD THERE BEEN SIGNIFICANT FINDINGS…�

We’ll assume moderation for actor and partner effects

The actor effects were stronger for females than males
The partner effects were stronger such that male depression had a greater negative impact on female relationship quality than the inverse

Gender moderated the actor effects such that the association between depressive symptoms and relationship quality were significantly stronger for women compared to men
Gender also moderated the partner effects. Specifically, the effects of men’s depression has a greater impact on the women relationship quality compared to women’s depression influence on male relationship quality

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APIMOM: WITHIN DYAD MODERATOR

First run the saturated model (fully unconstrained)

Get a sense of the parameter estimates (e.g., eye-balling them)

Next, constrain the effects to be equal

I recommend running the constraints one at a time to discern where the effects are -> a significant test with a chi-square value of 10.12 doesn’t tell us where the significant effects are

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MOVE INTO MPLUS

Data file: couples.dat
Syntax File: APIM within Dyad Moderation

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APIM MODERATION: BETWEEN DYAD MODERATORS

Within-Dyad moderation focuses on differences that occur with a dyad, on average, but differences also exist across dyads

Within dyad effects only vary at the individual level – all couples were cisgender heterosexual (between dyad) but constraining male and female differences

Between dyad moderator variables that are characteristic of the relationship and can help explain non-linear effects across dyads

We will examine variables that vary across dyads (but not individuals within)

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APIM MODERATION: BETWEEN DYAD MODERATORS

For indistinguishable dyads, there are two potential moderating effects

Actor Effects
Partner Effects
Recall that actor and partner effects are the same in indistinguishable dyads which is why we have 2 effects (actor and partner) rather than four (actor for dyad member 1, actor for dyad member 2, partner for dyad member 1, partner for dyad member 2)
There are four interaction pathways in the model, but the actor and partner effects are constrained to be equal

The magnitude of the actor effects (or X1 on Y1) depends on the level of the between dyads variable
The magnitude of the partner effects (or X1 on Y2) depends on the level of the between dyads variable

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APIMOM: BETWEEN DYADS MODERATOR FOR INDISTINGUISHABLE DYADS

Constraint AM1=AM2

Constrain: PM1 = PM2

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APIM MODERATION: DICHOTOMOUS BETWEEN DYAD MODERATORS FOR DISTINGUISHABLE DYADS

Example: sample of heterosexual couples (gender is the distinguishing variable) and we want to test age as a dichotomous moderator (female partner is older and male partner is older)

Female being the older partner (yes / no) is consistent across all dyads
Remember this is a sample of only cisgender heterosexual couples

We can test whether actor effects in male and female partners are the same in dyads where the male is older and the female is older

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SAMPLE SPACE OF BETWEEN DYADS MODERATION

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APIMOM: BETWEEN DYADS MODERATOR

If the between dyads moderator is binary -> multiple group SEM with equality constraints would be an effective strategy

Example: test gender differences in same sex relationships
Do same sex male relationships differ from same sex female relationships?

Grouping variable is the binary moderator and constrain effects to be equal across groups

Run a model with male couples (with equality constraints in place)
Run a model with female couples (with equality constraints in place)
Constrain the actor and partner effects to be the same in male + female group -> chi-square difference / likelihood ratio test

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MPLUS

Data File. Couples2.dat
Syntax File: apimom with between dyad binary moderator

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

Freely Estimated Multiple Group Model

All actor and partner effects are free

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APIM MODERATION: DICHOTOMOUS, BETWEEN DYAD MODERATOR

To test two within dyad moderators within SEM, we will implement a multiple group analysis
Simultaneously estimate a separate model for female older and male older

Label constrains path to be equal across groups (NOT across dyad members)

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

Actor effects are constrained to be equal across grouping variable (age)

Partner effects are still freely estimated

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RESULTS + INTERPRETATION?

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

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INTERPRETATION

Constraining the actor effects for males across the age groups significantly worsened the model-data fit, indicating moderation

The slopes of the lines from dyadic coping to relationship quality were significantly weaker in the group where the male was the older partner compared to the group where the female was the older partner
For males who are the older partner in relationships, dyadic coping has a significantly weaker effect than when women are the older partner

I was just “futzing” around but how interesting is this finding!?!?!

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

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INTERPRETATION

Constraining the actor effects for females across the age groups did not significantly worsened the model-data fit, indicating no moderation

The slopes of the lines from dyadic coping to relationship quality were not significantly different among women
The effects of dyadic coping on relationship quality on women does not vary across age groups
It doesn’t appear to matter if women are the older or younger partner, dyadic coping is a robust predictor of relationship quality

Coefficients are quite different but sample size for women being older was 33 and male being older was 166 – we MAY have moderating due to unequal sample sizes

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

Constrained partner effects

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We have a significant difference across groups among female’s dyadic coping impacting males relationship quality

Delta CFI < .01 is also used as an indicator of significance in change model-data fit

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Female

Male

Female

Male

Female Older Dyads

Male Older Dyads

Constrained partner effects

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We have a non- significant difference across groups among female’s dyadic coping impacting females relationship quality

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INTERPRETATION

Regarding the partner effects, we have evidence that the effects of female dyadic coping on male relationship quality across age groups are different, but no differences were found from male dyadic coping to female relationship quality
When the female is the older partner in the relationship, the effect of male dyadic coping on male relationship quality is significantly stronger compared to dyads where the male is the older partner

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APIMOM: BETWEEN DYADS MODERATOR WITH CONTINUOUS VARIABLE

What if the between dyad variables are continuous
Many times, researcher use a median split or dichotomize a continuous variable to be absent (0) or present (1)

Reduces variability and doesn’t allow nuance in your research question
Due to distribution or model issues, it can be acceptable but this needs to be clearly articulated to the reader
Median splits are rarely defensible (Preacher and colleagues)

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APIMOM: BETWEEN DYADS MODERATOR WITH CONTINUOUS VARIABLE

We now have four interaction effects (2 variables X 2 outcomes)
Example: relationship length as a moderator within opposite sex heterosexual couples

Interaction 1: woman’s actor effect moderated by between-dyad variable
Interaction 2: women’s partner effect moderated by between dyad variable
Interaction 3: male’s actor effect moderated by between dyad variable
Interaction 4: male’s partner effect moderated by between dyad variable

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APIMOM: BETWEEN DYADS MODERATOR WITH CONTINUOUS VARIABLE

Wanna get really wild: constrain the interaction effects to equality across partners in the between-dyad distinguishable dyad APIMoM

We can test whether the magnitude of actor moderation is the same for both partners

Constraint AM1=AM2

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APIMOM RESEARCH QUESTIONS

Are the levels of spillover from conflict to parenting behavior greater in gay or lesbian couples?
Does race (Hispanic vs African American) moderate the relationship between friendship reciprocity and aggression
Does marital length moderate the effects of conflict on marital satisfaction in heterosexual newlyweds

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MPLUS

Data file: couples2.dat
Syntax File: APIMoM with between dyad moderator continuous
The solution provided is an inadmissible solution due to very high multicollinearity, but we will interpret it as if it were admissible

Likely due, in part, to a smaller sample size

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APIMOM WITH MIXED VARIABLES

Most of the variables in family science, psychology, public health and other social and medical sciences will be mixed variables

Values of the variable will be different for both members of the dyady
Values of the variable will be different across dyads

Occurs when we have a moderating variable that varies within and between dyads and is reported by each person
Substantially increases the complexity

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APIMOM WITH MIXED VARIABLES

Key Terms

Actor Moderator: an individual’s own score on the moderating variable
Partner Moderator: the other dyad member’s score on the moderating variable
The same variable (in SEM) can be an actor or partner moderator depending on the outcome

There are then two (each person’s moderator) potential moderators of the actor and partner effects

Actor moderator is the person’s own score on the moderator (or when the mixed moderator variable and outcome variable are from the same person in SEM models),
Partner moderator, which is the person’s partner’s score on the moderator (or when the moderator variable and outcome variable are from different persons in SEM models).

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APIMOM WITH MIXED VARIABLES

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APIMOM WITH MIXED VARIABLES

There are two primary ways we can test moderation with mixed variables within an SEM framework
Actor by Partner Effect – when the strength of an actor effect is dependent on the partner effect of the other dyad member
Introduce a new variable all together that varies within and between dyads

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ACTOR BY PARTNER EFFECT

We can test to the extent to which each dyad member influences the relationship for each dyad member’s actor effect
Are the actor effects contingent on the level of the partner

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APIM MODERATION: ACTOR X PARTNER INTERACTION

Female

Male

X1 * X2

This the statistical model that depicts the interaction predicting unique variation in each dyad member’s outcomes

Since interaction terms are created via multiplication, order doesn’t matter -> 1 interaction term

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PERHAPS A MORE INTUITIVE WAY TO VISUALIZE AN ACTOR X PARTNER INTERACTION

Female

Male

This is the conceptual model on what the interaction term is doing

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APIM MODERATION

Examined trauma history (dichotomous yes/no) and positive and negative marital exchanges among couples (> 2,000)
Interaction effects indicated that couples where both members had a history of trauma reported more positive and less negative interactions compared to couples where only one partner had trauma

Shared experiences -> bonding and intimacy

Whisman, M. A. (2014). Dyadic perspectives on trauma and marital quality. Psychological Trauma: Theory, Research, Practice, and Policy, 6(3), 207–215. https://doi.org/10.1037/a0036143

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COUPLE DATA

Using the couple data (couple.dat), we can test whether there is an actor by partner interaction

Example: when males are low on depressive symptoms, but their partners are high on depressive symptoms, what is the effect on relationship quality?
Example: when males are high on depressive symptoms, and their partners are also high on depressive symptoms, what is the effect on relationship quality?

This is an actor X partner interaction

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CONCEPTUAL MODEL

Relationship Quality

Relationship

Depression

Female

Male

Int

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INTRODUCING THE DEFINE COMMAND

In Mplus the define command allows you to create and rescaled variables
In the APIMoM, we need to create interaction terms (we can bring them in as well)

Created interaction terms in red box
Standard = creating z scores prior to analysis

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MPLUS CODE FOR LABELING PARAMETER ESTIMATES

Independent Variable

Moderator

Interaction

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INTRODUCING THE MODEL CONSTRAINT COMMAND

Create Simple Slopes Equations

Graphing Slopes

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RESULTS

Unstandardized results
We have a significant interaction for males, and an interaction “approaching” significance
We’re only going to plot the male interaction that is significant -> likely that the simple slopes would find a significant simple slope among females (not guaranteed)
Significance test reflects significance variance account for

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SIMPLE SLOPES OUTPUT

An actually accurate interpretation of a p value: We have a .1% chance of observing the effect of -.64 or larger given that the null hypothesis is true

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SIMPLE SLOPES – MEDMOD

Red lines = estimate

Blue lines = confidence interval

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SIMPLE SLOPES – LOW MOD

Red lines = estimate

Blue lines = confidence interval

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SIMPLE SLOPES – VERY LOW MOD

Red lines = estimate

Blue lines = confidence interval

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REPRESENTATION OF THE OVERALL INTERACTION EFFECT

In the red box are where the adjusted means in relationship quality are significantly different from one another (simple slopes)

As male depression increases the partner effect of female depression become less pronounced but relationship quality continues to decrease

Note: These are NOT the same simples slopes as previously seen in the mplus output. These effects are +1, 0, and -1 SD – for illustrative purposes

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CONCLUSIONS

When males are not depressed, there is significant variability in how satisfied they are and it is dependent on the impact of female depression on male relationship quality (partner effect)

When males have low levels of depression, there are significant differences in their level of relationship satisfaction and it depends on level of female partner depression

When males have low levels of depressive symptoms, when female depression is high, males are less satisfied compared to when the female partner also have low levels of depression

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CONCLUSIONS

As depressive symptoms in male partner increase, relationship quality continues to decline
But the effects on female depression on male relationship satisfaction weaken

When males are really depressed, the impact of high, medium, and low female depression doesn’t contribute to how satisfied males are

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APIMOM IN MPLUS

You will use 2 files for Actor X Partner APIMoM with Distinguishable Dyads:

Syntax File: APIMoM with Actor by Partner Mixed Interaction.inp
Data File: Couples.dat

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THE SECOND WAY TO TEST A MIXED MODERATOR

You can also introduce a moderating variable that is not already in the base APIM
The change in effects is not due to the levels of an individual’s dyad member, but to a third variable all together

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APIMOM WITH MIXED VARIABLES

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APIMOM WITH MIXED VARIABLES

Actor-actor interaction effect (x2): the IV and the moderator are both measured by dyad member 1 (or 2) predict the outcome of dyad member 1 (or 2)
Actor-partner interaction effect (x2): the IV is reported by dyad member 1 ( or 2) and moderator is reported by dyad member 2 (or 1)
Partner actor interaction effect (x2): the IV and moderator are both reported by dyad member 2 (or 1) and predict the outcomes of person 1 (or 2)
Partner-actor interaction effect (x2) the IV is reported by the dyad member 2 and the moderator is reported by dyad member 1 ad predict outcomes for person 1 (or 2)

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APIMOM WITH MIXED VARIABLES: EXAMPLE

Using two wave of data from the MIDUS study, we tested sibling abuse, family strain and negative affect among dizygotic twins
Tested contemporary family strain as a moderator of sibling perpetrated abuse and negative affect over a 10 year period

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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APIMOM

Female Twin

IV: male twin perpetrated abuse
Moderator: current family strain
DV: negative affect

Male Twin

IV: female twin perpetrated abuse
Moderator: current family strain
DV: negative affect

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ACTOR-ACTOR INTERACTION

Interaction 1: female twin abuse victimization, female twin family strain, female negative affect

Actor-actor for female

Interaction 2: male twin abuse victimization, male twin family strain, male negative affect

Actor-partner for male

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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ACTOR-PARTNER INTERACTION

Interaction 3: Female twin abuse victimization, male twin reported strain, female twin reported negative affect

Actor-Partner interaction for female

Interaction 4: Male twin abuse victimization, female twin reported strain, male twin reported negative affect

Actor-Partner interaction for male

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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PARTNER-PARTNER INTERACTION

Interaction 5: female twin victimization, female reported family strain, male negative affect

Partner-Partner interaction for males

Interaction 6: male twin victimization, male reported family strain, female negative affect

Partner-Partner interaction for females

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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PARTNER-ACTOR INTERACTION

Interaction 5: female twin victimization, male reported family strain, male negative affect

Partner-actor for males

Interaction 6: male twin victimization, female reported family strain, female negative affect

Partner-actor for females

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

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MPLUS CODE (PRIMARY MODEL)

FIML Line to Account for Missing Data

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MPLUS CODE (COVARIATES)

Fixing covariates partner effects to be 0 – theory to link male twin level of education to impact female twin mental health over a 10 year period

Added bonus of allowiing us to test model-data fit

These are restrictions placed on the model, giving us one degree of freedom per restriction

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Female Twin

Male Twin

Family Strain

Brother Perpetrated Abuse

Family Strain

Sister Perpetrated Abuse

Sister Abuse X Brother Strain

Brother Abuse X Sister Strain

Negative Affect

.06*

Brother Abuse X Brother Strain

Sister Abuse X Sister Strain

-.05

-.00

-.03

.66

.13

-.17

-.05

-.03

-.08

1.03*

.24***

-.08

.05

-.16

Covariates:

Parental Abuse

Educational Achievement

Income

Age

MIDUS 1 Negative Affect

Unstandardized effects

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APIMOM

We have a significant interaction (partner-partner) where men who reported higher levels of sister abuse in childhood and reported higher levels of family strain predicted greater female negative affect
Next, we conducted simple sloes to determine at what levels of male family strain influences the relationship between sister perpetrated abuse and female negative affect

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MANUALLY CODING SIMPLES SLOPES

Everything between !* and *! Is not read by Mplus

Analogous to # in lavaan and /* and */ in SAS

Standard deviations taken from descriptive statistics (square root of variance)

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UNSTANDARDIZED OUTPUT

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SIMPLE SLOPES

X-Axis = male report of sister perpetrated abuse in childhood Red Line = Male report of Family strain

Y-Axis = sister current levels of negative affect

Red line = estimate

Two blue lines indicate 95% CI

Either blue line crosses 0 then the path is not significant

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INTERACTION PLOT FOR -2, 0, AND + SD

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IMPLICATIONS

We found that there was an partner-partner interaction between male abuse by sister in childhood and male family strain in the prediction of women’s negative affect

This only occurred when family strain was 2 standard deviations above the mean
The synergistic interaction appears to be influential in cases at the tail end of the distribution -> good news from a public health perspective!
Could also be clinically informative (despite sample) because clinicians tend to see the tail ends of the distribution

When testing moderation, always include a pictorial representation of the interaction (graph it!)

I will begrudging admit that R’s graphics capability is FAR superior to Mplus

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IMPLICATIONS

We need to be mindful that +2 and -2 (and other larger values) of moderator may not be within the empirical range of the scale and/or the sample

For example, the mean and standard deviation can be close to the same value so testing an interaction with +/- 2 SDs is outside the sample range and values may not be possible (e.g., depression values would be negative at -2SDs)

Our Example:

Family strain scale ranged from 1-4, mean was 2.07, SD ~ .60
Two standard deviations above the mean is still within the empirical range of the scale and we did have people report values of 4.
If SD was 1.5 then + 2 SD would not be a possible value (score of ~ 5)

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ANOTHER EXAMPLE

I tested depressive symptoms as a moderator of partner’s depressive symptoms and relationship quality
-2SDs would be out of the empirical range -> -2SD = a score of -2 on depression
Trimmed the interaction to -1.5

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SNEAK PEEK: LATENT VARIABLE ACTOR PARTNER INTERDEPENDENCE MODEL WITH ACTOR BY PARTNER LMS INTERACTION

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SYNTAX PART 2

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LATENT VARIABLE APIM WITH 2 INDICATORS (RESIDUAL COVARIANCES NOT SHOWN)

x1A

x2A

x1B

x2B

y1A

y2A

y1B

y2B

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Notice Male slopes are increasingly negative while female slopes are decreasingly negative

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SCREENING FOR OUTLIERS AND INFLUENTIAL OBSERVATIONS

Potential influential observations – run sensitivity analyses

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APIM MODERATION WITH LATENT VARIABLES

Define:

Standardize Copare1M Copare4M Copare7M Copare8M EmoM ConductM HyperM

Copare1F Copare4F Copare7F Copare8F EmoF ConductF HyperF;

Analysis:

Algorithm = Integration;

Type = Random;

ESTIMATOR IS MLR;

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APIM MODERATION WITH LATENT VARIABLES

!Create latent variables: MCopare by Copare1M Copare4M Copare7M Copare8M;

FCopare by Copare1F Copare4F Copare7F Copare8F;

MSDQ by EmoM ConductM HyperM;

FSDQ by EmoF ConductF HyperF;

!Specifying latent variable interaciton

INT | FCopare xwith MCopare;

!structural portion of model with labels to be used for plotting interactions – parameters without labels are not used in calculating the simple slopes

MSDQ on FCopare; MSDQ on MCopare(b0);

FSDQ on FCopare(b1); FSDQ on MCopare; FSDQ on INT(b3);

MSDQ on INT(b4); MSDQ with FSDQ;

MCopare with FCopare;

!covariances on indicators of the IV

Copare1M with Copare1F; Copare4M with Copare4F; Copare7M with Copare7F; Copare8M with Copare8F;

!covariances on indicators of the DV

EmoM with EmoF; ConductM with ConductF; HyperM with HyperF;

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A TEASER IN MODERATED MEDIATION WITHIN THE APIM (TIME PERMITTING)

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MODERATED MEDIATION

You can integrate the APIMeM and APIMoM to test moderated mediation
We can test actor and partner effects as moderators within the APIMeM
Example

Using the couple data, we can expand the interaction between depression and relationship quality by adding in ACEs as a predictor!

Exact same model as before (actor X partner interaction), but added childhood maltreatment as a predictor

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MODERATED MEDIATION

Maltreatment

Depression

Relationship Quality

Male

Female

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PRE-ANALYSES SYNTAX

Couple level ID variable to identify outliers

Specify missing values in the data set

Useob = use observations (cases)

NE = not equal

Removed dyads 159, 288 and 258 from analysis (both partners)

1 Comments out all text on that line– mplus ignores

Create interaction term, MUST go at the end of the variables list in USEVARIABLES COMMAND

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MODEL SYNTAX

For moderation

For mediation

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COVARIATE SYNTAX

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PLOT SYNTAX

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FINAL SYNTAX (PROBING INTERACTIONS + SPECIFYING MAGNITUDE OF SPECIFIC INDIRECT EFFECT) FOR MODERATED MEDIATION TO COME�

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UNSTANDARDIZED RESULTS OF MODERATED MEDIATION

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VARIANCE ACCOUNTED FOR

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LOOP PLOT OF INDIRECT EFFECTS AT LEVEL OF MODERATOR (FEMALE DEPRESSION)

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LOOP PLOT OF INDIRECT EFFECTS AT LEVEL OF MODERATOR (FEMALE DEPRESSION)

As the levels of female depression gets bigger, bigger, and bigger, the indirect effect gets smaller, smaller, and smaller
The indirect effect remains significant (look at the confidences bands) until depression levels reach about a 10 on the CESD, but the more the female is depressed, the less male depression impacts his views on the relationship

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WITHIN-PERSON ANALYSES IN THE APIM

MERGING TWO STATISTICAL METHODS TOGETHER: RANDOM INTERCEPT CROSS LAGGED PANEL MODEL AND THE CROSS LAGGED ACTOR-PARTNER INTERDEPENDENCE MODEL

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READINGS IN THE APIM (MEDIATION AND MODERATION)

Garcia, R. L., Kenny, D. A., & Ledermann, T. (2015). Moderation in the actor–partner interdependence model. Personal Relationships, 22(1), 8-29.
Ledermann, T., Macho, S., & Kenny, D. A. (2011). Assessing mediation in dyadic data using the actor-partner interdependence model. Structural Equation Modeling: A Multidisciplinary Journal, 18(4), 595-612.

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WITHIN-PERSON ANALYSES IN THE APIM

The cross-lagged APIM test between-dyad effects, but our theories (e.g., family systems theory) are largely focused on within family processes

Using the APIM doesn’t actually test the proposition of FST
Between person analyses smush together within person and between person effects over time

How do we capture within-person variability over time? Enter the Random-Intercept Cross Lagged Panel Model

Multilevel modeling can test within-person variability but has 3 major limitations

1 dependent variable
Cannot examine cross lagged effects
No formal tests of mediation (not as important here)

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RANDOM INTERCEPT – CROSS LAGGED PANEL MODEL

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CHANGE OVER TIME

Panel models are often used to examine longitudinal data / predicting change over time

Cross lagged APIM (CL-APIM)
Random Intercept CLPM (RI-CLPM)

Panel Models also exist in dyadic data

Cross lagged APIM (CL-APIM)
Random Intercept CLPM APIM (RI-CLPM-APIM)

CL-APIM can be used estimate reciprocal associations across two or more people

Focusing analysis on the covariance structure with less attention to means

CLPM estimates change in the dependent variable (y) being predicted by an individual’s prior deviation from the group mean (x) while controlling for change in y and that same individual’s prior deviation of the group mean on y

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THE CROSS LAGGED PANEL MODEL

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THE CROSS LAGGED PANEL MODEL

Using observed variables, the CLPM is just identified with 2 waves of data (df = 0)

In a 2 variable model (T1X, T2X, T1Y, T2Y) you have 2 covariances, 2 autoregressive paths, and 2 crosslagged paths
Visually analogous to the APIM

Just identified models provide perfect fit (CFI = 1, TLI =1, RMSEA = 0, df = 0, x² = 0)

Not meaningful – could be a terrible model but with 0 df it is impossible to determine
When you have 0 df then you have no statistical courage

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TRADITIONAL CLPMS

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The goals of traditional CLPMs are to determine:

How stable are two (or more) constructs (autoregressive effects)?
If variable X predicts changes in variable Y over time (cross-lagged effects)?
Is there a bi-directional relationship between variables X and Y?
Does variable X have causal predominance over variable Y (Granger causality)?

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TRADITIONAL CLPMS

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The goals of traditional CLPMs are to determine:

How stable are two (or more) constructs (autoregressive effects)?
If variable X predicts changes in variable Y over time (cross-lagged effects)?
Is there a bi-directional relationship between variables X and Y?
Does variable X have causal predominance over variable Y (Granger causality)?

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TRADITIONAL CLPMS

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The goals of traditional CLPMs are to determine:

How stable are two (or more) constructs (autoregressive effects)?
If variable X predicts changes in variable Y over time (cross-lagged effects)?
Is there a bi-directional relationship between variables X and Y?
Does variable X have causal predominance over variable Y (Granger causality)?

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TRADITIONAL CLPMS

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The goals of traditional CLPMs are to determine:

How stable are two (or more) constructs (autoregressive effects)?
If variable X predicts changes in variable Y over time (cross-lagged effects)?
Is there a bi-directional relationship between variables X and Y?
Does variable X have causal predominance over variable Y (Granger causality)?

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THE CROSS LAGGED PANEL MODEL

Answers the following types of research questions

Does maternal warmth and involvement reduce children’s behavioral and emotional problems?
Does forgiveness enhance marital quality or does marital quality enhance forgiveness?
What is the directional relationship between mental health and substance use?

Each question can be answered using a single reporter – do not need dyadic data

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TRADITIONAL CLPMS

Pros of the traditional CLPM:

Requires relatively few waves of data (two or more)
Accounts for the stability of constructs over time
Uses one variable to predict changes in another variable
Examines reciprocal associations / “causal predominance”

Cons of the traditional CLPM:

Researchers often test models that are just-identified
Yields biased cross-lagged effects when constructs have a trait-like component
Confounds between-person stability and within-person change

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THE CROSS LAGGED PANEL MODEL

Using observed variables, the CLPM is overidentified with 3 waves of data (df = 4)

In a 2 variable model (T1X, T2X, T1Y, T2Y) you have 2 covariances, 2 autoregressive paths, and 2 crosslagged paths
Paths from T1 of X1 and X2 are not typically thought to predict Y1 and Y2 at T3

Will not be a just identified models and you will have fit statistics

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THE CROSS LAGGED PANEL MODEL

X1_T1

X2_T1

M1_T2

M2_T2

Y1_T3

Y2_T3

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THE CROSS LAGGED PANEL MODEL

Often Presumed to be 0

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CL-APIM

CRPM should be estimated using latent variables when possible (reduces measurement error)– you must do this to compare latent scores over time

Must established longitudinal invariance to estimate CLPM

Levels of invariance

Configural (factor structure) - needed
Metric (loadings) – needed
Scalar (means) – not needed
Strict (residuals) – not needed

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CL-APIM

The CLPM can easily be applied to dyadic data
Instead of measuring two different constructs within one individual, you measure one construct (or more) within one dyad
Example: Reciprocal associations in parent-adolescent relationship quality across adolescence

Gather dyadic data from parents and adolescents at multiple time points

Can also gather multiple constructs

Time 1: Parent-Adolescent Relationship Quality
Time 2: Parent and Adolescent depression
Time 3: Parent-Adolescent Relationship Quality
Time 4: Parent and Adolescent depression

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THE CROSS LAGGED PANEL MODEL

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Various models separate between-person and within-person variance:

ALT-SR (Berry & Willoughby, 2017, Child Development)
Latent State-Trait model (Kenny & Zautra, 2001, Book Chapter)
Dynamic SEM (McNeish & Hamaker, 2020, Psych Methods)
RI-CLPM (Hamaker et al., 2015, Psych Methods)

Pros of the RI-CLPM:

Conceptually similar to the traditional CLPM
Requires relatively few waves of data (three or more)
Fits data better than traditional CLPMs (CLPM is nested under RI-CLPM)
Least likely to encounter convergence issues

Solutions to the shortcomings of the CLPM

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RANDOM INTERCEPT – CROSS LAGGED PANEL MODEL

The RI-CLPM attempts to go beyond the CLPM by estimating a random intercept

Accounts for trait-like, time invariant stability (over the study period)

This is discussed inappropriately in many applications – inferences on what the trait component is that are beyond the scope of the study (e.g., genetics)
Partials out between person variability (random intercept) from within-person variability

To quote Lesa Hoffman, within and between person effects are “smushed together”

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RANDOM INTERCEPT – CROSS LAGGED PANEL MODEL

Between Person Effects: What is occurring at a group of people (e.g., individuals who use problem focused coping, people have fewer depressive symptoms)

Tests between person differences (e.g., stability)

Within-Person Effects: What is occurring within an individual (e.g., when I use problem focused coping, I have fewer depressive symptoms)

Tests interindividual differences in intraindividual residualized change

Most of our theories postulate individual level change but our statistical methods test between person change

Example: family systems theory focuses on how one individual influences another (e.g., defensiveness) within a family and is less focused on family to family differences in defensiveness

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WITHIN AND BETWEEN LEVEL EFFECTS: SMUSHING

In the CLPM, the estimates of within and between level effects are “Smushed”
In CLPM we cannot disaggregate the effects

Within Depression

Between Depression

RI-CLPM

CLPM

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WITHIN AND BETWEEN LEVEL EFFECTS: SMUSHING

On a depression variable at a single time-point in time a sample of 10 people have a score of 5/10 – somewhat depressed
The 5 is comprised of both a within and between component that sum to be 5

Between-person is the trait like tendency (between person) to be depressed and the sample averages a 3
One individual within component is then a 2 (they are more depressed than they usually are)

These intraindividual parameters will have a mean and a variance
“On average, individuals who….”

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WITHIN AND BETWEEN LEVEL EFFECTS

Time 1 Depression

Time 2 Depression

Time 3 Depression

Within Component

Between Component

There is a general tendency for an individual to be depressed across the study

At time 1, they may be happier (Within) than they usually are (Between) but at time 2, they are more depressed (within) than they usually are and at time 3 (between), they are at their average

Random intercept

Deviations from random intercept

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RI-CLPM�

RI-CLPM decomposes measured data into three components:

Grand mean for each measurement occasion
Between-person component (random intercepts)
Within-person component

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FROM USAMI (2021)

Reciprocal associations between variables using the RI-CLPM

X_it = u_xt + I _xi + x*_it
Y_it = u_yt + I _xi + y*_it
i = individual
U_xt and u_yt are the group means at time point t
I_xi and I_yiAre stable trait factors / time invariant factors (random intercepts that represent that individual’s trait like deviations from the temporal group means)
X* and Y* are temporal deviations from an individual’s expected scores

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FROM USAMI (2021)

Reciprocal associations between variables using the RI-CLPM applying it depression and coping using three waves of data

X_it = u_xt + I _xi + x*_it
Y_it = u_yt + I _xi + y*_it

i = individual

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FROM USAMI (2021)

U_xt and u_yt are the group means at time point t

Sample mean of coping at time 1 (U_x1) time 2 (U_x1) and time 3 (U_x3)
Sample mean of depression at time 1 (U_y1) time 2 (U_y1) and time 3 (U_y3)

I_xi and I_yiAre stable trait factors / time invariant factors (random intercepts that represent that individual’s trait like deviations from the temporal group means)

I_xi= time invariant mean of coping within the overall sample
I_{yi =}time invariant mean of depression within the overall sample

X* and Y* are temporal deviations from an individual’s expected scores

x*_itis an individual’s deviation from their underlying trait level of coping
y*_itis an individual’s deviation from their underlying trait level of depression

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RI-CLPM EQUATIONS

x*_ity*_itare temporal deviations from the means of that person because they are subtracted from the expected scores for that person (μ_xit = μ_xt + I_xi and μ_yit = μ_yt + I_yi )
x*_it = B_xX*_i(t-1) + Y_xY_i(t-1) + d_xit
y*_it = B_xY*_i(t-1)+ Y_yY_i(t-1) + d_xit

X_i1 and y_i1 are exogenous variables, d are residuals assumed to be normally distributed
B_x and B_yare autoregressive paths (actor effects)
Y_x and Y_y are cross-lagged parameters (partner effects)

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Reparametrize Random Intercepts

Freely Estimate Loadings

Multiple Indicator RI-CLPM

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Standard Approach Fixed Loadings

Parameterizing the random intercepts (for RI-CLPMs with EITHER manifest or latent variables)

Fix the RI means to 0

Estimate the RI means

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FROM USAMI (2021)

In the traditional RI-CLPM, the temporal group means (α)are excluded from the analysis
Group means can be included within the RICLPM

x*_it = α_xt + B_xX*_i(t-1) + Y_xY_i(t-1) + d_xit
y*_it = α_yt + B_xY*_i(t-1)+ Y_yY_i(t-1) + d_xit

α_yt= Intercept / group means at a specific time
α_xt= Intercept / group means at a specific time
X_i1 and y_i1 are exogenous variables, d are residuals assumed to be normally distributed
B_x and B_yare autoregressive paths
Y_x and Y_y are cross-lagged parameters

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SPECIFYING THE RI-CLPM

Repeated measure of X and Y at three time points
Use latent variables with factor loadings fixed to 1 or freely estimate by constrain to equality
Fix Error variances of measured variables to 0
Create within-person variables on the residuals of the observed variables using phantom latent constructs

Arrows on X2, Y2, X3, Y3 indicated residuals

RI X

RI Y

Within

Between

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Distinct interpretations:

Between-person component (random intercepts): Stability

Hamaker approach: Significant variance = individual differences in the construct

RI-CLPM

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Distinct interpretations:

Between-person component (random intercepts): Stability

Our approach: Recast as factor loadings² and recovered via model constraints

RI-CLPM

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Distinct interpretations:

(b) Within-person component = difference between observed and expected (grand mean + random intercept) score

Auto-regressive effect: Inertia or carry-over effects
Interpreted at my own deviation from my mean

RI-CLPM

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Distinct interpretations:

(b) Within-person component = difference between observed and expected (grand mean + random intercept) score

Cross-lagged effect: within-person deviation from expected X 🡪 deviation from expected Y

RI-CLPM

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Distinct interpretations:

(b) Within-person component = difference between observed and expected (grand mean + random intercept) score

Cross-lagged effect: within-person deviation from expected Y 🡪 deviation from expected X

RI-CLPM

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Interpreting Coefficients

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INTERPRETING COEFFICIENTS: RANDOM INTERCEPTS

Between person latent constructs (random intercept) capture a unit’s time-invariant deviation from the grand means and thus represent the stable differences between units.
A significant positive covariance between the random intercepts suggesting that individuals who have higher levels on variable A also have higher levels on variable B
If the variance of the random intercept does not significantly differ from 0, then there are little to no stable between-unit (dyads) differences, and that each unit fluctuates around the same grand mean over time

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INTERPRETING COEFFICIENTS: WITHIN PERSON PHANTOM VARIABLES

Within person effects are the differences between a unit’s observed measurements and the unit’s expected score based on the grand means and its random intercepts
If αt is positive, this implies that an individual who experiences elevated levels relative to his/her own expected score

Is likely to experience elevated depressive symptoms problems relative to his/her own expected score at the next occasion as well (inertia).

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INTERPRETING COEFFICIENTS

The cross-lagged effects in the model represent the spill-over of the state in one domain into the state of another domain
A positive slope (B) implies that a positive deviation from an individual’s expected level of variable a (e.g., depression) will likely be followed by a positive deviation in the individual’s expected level of variable b at the next occasion in the same direct (controlling for prior levels of b)
A significant variance implies that there are stable, trait-like differences between variable A and variable B

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EXAMPLE 1: DYADIC EXAMPLE

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PURPOSE OF RI-CL-APIM

Why do we need a RI-CL-APIM anyways? We have intensively longitudinal methods (e.g., Laurenceau & Bolger)
MLM is effective, but estimation of multivariate outcomes and lagged effects is difficult if not possible

SAS, for example, has error structures related to lagged effects, but not parameter estimates

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EXAMPLE OF CL-APIM VS RI-CL-APIM

We’re going to examine reciprocal reports of parent and adolescent reports of relationship quality across adolescence

Dyadic data
Biological parent only

3 waves: 12, 14, and 16 from LONGSCAN data (37% White, 63% Black)

N=479

Already tested and found measurement invariance across reporter, time, and race

Configural, Metric, and Scalar

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EXAMPLE OF CL-APIM VS RI-CL-APIM

Cross lagged panel model (cross lagged APIM)

Do adolescents and mothers reports of relationship quality mutually influence each other across adolescence?

Random Intercept CLPM

Are departures from the trait-like perceptions of relationship quality at time 1 and 2 associated with greater departures in their own and the other dyad member’s departures at waves 14 and 16, respectively, controlling for trait like responses

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CROSS LAGGED ACTOR PARTNER INTERDEPENDENCE MODEL (CLPM)

ARQ12

PRQ12

ARQ14

PRQ14

ARQ16

PRQ16

.45***

.23***

.53***

.05

.58***

.01

.19***

.18***

.20***

.40***

Standardized Effects

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RI-CL-APIM

ARQ12

PRQ12

ARQ14

PRQ14

ARQ16

PRQ16

-.28

.31

-.07

.14

.02

Between RQ

.41*

.53***

.12

.68***

.22

.15

.18*

Standardized Effects

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RI-CL-APIM FAQ

Question: The standard errors are not pictured, but they can often be larger than we are accustomed to
Answer: The RI-CLPM is more complex than the CLPM so the parameter estimates are less certain
Question: Can we incorporate predictors and outcomes into the model?
Answer: Absolutely! Variation in within and between components can be predicted by covariates and can predict distal outcomes
Question: Can we incorporate growth into the RI-CL-APIM?
Answer: Yes! But the hitch is that it is done within a dyadic growth curve model with structured residuals (Fitzgerald & Ledermann, forthcoming)�

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P1 S

P1 I

P2 I

P2 S

P1 T1

P2 T1

P1 T2

P2 T2

P1 T3

P2 T3

P1 T4

P2 T4

Dyadic Growth Model with Structured Residuals

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IMPLICATIONS

In the CL-APIM, we found that across dyads, there tends to be actor and partner effects
When looking at what is occurring within dyads, we found substantial differences in the RI-CL-APIM

Notice the negative effect among mothers (not sig), but reflects change in RQ at the onset of puberty
Moms and adolescents don’t have a strong influence on each other over time, but adolescents have a positive “trajectory”
When adolescents have higher within-person deviations from their general tendency then they do the same two years later (both 14 and 16)

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MOVING INTO MPLUS

Mplus

Input File: Baseline RICLPM RI Loadings Constrained to Equality
Data File: RICLAPIM
Input File: Baseline RICLPM RI Loadings Fixed to 1
Data File: RICLAPIM

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MPLUS CODE

USEVARIABLES ARE PRQ12 PRQ14 PRQ16 ARQ12 ARQ14 ARQ16 ;

Analysis:

ESTIMATOR IS MLR;

MODEL = NOCOV; !do not estimate residual covariances by default

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MPLUS CODE

RANDOM INTERCEPT PORTION

Model:

! Between person latent variables or the random intercept

RI_Adol by PRQ12@1 PRQ14@1 PRQ16@1 ;

R1_Mom by ARQ12@1 ARQ14@1 ARQ16@1;

!covariance between RI latent variables

RI_Adol with R1_Mom;

!fix measurement error of observed indicators to 0

PRQ12-ARQ16@0;

WITHIN PORTION OF MODEL

!within person effects (phantom variables)

WithAd1 by ARQ12@1; WithAd2 by ARQ14@1;

WithAd3 by ARQ16@1;

WithMom1 by PRQ12@1; WithMom2 by PRQ14@1; WithMom3 by PRQ16@1;

! lagged effects

WithAd2 on WithAd1; WithAd2 on WithMom1;

WithMom2 on WithAd1 WithMom1; WithAd3 on WithAd2; WithAd3 on WithMom2; WithMom3 on WithAd2 WithMom2;

!covariance between within person effects at time 1

WithAd1 with WithMom1;

!covariances between residuals of within person components

WithAd2 with WithMom2; WithAd3 with WithMom3;

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SYNTAX

You may see different notation in the syntax when looking at others syntax for models using phantom variables
Starting in version 8.7 (I believe), Mplus introduced the residual syntax
Using the symbol “^”, Mplus will automatically add a residual latent variable

Eliminates the need for the by statements of (for example, X1Within by X1@1;)

I haven’t yet integrated this function into my syntax and hard coding the values and variables forces you to know what you’re doing

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Random Intercept Mom

Random Intercept Kid

Factor Loadings

Phantom variables

Within Dyad Actor Effect in Ugly Texas Orange

Within Dyad Partner Effects in Yellow

Correlation of mothers and adolescent random intercepts

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READINGS AND RESOURCES FOR APIM AND RI-CLPM

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READING FOR RI-CLPM

Mulder, J. D., & Hamaker, E. L. (2021). Three extensions of the random intercept cross-lagged panel model. Structural Equation Modeling: A Multidisciplinary Journal, 28(4), 638-648.
Usami, S. (2021). On the differences between general cross-lagged panel model and random-intercept cross-lagged panel model: Interpretation of cross-lagged parameters and model choice. Structural Equation Modeling: A Multidisciplinary Journal, 28(3), 331-344.
Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. (2015). A critique of the cross-lagged panel model. Psychological methods, 20(1), 102.
https://jeroendmulder.github.io/RI-CLPM/mplus.html

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READING FOR RI-CLPM

Berry, D., & Willoughby, M. T. (2017). On the practical interpretability of cross‐lagged panel models: Rethinking a developmental workhorse. Child Development, 88(4), 1186-1206.
Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. P. P. (2015). A critique of the cross-lagged panel model. Psychological Methods, 20(1), 102-116.
Mulder, J. D., & Hamaker, E. L. (2021). Three extensions of the random intercept cross-lagged panel model. Structural Equation Modeling, 28(4), 638-648.
Zyphur, M. J., Voelkle, M. C., Tay, L., Allison, P. D., Preacher, K. J., Zhang, Z., Hamaker, E. L., Shamsollahi, A., Pierides, D. C., Koval, P., & Diener, E. (2020). From data to causes II: Comparing approaches to panel data analysis. Organizational Research Methods, 23(4), 688-716.

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APIM READINGS

Kim, H., & Kim, J.-S. (2022). Extending the actor-partner interdependence model to accommodate multivariate dyadic data using latent variables. Psychological Methods. Advance online publication. https://doi.org/10.1037/met0000531

Simulation Research with Latent Variables

Ledermann, T., Rudaz, M., Wu, Q., & Cui, M. (2022). Determine power and sample size for the simple and mediation Actor–Partner Interdependence Model. Family Relations, 71(4), 1452-1469.

Simulation Research estimating approximate power and sample size requirements for dyadic research

Ledermann, T., & Kenny, D. A. (2017). Analyzing dyadic data with multilevel modeling versus structural equation modeling: A tale of two methods. Journal of Family Psychology, 31(4), 442–452. https://doi.org/10.1037/fam0000290

Estimating the APIM using varying methods

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APIM READINGS

Loeys, T. O. M., Cook, W., De Smet, O., Wietzker, A., & Buysse, A. N. N. (2014). The actor–partner interdependence model for categorical dyadic data: A user‐friendly guide to GEE. Personal Relationships, 21(2), 225-241.

APIM with categorical data

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APIM READINGS

Fitzgerald, M., Esplin, J., Wright, L., Hardy, N., & Gallus, K. (2022). Dyadic parent–adolescent relationship quality as pathways from maternal childhood abuse to adolescent psychopathology. Journal of Marital and Family Therapy, 48(3), 827-844.

Substantive manuscript with dyadic patterns

Fitzpatrick, J., Gareau, A., Lafontaine, M. F., & Gaudreau, P. (2016). How to use the actor-partner interdependence model (APIM) to estimate different dyadic patterns in Mplus: A step-by-step tutorial. The Quantitative Methods for Psychology, 12(1), 74-86.

Estimation of dyadic patterns

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WITHIN-DYAD PROCESSES READINGS

Laurenceau, J.-P., & Bolger, N. (2005). Using diary methods to study marital and family processes. Journal of Family Psychology, 19, 86–97. doi:10.1037/0893-3200.19.1.86
Laurenceau, J.-P., & Bolger, N. (2012). Analyzing diary and intensive longitudinal data from dyads. In M. R. Mehl & T. S. Conner (Eds.), Handbook of research methods for studying daily life (pp. 407–422). New York, NY: Guilford Press.

Focused on intensive longitudinal dyadic data

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INTERMEDIATE AND ADVANCED APIM (TIME PERMITTING)

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INTERMEDIATE AND ADVANCED APIM

Implementing Greek Notation
Diving into matrices and equations of observed variable APIMs
Diving more into model identification
APIM with latent constructs (using CFA)

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APIM

X1A

X2B

Y1A

Y2B

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APIM

X1A

X2B

Y1A

Y2B

X1A Mean

X1A Variance

X2A Mean

X2A Variance

Y1AMean

Y1A Residual Variance

Y2A Mean

Y2A Residual Variance

1 Actor Effect

1 Partner Effect

1 Actor Effect

1 Partner Effect

1 Covariance

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CALCULATING THE NUMBER OF PARAMETERS

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MODEL FIT IN THE APIM

The observed variable path model is just identified (e.g., 0 df) so model data fit cannot be evaluated

Just identified models have only one solution
Solution of a just identified model consists of a set of parameter estimates that perfectly reproduces the observed covariance matrix

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MODEL FIT IN THE APIM

Can evaluate model-data fit of a saturated model in a couple of ways
Remove non-significant parameter estimates and conduct an LRT – needs to be theoretically justified

Remember suppression and overlapping variance are a thing -> path removal can actually have an impact
Results can also be changed if there is a model-misspecification and removal of the path highlights the misspecification via parameter estimates or fit

Constrain specific actor and partner paths to be equal – needs to be theoretically justified
Residual matrix – not an omnibus indicator but can provide information about possible local mistfit

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MULTIVARIATE APIM

The APIM can be extended to include multiple independent variables and multiple dependent variables (multivariate APIM)
Actor and Partner effects are estimated across two (or more) constructs
X1 and X2, Y1 and Y2 are designated as variables, respectively and A and B specify the dyad member

X1A = value of the first predictor for partner A (e.g., husband)
Y1A = value of the first outcome for partner A (e.g., husband)

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X1A

X1B

Y1A

Y1B

X2A

X2B

Y2A

Y2B

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MULTIVARIATE APIM WITH LATENT CONSTRUCTS

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APIM WITH LATENT VARIABLES

Implicit to previous discussion, the APIM was measured using observed variables

Presumed to have no measurement error (e.g., all variance is true score variance), demonstrate tau-equivalence, variables are normally distributed (multivariate normal), variables are continuous, measurement is assumed to be invariant across dyad members

Assumption: x variables distributions are fixed and known (e.g., no measurement error) - regression

Results in parameter estimates that are over or under estimated resulting in possible different substantive conclusions (Cole & Preacher, 2014)

Utilizing latent constructs addresses many these concerns

Estimators can address categorical items (WLSMV) and address non-normality (MLR)
Addresses measurement error via utilization of latent constructs
Measurement invariance can be explicitly tested

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APIM WITH LATENT VARIABLES

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LATENT VARIABLE APIM WITH 2 INDICATORS

x1A

LATENT APIM NOTATION

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COVARIANCE MATRIX AND MODEL SPECIFICATION

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COVARIANCE MATRIX AND MODEL SPECIFICATION

The latent variable APIM with two indicators poses some identification issues

Freely estimated loading

Measurement model is under identified when factor loadings are freely estimated

In CFA we need three indicators for model to be identified (df = 0 or > 0)

Current measurement model with 2 indicators is borrowing degrees of freedom from other latent factors
Would can fix factor loadings to be 1 for both indicators for it to be locally identified

Structural portion of your model is saturated (df = 0)

If your CFI, TLI, RMSEA and chi-square are problematic then it’s because your measurement model is problematic, not you structural portion -> if you have bad fit statistics you know where to look

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LATENT VARIABLE APIM WITH 2 INDICATORS

x1A

x2A

x1B

x2B

y1A

y2A

y1B

y2B

Mean, variance and covariance for each factor = 14 pieces of information

Variances and covariances of factors are estimated as well as their means = 14

14-14 = 0 df in structural portion of the model

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POWER IN THE LATENT VARIABLE APIM

When there is very low reliability within the measures, there is substantial threat to statistical power

Increasing sample size or number of indicators will not address or attenuate measurement concerns

Difficult to detect partner effects using measured variable models – extremely large sample sizes (> 200) are needed

Sample size is in dyad units
Sample size of 200 dyads is 400 people

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COMMON FATE MODELS

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COMMON FATE MODELS

A second method of dyadic analyses is the common fate model
Answers a completely different set of question compared to the APIM – highly complementary analysis
The APIM is focused on mutual influence at the person level while the CFM is focused on the dyad as the unit of analysis affecting each individual

Focus on latent, unmeasured processes operationalized as shared variation between dyad members

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INTERDEPENDENCE IN THE COMMON FATE MODEL

How Interdependence is Modeled

APIM: Mutual influence and correlated residuals
CFM: Dyad level latent variable causing each partner’s observed indicators

The APIM allows researchers to model shared variance between individual reports of depression, but presumes the shared variance is due to other processes (e.g., assertive mating in couples) rather than a shared variable

Couples having a high education level is a function of one or both partners (unlikely to be purely random) is due to homogamy or dating people with similar background characteristics

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INTERDEPENDENCE IN THE APIM

In the CFM, dyad members are presumed to be interdependent due to the influence of a shared variable

External to relationship (e.g., neighborhood)
Characteristics of the relationship itself (e.g., satisfaction)

Another way of thinking about a covariance is that there is an unmeasured common cause

In the APIM, the covariance between dyad members reports can be thought of as an unmeasured dyadic process
In the CFM, the common cause is the dyad level process (level 2 for you multilevel modelers)

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ACTOR PARTNER INTERDEPENDENCE MODEL

Depression

Husband

Wife

Unmeasured Common Cause(s)

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COMMON FATE MODEL

The dyad is the unit of analysis

In the APIM, the individuals are the unit of analysis, but within a dyadic context
Many use APIM when CFM is more appropriate (Galovan et al., 2017)

Rather than using each person’s reports of IVs and DVs, the CFM focuses on the dyad as the unit of analysis

Focuses on similarity between partner

Dyad as unit of analysis

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THE COMMON FATE MODEL

First describe by Kenny & LaVoie (1985) and aimed to disaggregate individual and group effects.
Is particularly helpful partialing out what is shared by members of a dyad and what is unique to each partner
Dyad becomes the primary unit of analysis

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THE COMMON FATE MODEL: BRIEF RETURN TO THEORY

The APIM was highly effective for address the theoretical proposition of interdependence and mutual within FST, but doesn’t fully capture other concepts
FST also proposes that the whole is greater than the sum of the parts: 1 + 1 = 3

APIM cannot assess this domain

CFM assesses what is shared between partners
CFM are APIM are not mutually exclusive – we need both models to test FST

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MEASUREMENT IN COMMON FATE MODEL

Adopt a traditional causal modeling interpretation: The latent variable can be though as a dyad level variable causing each indicator (each person’s response to the commonly measured variable)-> Dyadic / Relationship Level processes CAUSE each individual’s scores
Measurement is about the content of the questions NOT the correlation between the variables

High correlation doesn’t necessarily mean that there is a shared relational or dyad level process

Measures MUST report on same variable AND variable must occur at the dyad level (less flexible than the APIM)

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COMMON FATE MODEL

Shared latent variable predicts a shared latent outcome in the CFM

You can predict individual outcomes in a hybrid model

Why not just create a dyadic average by summing two observed variables together?

Latent variables account for measurement error and in the CFM, measurement error encompasses both random error AND individual perceptions that are not shared with your partner
Using a dyadic average assumes no measurement error and that 100% of the variance between partners is shared -> over estimation of dyad level effects
In the CFM, you can partition out individual vs couple level perceptions

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PATH DIAGRAM EXAMPLE

Dyad Level

Individual Level

Unstandardized Coefficients

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COMMON FATE MODEL

We assume that dyad members are similar to each other because of the interdependence

Can be an attribute of the relationship itself OR external to the relationship

Relationship Example: Coparenting Styles
External Example: Neighborhood factors

If correlations are negative or small then this suggests that the proposed variables are occurring at the individual level -> APIM
In the CFM, the unit of analysis is the dyad, therefore, we must have a level of measurement that is also at the dyad level

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DYADIC COPING INVENTORY EXAMPLE

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WHAT ABOUT PARTNER REFERENTIAL QUESTIONS? WOULD YOU USE THEM IN A CFM

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WHAT ABOUT THIS SUBSCALE?

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RELATIONSHIP LEVEL VARIABLES

Depression

Family Emotional Climate

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RELATIONSHIP LEVEL VARIABLES

Trait Mindfulness

Relationship Mindfulness

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RELATIONSHIP LEVEL VARIABLES

Single Reporter on Neighborhood Cohesion

Dyadic Reports of Neighborhood Cohesion

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IS THE COMMON FATE APPROPRIATE?

Say you are interested in changes in the posttraumatic stress symptoms in couples following the unexpected death of a child.

Secondary traumatic stress theory (Figley) -> psychological symptoms are communicable due to exposure to partner’s symptoms

You measure each dyad member’s posttraumatic stress symptoms (anger, depression, anxiety, intrusive thought, nightmares)

Self-report (e.g., ‘I feel anxious’)

Can you use the common fate model with posttraumatic stress as a common fate variable?

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IS THE COMMON FATE APPROPRIATE?

Although there is a common event that happened to both dyad members, the measurement remains at the individual level

Competing explanation with secondary traumatic stress theory: FST talks about overfunctioning and underfunctioning -> actor partner model may be a better choice

You would really need a dyadic measure of posttraumatic stress
I might estimate a CFM, but wouldn’t expect the factor loadings to be high (perhaps .30-.50)

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ANOTHER SCENARIO

You have two partners within a couple and assess depression using the CESD-10
You run a common CFM model to see what happens
You find that the factor loadings are above the recommended threshold for the CFM

Theory says this likely isn’t a good way to go
Empirically, this is more than sufficient

Do you run a CFM? Do you run an APIM?

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VERDICT: WHAT WOULD I DO?

If you use an APIM, you may not be accurately capturing the dyadic or shared nature of depression symptoms

Depressive symptoms are contagious! (this is a phenomenal title for a paper so don’t you dare steal it - it’s mine)
There is theory to support mutual influence

Can we really feel comfortable about using a dyad-level analysis on an individual level outcome?

Also theory to support a family level process (e.g., family emotional climate)
Do we feel comfortable extrapolating a family level process from shared individual symptoms GIVEN that there is theory to suggest that depressive symptoms are communicable (which is influence – not a characteristics / trait of the family)

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VERDICT: WHAT WOULD I DO?

Present both model and theoretically defend both of them
Advocate for the development of a family emotional climate scale (again – this is mine, but feel free to collaborate)
Let the reader evaluate which is more theoretically consistent with them
One paper is going to solve the problem – but presenting both can spark future research

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COMMON FATE MODEL

The model with 2 latent constructs has 13 parameters estimated

1 variance of exogenous
1 residual error variance of endogenous
1 direct path or covariance from exogenous to endogenous
2 error variances
4 intercepts for indicators
4 error variances for indicators

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

A latent variable with three indicators is “just identified” (you’re broke)

You want to estimate 9 things and you have 9 pieces of information
P +( P(P+1))/2
3 + (3(4))/2 -> factor mean, factor variance, 2 loadings, 2 intercepts, 3 residuals

A latent variable with two indicates is under identified

2 + (2(2+1)/2)
2 + 3 = 5 -> factor mean, factor variance, 1 loading, 1 intercept, 2 residuals = -1 df

You’re in debt!

To estimate the CFM, you generally have to constrain the factor loadings to be 1 on all factors

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

Recall path tracing rules!
Trace backwards, change direction at a two-headed arrow, then trace forwards or simply go forwards
You cannot go forward and then backward
You can only have 1 variance or covariance
You cannot go through the same variable twice in a single trace
Relationship between X1 and X2 -> ?

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

Recall path tracing rules!
Trace backwards, change direction at a two-headed arrow, then trace forwards or simply go forwards
You cannot go forward and then backward
You can only have 1 variance or covariance
You cannot go through the same variable twice in a single trace
Covariance between X1 and X2 -> ?

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

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COMMON FATE MODEL: STATISTICAL IDENTIFICATION

The disturbance (error) term for the latent variable (when a dependent variable) reflects dyad specific deviations
The error terms on the individual, observed scores reflects person-specific deviations from the dyadic score

How do we normally think about and define residuals? Are those reasons applicable here?

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PATH DIAGRAM EXAMPLE

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PATH DIAGRAM EXAMPLE

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PATH DIAGRAM EXAMPLE

Family Violence

Co-parenting

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INTERPRETATION

The more the couple endorsed family violence as a reason for the divorce, the lower quality of the coparenting relationship
A one one unit increase in family violence being a reason for divorce there is -1.324 unit decrease in the coparenting relationship

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COMMON FATE EXAMPLE

Lets Dive into Mplus and estimate the common fate model
Use the following input and data files

Input: couples.dat
Data: Common Fate Baseline

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COMMON FATE WITH INDISTINGUISHABLE DYADS

In indistinguishable dyads (arbitrary assignment), additional constrained are imposed in the model
Imposed Constraints

Factor loadings are all constrained to 1 (same)
All the item intercepts are constrained to equality (unique to indistinguishable dyad)
Error covariances are constrained to equality across member (unique to indistinguishable dyad)

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COMMON FATE WITH INDISTINGUISHABLE DYAD MEMBERS

Lets Dive into Mplus and estimate the common fate model
Use the following input and data files

Input: CFM indistinguishable
Data: twins.dat

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COMMON FATE MEDIATIONAL MODEL

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CFM MEDIATIONAL MODEL

We can extend the CFM into testing indirect effects / mediation at the dyad level
Examines how one dyad level process is associated with another dyad level process through an intervening dyadic level process
When “we” do this, then this happens in “our” relationship and affects “us” as a dyad

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ASSUMPTIONS OF THE CFMEM

1) there is a mediating process occurring and it occurs at the dyadic level
2) observed indicators for both dyad members are reliable
3) dyads are affected by a common cause. Additionally, the assumptions of structural equation modeling must be upheld (e.g., multivariate normality, exogeneity, correctly specified model)

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COMMON FATE MEDIATIONAL WITH HETEROSEXUAL COUPLES

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RESEARCH QUESTIONS WITH CFMEM

Research Questions:

Do couple’s levels of dyadic coping mediate the relationship between couple communication and marital satisfaction,
Does dyadic communication mediate the relationship between family chaos and dyadic closeness in a sample of single parent-adolescent dyads.

The research questions do not appear to be much different than mediation questions that are asked, but emphasize the level of measurement within the question (e.g., dyadic closeness and dyadic closeness). Alternatively, research questions can specify dyadic for each variable to increase precision.

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SPECIFYING THE CFMEM

Similar to base CFM
Fix all loadings to be 1
Estimate variances among within-person variables (this does not cause estimation problems as it typically would when you have a simultaneous predictive relationship and covariance of error terms
Path from X -> M, M->Y, and X->Y depending on theory / hypothesis of full or partial mediation.

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COMMON FATE MEDIATIONAL MODEL

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Non a part of the CFM – this is what makes it a hybrid model

Erratum: Arrows from indicators to latent variables are in the wrong direction

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INTERPRETATION

The more ex-partners endorsed parenting and family violence as reasons for the divorce, the more negative their coparenting relationship which, in turn, as associated with greater behavioral and emotional problems
Coparenting relationship quality is a possible mechanism by which reasons for divorce are associated with greater behavioral and emotional problems among children whose parents are separated or divorced

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CFMEM

The common fate mediational model (CFMeM) is an extension of the CFM and operates in a similar manner to traditional mediational analyses (Ledermann & Macho, 2009).
Criteria for mediation

X does NOT have to be associated with Y (cf. Baron and Kenny, 1986)

Cascades framework -> IV predicts small deviation -> m and m predicts small deviation -> y and multiplying them together can be of great consequences!

The multiplicative effects from X->M and M-> Y is significantly different from 0

Estimation procedures

Bootstrapping
Robust estimators with delta method
Sobel test / delta method

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GOING INTO MPLUS

Input File: Common Fate Mediation
Data File: couples.dat

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INTRAPERSONAL AND DYADIC CFM

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL

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COMMON FATE MODEL

Low Loadings in CFM yield Several options
Investigate mutual influence using the APIM -> requires theoretical adjustment
Test the extent to which the dyad differs on the construct (Latent Congruence Model)
Keep the common fate model, but utilize phantom variables to model the residuals of the X predicting the residuals of the Y (within-person effects)

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PATH DIAGRAM EXAMPLE

Family Violence

Co-parenting

This is the previous model we’ve seen – we see the individual but how do we model the variability?

Unstandardized loadings are presented and standardized loadings are in parentheses

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL

Residuals are typically not of great interest to researchers (e.g., random measurement error)
Upon closer investigation, residuals are of great interest (Curran et al., 2014), particularly in dyadic research

Dyads have shared perceptions, behaviors, and attitudes, but there is rarely complete agreement or disagreement (overlapping variance)
Individuals will differ in how satisfied they are in their marriage
In the CFM, the residuals are not just random measurement error but also where the individual perceptions are

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PATH DIAGRAM EXAMPLE

Family Violence

Co-parenting

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL: PHANTOM VARIABLES

Phantom variables were developed by Rindskopf (1983, 1984)
Originally developed to overcome problems with model constraints in early SEM software (LISREL)
Phantom variables are latent constructs that do not have any manifest indicators
Implement factor loadings of 1 and residual variances of 0
Within the CFM, we use them on the observed indicators

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL: PHANTOM VARIABLES

For phantom variables, we need to “pin them down” since they are not observed variables
To specify the variance, we constrain the loading of the phantom variable to be 1

We also fix the error variance of the observed indicator to be 0 -> pushes information into the phantom construct

Latent variables have a mean, too

Fix the means of the observed indicator to be 0 -> pushes information about the mean into the phantom construct

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THINKING ABOUT LATENT VARIABLES

When you were in undergrad, latent variables were the people who show up to your party but didn’t bring any beer…so they drink yours

They drink your mean (it’s crappy keystone) – we don’t care all that much about means in SEM
They drink your IPAs (these are fightin words) – we do care about variances

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THINKING ABOUT LATENT VARIABLES

You can also think about latent variables being gold diggers
They are very alluring…and they’re pretty…and other people are impressed when they see your model
But you pay a hefty price…you have to give them your money (means + variances)

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL: PHANTOM VARIABLES

I think of paths (and model constraints) as allowing or stopping the flow of information between variables
Constraining the residual variance and mean to be 0 gives information a green light to move into phantom variable

When phantom variables are not there, the residual is a dead end / culdasac and you must turn around

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL

Partner 2 / Individual Level

Dyad Level

Partner 1 / Individual Level

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MODELING INTRAPERSONAL DYNAMICS IN THE COMMON FATE MODEL: NOTATION

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EXAMPLE

Using a sample of 926 heterosexual dyads going through a separation or divorce
Research Question: Do dyad level reasons for divorce predict the quality of the coparenting relationship

Parenting as a reason for divorce (I item on 4 point likert scale)

Common event allow us to use the CFM

Coparenting relationship quality (4 items on 5 point likert scales)

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MODEL-DATA FIT

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MPLUS OUTPUT: UNSTANDARDIZED

Dyad Level

Individual Level

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STANDARDIZED PATH MODEL OF RESULTS

-.162

-.113

Male / Individual Level

Dyad Level

Female / Individual Level

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STANDARDIZED PATH MODEL OF RESULTS

-.162*

-.113*

Male / Individual Level

Dyad Level

Female / Individual Level

Notice how the phantom variable on X1 and X2 doesn’t have it’s own residual -> this is completely consistent with OLS regression and SEM: we assume perfect reliability in our exogenous variables

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ANOTHER EXAMPLE TESTING COMMON FATE MEDIATION

Cross sectional study of heterosexual couples going through separation or divorce
Testing the coparenting relationship as a mediator linking reasons for divorce (family violence and parenting)
926 heterosexual couples – coparenting for resilience program
Each of the variables were common fate variables

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Raise

Family Violence

Coparent

Emotional

Conduct

Mother

Father

Mother

Father

Mother

Father

Mother

Father

Mother

Father

Remember this model? There’s More

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MODEL FIT STATISTICS

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RESIDUAL MATRIX

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Parenting

Family Violence

Co-parenting

Emotional Problems

Conduct Problems

Mother

Father

Mother

Father

Mother

Father

Mother

Father

Mother

Father

e10

-.39

-.29

-.85

-.95

-.11

.27

.16

.15

-.41

-.42

-.37

Note. Unstandardized regression coefficients are presented. Only significant paths are presented for ease of presentation; non-significant paths and covariances were omitted. Terms that start with e represent individual perceptions of the construct for that dyad member. For example, e2 is mothers individual perceptions of family violence contributing to divorce. All paths were significant at p < .05.

Results

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Parenting

Family Violence

Co-parenting

Emotional Problems

Conduct Problems

Mother

Father

Mother

Father

Mother

Father

Mother

Father

Mother

Father

e10

-.39

-.29

-.85

-.95

-.11

.27

.16

.15

-.41

-.42

-.37

The coefficient in the orange box may be over estimated as indicate by significant z score in the residual matrix

Use path tracing rules to go from Male IPV to female coparenting

Could indicate a partner effect or residual covariance

Orange pathways represent significant indirect effects at the individual level

Blue pathways represent significant indirect effects at the dyadic level

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STANDARDIZED INDIRECT EFFECTS AT THE DYAD LEVEL

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STANDARDIZED INDIRECT EFFECTS AT THE DYAD LEVEL

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STANDARDIZED INDIRECT EFFECTS AT THE INDIVIDUAL LEVEL

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STANDARDIZED INDIRECT EFFECTS AT THE INDIVIDUAL LEVEL

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FINDINGS

Couples where family violence and parenting problems contributed to divorced was associated with greater divorced-related behavioral and emotional problems via a lower quality coparenting relationship
Mothers perceptions of family violence and parenting independent of what was shared with her ex-partner were both indirectly related to mother’s individual perception of their child’s behavioral and emotional problems through their independent perceptions of relationship quality

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COMMON FATE MODEL WITH DYADIC AND INDIVIDUAL PARAMETERS

Moving Into Mplus
Input File. Common Fate Mediation Individual parameters
Data File. Couples.dat

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INTERPRETATION

At the dyad level, ex-spouses who reported that they got divorced because of parenting differences also reported a poorer quality coparenting relationship

Notice language: they + coparenting relationship refer to a dyad level process

At the individual level, both mothers and fathers who reported that parenting was a reason for divorce, also reported that they perceived the coparenting relationship, independent on couple level agreement, to be of a lower quality

Notice language: they perceived reflects an individual perceptions

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CAN’T WE DO MORE?

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WE CAN DEFINITELY DO MORE

Both mothers and fathers reported significant effects and we can actually test whether they are significantly different than one another
Create labels for the mother and father paths
Use model constraint section to create a difference score and we will test whether that is different from 0
We can also constrain paths to equality and to a chi-square difference test

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AND CAN DO EVEN MORE!!

We can also add in partner effects into the model

Unlikely to be significant since the CFM is accounting for clustering via latent construct

Constrain the actor and partner effects to be equal
This time, we use a chi-square diff test
Individual perceptions became non-significant -> which model do you use? Consult theory!

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MODERATION IN THE CFM

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MODERATION IN COMMON FATE MODEL

Moderation is a key process in developmental research

Answers questions regarding for whom and under what circumstances do association vary
Key process in most psychological theories – rarely do we actually care about main effects

Moderation in structural equation modeling provides more options

Traditional product approach (you don’t need to center BTW)

Centering only reduces multicollinearity

Multiple group structural equation models

Dichotomous or trichotomous variables and constrain the paths to be equal
My personal preference: freely estimates all parameters in the model (main effects and covariates have separate estimates across the groups

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BUT THE CFM USES LATENT VARIABLES…

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LATENT VARIABLE INTERACTIONS

Scholars have been trying to solve this vexing problem for several decades

Textbooks have been written but provide no singular recommended approach was ever proposed

Some approaches are more trial and error or pragmatic solutions while others are more elegant and creative
We’ll think about the pragmatic approaches from a rage hulk perspective and the more elegant perspectives as sexy hulk: Dr Jekyll and Mr Hyde is Way Overplayed

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RAGE HULK

Product-indicator approach: create a third latent variable that has all possible interaction terms between the observed indicators loaded onto the latent construct

Two latent constructs with 3 indicators each would have 9 product indicators loaded on the latent interaction variable.
Exceptionally Cumbersome

In the CFM, we have four indicators – not terrible

Husband X1 * Wife X1
Husband X2 * Wife X1
Husband X1 * Wife X2
Husband X2 * Wife X2

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RAGE HULK

For those of you who use lavaan, you can estimate the common fate latent variable interactions

This is also entirely possible in Mplus, depending on your preferences

Use the double mean centered approach (Lin et al., 2010)

Mean center the independent variables then create interaction terms
Mean center the interaction terms again

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PRODUCT INDICATOR APPROACH

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DOUBLE MEANING CENTERING IN LAVAAN

https://youtu.be/gAV2fmHrPPw?si=WXbS-ppoiG3HFIN0

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IT WAS SURPRISINGLY EASY TO FIND THESE IMAGES AND I DON’T KNOW HOW TO FEEL ABOUT THIS…�

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SEXY HULK

Nearly all scholars have tried to address the moderation problem using some form of product indicactor
Luckily, latent moderated structural equations (LMS) was developed (Klein & Moosbrugger, 2000)

veered sharply away from the product indicator approach
Uses Latent Class Analyses

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UNDER THE HOOD OF LATENT VARIABLE INTERACTIONS

Distributions of interaction terms are known to be severely non-normal
Product indicator approach would use Robust ML to address non-normality
LMS uses latent class analyses to represent the non-normal distribution as a series of normal distributions

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MODERATION IN COMMON FATE MODEL: LMS LIMITATIONS

It does not provide fit statistics to evaluate model-data fit (uses the integration algorithm)

Just like SEM with discrete outcomes, moderated non-linear factor analysis

Makes testing model-data fit more difficult

Can do a likelihood ratio test (Maslowsky et al., 2015)

Robust methods to non-normal data are not fully developed for LMS and there can be bias inserted when data (outcome) are not normally distributed

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MODERATION IN COMMON FATE MODEL – MODEL FIT IN LMS

First run main effects model without the interaction and assess model-data fit (chi-square, CFL, TLI, RMSEA) and to obtain a loglikelihood value (Model 0)
In Mplus output, the log-likelihood values needed to perform this calculation are labeled “H0 Value.”

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MODERATION IN COMMON FATE MODEL – MODEL FIT IN LMS

Next run the model with LMS interaction and obtain the loglikelihood
D = [(log-likelihood for Model 0) – (log-likelihood for Model 1)]

D = test statistic for likelihood ratio test; distributed of D is very similar to a chi-square distribution

The degrees of freedom (df ) to determine the significance of D is calculated by subtracting the number of free parameters in Model 0 from the number of free parameters in Model 1.

When adding one interaction then you would have a df = 1
Use raw differences rather than corrected difference score (e.g., Satorra & Bentler, 2001) -> don’t use MLR at this point

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MODERATION IN COMMON FATE MODEL – MODEL FIT IN LMS

Example:

Model 0 LL = -7280.014
Model 1 LL = -7233.039
LL difference = 46.975

P < .001

Df =1
Critical value for df =1 in a chi-square distribution = 3.84

Also use AIC and BIC

Only good for nested model – not helpful when comparing two completely different models
Used quite rarely in routine SEM with the exception of mixture modelin

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MODERATION IN COMMON FATE MODEL

Traditionally, I prefer raw/unstandardized effects when testing interaction

I don’t exactly know how to interpret a product term that is created by multiplying two variables in standard deviation units

Previously, LMS did not provide standardized estimates in Mplus (I believe), but now they do

Or, you can standardize all your variables before you run your analysis and the unstandardized output is actually standardized

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YOU CAN STANDARDIZE VARIABLES BEFORE ANALYSIS

If doing this then consult unstandardized results and interpret coefficients as standardized

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COMMON FATE INTERACTION WITH LMS

Coparenting

Mother

Father

Conduct Problems

Mother

Father

Reasons For Divorce

Mother

Father

Coparenting X Reason for Divorce

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COMMON FATE INTERACTION

New commands used in Mplus
Create the latent interaction term using a vertical pipe and XWITH statement

INTER | IV xwith Moderator;

Do not correlate INTER with the IV and moderating variable as you would in an observed model -> LMS calculates the latent factor estimates that are independent with X and moderator. If you do, Mplus will yet at you.

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COMMON FATE INTERACTION

Now that your LMS interaction has run, lets test simple slopes
More New Mplus Commands

Model Constraint

New Parameters creates the simple slopes
-1, 0, and 1 are in SD units
Use basic formulas to create slopes (labels from previous slide are used)

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COMMON FATE INTERACTION: VISUALIZATION

More New Mplus Commands
Plot (LOmod MEDmod Himod) graphs the interacitons
Loop(xval -3, 3, .5) tells Mplus that x and y axes should stop at 3 (remember units are in SD) and we want increments of .5 units (this can be adjusted and is not a hard and fast criteria

Same formula as before but adding the xvalues specified

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RESULTS OF THE CFM WITH LATENT VARIABLE INTERACTIONS

You don’t get any fit statistics
H0 value can be an indicator of model misfit when compared to the model without the interaction terms
Conduct likelihood ratio test

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UNSTANDARDIZED RESULTS

We have two significant main effects, but the interaction between them is not significant (or anywhere close)

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UNSTANDARDIZED RESULTS

You do not get output for new/additional parameters in a standardized metric

Somethings off here….

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SIMPLE SLOPE -> LOW MOD

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SIMPLE SLOPE -> MED MOD

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SIMPLE SLOPE -> HIGH MOD

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MOVING INTO MPLUS

Data File: Couples3.dat
Syntax File: CFMoM LMS

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WHEN TO USE SPECIFIC MODELS

APIM

When research questions focus on mutual influence
When there is little variability at level 2 (random intercept)
When individual level variables are of interest

CFM

When the unit of analysis is at the dyadic level
When you hypothesize both individual and dyadic effects
When you want to consider rates of change over time

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TIME TO GROW UP: COMMON FATE GROWTH MODEL

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COMMON FATE GROWTH MODEL

CFM with mediation, moderation, and individual and dyadic effects can test longitudinal models, but are limited to examining change in a piecewise manner
CFM can also be applied to growth modeling where growth at the dyad level is examined
The CFM is a curve of factors model (latent variable indicates of a growth process)

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COMMON FATE GROWTH MODEL

Growth models estimate rates of change over time

Typically test change in individual constructs over time (e.g., dyadic growth curve)
We can also test dyad level change

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1 SLIDE REFRESHER / INTRODUCTION TO MEASUREMENT

Configural Invariance: is the factor structure (e.g., number of latent variables) the same across group or over time the same
Metric (Weak) Invariance: are the factor loadings of the indicators equal across group or over time the same
Scalar (Strong) Invariance: are the item intercepts of the indicators equal across group or over time the same
Strict Invariance: are the residual variances of the indicators equal across group or over time the same

Not needed, rarely found, using strict invariance can create problems in other parts of your model if there is a misspecifacation

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MEASUREMENT IN CFGM

Measure is absolutely critical in all of research, but especially in growth modeling

Requires configural, metric, and scalar invariance across partner and across wave

In variable variable modeling we need only metric invariance (and really only one indicator that is invariant)
Without invariance, you’re modeling a growth process of measurement error

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COMMON FATE GROWTH MODEL

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SPECIFYING THE COMMON FATE GROWTH MODEL

A measurement model is specified (intercept and slope factors only)

There are no differences between distinguishable and indistinguishable dyads in the measurement portion

Intercept Factor – each of the factor loadings predicting the repeated measures common fate variables are fixed to 1
Slope factor – each of the factor loadings predicting the repeated measures common fate variables are fixed to indicators of time between wave (e.g., 0, 1, 2, 3)

(For our purposes) Time has to be linear, but waves to not need to be evenly spaced

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SPECIFYING THE COMMON FATE GROWTH MODEL: DISTINGUISHABLE DYADS

The factor loadings of the dyad members on the repeated measure common fate variables are fixed to equality across all measurement occasions
Fix the intercepts for one of the dyad members to be 0 at all three measurement occasions

The other partner’s intercepts are freely estimated but constrained to equality

Latent variables do not have a natural metric so we need to give them a mean and variance via factor loadings, and intercepts
Specify covariances across dyad member

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Time 1

Time 2

Time 3

Mother

Father

Mother

Father

Mother

Father

Intercept

Slope

Intercepts fixed to 0 for one partner

Loadings

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COMMON FATE GROWTH MODEL: INDISTINGUISHIBLE DYADS

The factor loadings of the dyad members on the repeated measure common fate variables are fixed to equality across all measurement occasions
Fix all the intercepts for one of the dyad members to be 0 at all three measurement occasions
Constrain the residual variances to be equal across partners within wave

Dyad member A residual variance at time 1 = Dyad member B residual variance at time 1

Specify covariances across dyad member and constrain them to be equal across partner

Covariance between dyad 1 member score at wave 1 and wave 2 is constrained to be equal to dyad

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MODEL BUILDING STRATEGY

0 – establish strong (scalar) measurement invariance
1 – No growth model (intercept only) – if the model has a bad fit (which would expect) then you estimate the growth
2 – intercept and linear growth model
3 – intercept, linear, and quadradic growth
4 – estimate final model then test to see if dyad members are indistinguishable

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MPLUS CODE FOR DISTINGUISHIBLE DYADS

Analysis:

Estimator IS MLR;

Model:

!Measurement Portion of the model

RQ12 by !common fate latent construct) at time 1

ARQ12@1(a1)

PRQ12@1(a1);

RQ14 by ! by !common fate latent construct) at time 2

ARQ14@1(a1)

PRQ14@1(a1);

RQ16 by by !common fate latent construct) at time 3

ARQ16@1(a1)

PRQ16@1(a1);

!constrain means of observed indicators to be 0

[ARQ12@0](a2); [ARQ14@0](a2); [ARQ16@0](a2);

[PRQ12*](b1); [PRQ14*](b1); [PRQ16*](b1);

!covary within person reports over time

ARQ12 with ARQ14 ARQ16; ARQ14 with ARQ16;

PRQ12 with PRQ14 PRQ16; PRQ14 with PRQ16;

!Growth Portion of the model

!Growth factors. I = intercept, S = Slope

S by RQ12@0 RQ14@1 RQ16@2;

I by RQ12@1 RQ14@1 RQ16@1;

s with i;

[i*]; [s*]; !freely estimate factor means

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SNEAK PEAK OF WHAT IS UNDER DEVELOPMENT

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AN EXTRA WAVE, BUT NOTHING NEW

Time 1

Time 2

Time 3

Mother

Father

Mother

Father

Mother

Father

Intercept

Slope

Time 3

Mother

Father

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Time 1

Time 2

Time 3

Mother

Father

Mother

Father

Mother

Father

Intercept

Slope

Time 3

Mother

Father

Phantom Variables

Within Dyad APIM

Use residuals of growth process (individual perceptions) to model a within-dyad APIM

Occurs independently of the growth curve