Multilevel modeling and ego network dynamics

Brea L. Perry

Indiana University

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MLM for social networks

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MLM for social networks

Like students in schools or obs over time in people

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MLM for social networks

When to use MLM for ego SNA: Formal requirements

1. DV is an alter or tie-level variable (level-1)
• If you are interested in predicting a characteristic of ego (e.g., health, employment outcomes, movement participation), MLM is not appropriate
• IVs can be alter, tie, network, or ego-level variables (level-1 or 2)
2. Personal networks of egos do not overlap (or overlap is negligible)
3. Ego observations are independent of one another

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MLM research questions

• What affects formation of ties to alters with particular attributes?
• What kinds of egos are more likely to form cross-racial ties?
• What affects alter behavior or contributions?
• What kinds of ego and network characteristics affect alter provision of instrumental support?
• What affects characteristics of dyads/relationships?
• What ego and alter characteristics affect likelihood of sexual activity between members of a dyad?
• What ego, alter, and tie characteristics affect level of conflict in a dyad?

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What is MLM?

Same as other terms you have heard:

• General Linear Mixed Model
• Random Effects Model
• Fixed Effects Model
• Variance Components Model
• Hierarchical Linear Model
• Growth Curve Model (if using longitudinal data)

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Two major parts of any model

Part I: Model for the means

• AKA fixed effects part of the model (i.e., fixed parameters)
• What you are used to caring about for testing hypotheses
• How the expected outcome for a given observation varies on average as a function of values of predictor variables

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Two major parts of any model

Part II: Model for the variances

• AKA random effects and residuals (i.e., stochastic or varying parameters)
• What you are used to making assumptions about
• How residuals are distributed and related across observations (persons, groups, time, etc.) are the primary way that multilevel models differ from general linear models (e.g., regression)

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Dependency

Model dependency or autocorrelation across observations

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Why use MLM?: Dependency

Where does dependency come from? Two sources:

1. Mean differences across sampling units (e.g., egos)
• Represented by a random intercept in our models

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• Differences in effects of predictors across sampling units
• Longitudinal: individual differences in growth or change
• Clustered: group differences in slopes of predictors
• Represented by random slopes in our models

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Why use MLM?: Dependency

• Ignoring multi-level structure has implications for statistical validity
• Depresses standard errors, makes it easier to find significance when there really is none
• Can affect parameter estimates when extreme

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• Multilevel model accounts for clustering (non-independence) and allows you to explicitly model it rather than just control for it

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Review of General Linear Model (GLM)

• The error term is a random parameter
• Try to minimize error by adding covariates that explain as much variance as possible

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What we’re doing with a GLM

Each red arrow represents error - the line of fitted values does not go through every data point, so there is error in the estimate

We want to draw a regression line that minimizes error…this is what OLS does

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Review of GLM

• Problem is that error terms for observations clustered within an individual are probably correlated
• Violates OLS assumptions

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Random-intercept model

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• Explicitly model the error dependence by splitting up the error term into level-1 and level-2 components
• Have a random intercept for level-2 person j that is constant across all level-1 observations
• Have an error term for each obs i clustered within person j

zeta

epsilon

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Random-intercept model

Think about this as simply splitting up a pile of variance (i.e., differences from the mean) into two types:

• Between-cluster (BC) variation
• Level-2 “INTER-cluster differences” (zeta)
• How (and why) do ego nets differ from other ego nets?
• Within-cluster (WC) variation
• Level-1 “INTRA-cluster differences” (epsilon)
• How (and why) do alters within the same ego differ from other alters in the same network, on average?

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Random intercept model

We are just making piles of variance, not reducing overall variance

OLS

Random-intercept MLM

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Random-intercept model

psi

theta

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Intraclass correlation

• Rho is a measure of between-cluster heterogeneity OR as within-cluster homogeneity (two sides of the same coin)
• Typically call it the intraclass correlation, which is a measure of within-cluster correlation

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rho

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Intraclass correlation

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Intraclass correlation

ICC is a standardized way of expressing how much we need to worry about dependency due to cluster mean differences

• Bigger ICC 🡪 more messed up standard errors

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Sexual contact example

Suppose I want to predict how often couples have sex…

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Sexual contact example

Two egos Jane and Joe and five sex partners (dyads)

• What can we say about Jane and Joe’s sex lives?
• What can we say about within and between variation?

Variation within

Variation within

Variation between

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Sexual contact example

• Both Jane and Joe get their own random intercept

Jane’s regression line

Joe’s regression line

y = # sexual contacts

x = number of instruments

3

2

1

0

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Sexual contact example

Egos get their own random intercept based on their alter/tie observations

Every ego gets their own regression line

• Intercept is “random” (varies)
• Slope is constant

y = # sexual contacts

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Sexual contact example

What if the effects of x on y within egos are different?

Jane’s regression line

Joe’s regression line

y = # sexual contacts

x = number of instruments alter plays

3

2

1

0

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Random-coefficient model

The random-coefficient linear regression model:

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Sexual contact example

Jane’s regression line

Joe’s regression line

y = # sexual contacts

x = number of instruments alter plays

2

1

0

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Sexual contact example

Egos get their own random intercept and slope based on their alters

Every ego gets their own regression line

• Intercept is “random” (varies)
• Slope is “random” (varies)

y = # sexual contacts

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Random coefficient model

We are still just making piles of variance, not reducing overall variance

OLS

Random-intercept MLM

Random-coefficient MLM

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Random effects, between effects, and fixed effects

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Cluster confounding: A threat to RE models

• The RE model assumes that Level-1 covariates are uncorrelated with the random intercept
• Problematic because all Level-1 variables contain information about observations and groups (or individuals and individual change over time)
• Another major problem is that the process of pooling assumes that within-cluster and between-cluster effects are equal

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Cluster confounding example

• Let’s return to our example of the effects of instruments on sexual frequency in sexual networks…
• In typical random effects model, we are assuming that the within and between effects are the same:
• Purely between-cluster effect 🡪 Do egos with higher mean levels of instrument-playing sex partners have higher mean levels of sexual activity?
• Purely within-cluster effect 🡪 Do egos have more sex with partners in their network that play more instruments relative to other partners that play fewer?

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Sexual contact example

Number of instruments

Three egos:

Jane

Joe

Jalissa

OLS regression line (completely pooled)

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Sexual contact example

Number of instruments

Three egos:

Jane

Joe

Jalissa

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Sexual contact example

Number of instruments

Three egos:

Jane

Joe

Jalissa

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Sexual contact example

Number of instruments

OLS regression line (completely pooled)

Three egos:

Jane

Joe

Jalissa

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Sexual contact example

Number of instruments

RE regression line (partially pooled)

Three egos:

Jane

Joe

Jalissa

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Cluster confounding example

• Interpreting the weighted average is problematic in this example
• What is the comparison group?
• Coefficient is biased
• Significance tests will tell us there is no effect
• Meaningful and interesting effects are masked

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There are solutions…

1. Add a “contextual effect”
2. Split alter variance into within and between
3. Use a fixed effects or between effects model (depending on your research question and data properties)

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Contextual effects

Add a contextual effect of alter/tie-level variables by including the aggregated (usually mean) version of the variable

• This tests whether the within and between effect are equal. If the contextual effect is non-significant, you don’t have to worry about biased estimates due to cluster confounding
• Does the network context matter?

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Contextual effects

In sexual contact example, you would add mean number of instruments across all alters within a network when modeling the effect of number of instruments each alter plays

• Need to separate out the effects of ego being attracted to alters who play instruments (mean) from having Prince as a sexual partner (value for each alter)
• The ego or network context could affect the true parameter value of the alter-level effect

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Decomposing variance

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Decomposing variance

In sexual contact example…

• At Level-2, the variable is equal to the mean number of instruments played by alters in a network
• At Level-1, the variable is equal to each alter’s deviation from the network mean number of instruments

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Random vs. fixed effects models

• We can think about the difference between random effects and fixed effects models as variations in the degree of pooling
• Pooling = Degree to which each cluster mean is “pulled” toward the overall grand mean
• As ICC 🡪 0, RE looks more like OLS; As ICC 🡪1, looks more like FE

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No Pooling Partial Pooling Complete Pooling

OLS 🡪 Ignore clusters and assume they don’t matter

Random effects 🡪 Borrow information from the grand mean

Fixed effects 🡪 Within-cluster mean and variation are all that matter

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MLM in Stata

What does the output/interpretation look like?

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YOU CAN DO THIS!!!

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MLM in Stata

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MLM in Stata

Suppose we want to look at the effects of ego and alter gender on the number of support functions provided by an alter to an ego

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Empty RI model

Step 1: Run an “empty” model (without covariates) to assess the random parts of the model

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Empty RI model

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Empty RI model

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RI model with predictors

Model for the means is just business as usual

Most people ignore the model for the variances

Step 2: Add covariates and interpret as usual

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Add a contextual effect

. *Random intercept model with contextual effect

. xtmixed support egofem altfem netfem10 || EGOID: , mle variance

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Performing EM optimization:

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Performing gradient-based optimization:

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Mixed-effects ML regression Number of obs = 19,409

Group variable: EGOID Number of groups = 1,050

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Obs per group:

min = 2

avg = 18.5

max = 67

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Wald chi2(3) = 40.11

Log likelihood = -24624.611 Prob > chi2 = 0.0000

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support | Coef. Std. Err. z P>|z| [95% Conf. Interval]

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egofem | .0189512 .0216578 0.88 0.382 -.0234973 .0613998

altfem | .0761756 .0126522 6.02 0.000 .0513778 .1009735

netfem10 | -.0217038 .0073214 -2.96 0.003 -.0360534 -.0073541

_cons | .8996575 .0332023 27.10 0.000 .8345822 .9647329

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Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]

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EGOID: Identity |

var(_cons) | .0431032 .0037524 .0363419 .0511224

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var(Residual) | .7121663 .0074291 .6977535 .7268769

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LR test vs. linear model: chibar2(01) = 377.53 Prob >= chibar2 = 0.0000

Step 3: Add a contextual effect and interpret

*Could also decompose variance but not if your predictor is binary (interpretation doesn’t make sense)

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Ego network dynamics

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Social network dynamics

• Most research provides only a “snapshot” of networks
• However, networks are constantly changing
• Key is to disentangle “real” change from methodological artifact

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What we know about social network dynamics

• Structural properties of networks tend to remain stable over time
• E.g., homeostasis in networks; social signatures
• BUT lots of “turnover” or “churn” in the individuals that make up a network
• Loss of ties does not mean networks are getting smaller – may just be replacement

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What we know about social network dynamics

• Toronto, Ontario residents: only 27% of ties persist over a decade (Wellman et al. 1997)
• Caregivers’ support networks: 30% of ties persist over a decade (Suitor & Keeton 1997)
• Recent widows: 22% of ties persist over one year (Morgan et al. 1997)

(Cornwell et al. 2021)

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What we know about social network dynamics

At least three mechanisms of membership turnover:

• Selective activation of latent ties (e.g., job search)
• Internal dynamics (e.g., conflict)
• External dynamics (e.g., moving away)

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What we know about social network dynamics

Personal network dynamics operate similarly to biological evolution, which is to say that they undergo gradual, random ebbs and flows in membership punctuated by intense rapid shifts that correspond to significant events (Wellman)

• Divorce
• Major acute illness
• Natural disaster
• Migration
• Retirement
• Parenthood

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What we know about social network dynamics

Networks are comprised of two basic components:

• a smaller and more stable core
• Densely-knit, mostly kin, highly supportive
• a larger set of temporary or sporadic ties (the periphery)
• Most turnover occurs in periphery

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What we know about social network dynamics

Periphery is a problem for cross-sectional network studies

• People engage in periods of brief and sporadic periods of meaningful contact (e.g. old friend visits)
• The likelihood of these sometimes-inactive relationships being present in a snapshot of a network is essentially random
• When peripheral ties are not captured, they are assumed to be absent rather than inactive
• Instability does not mean real change

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How to measure network change

Problem 1: Real change or methodological artifact?

• Respondents forget to name alters from previous waves 5-10% of the time
• Respondents deliberately underreport alters in subsequent waves because they know each alter = more work (i.e., panel conditioning)
• Respondents give different names or spellings in subsequent waves (i.e., error)

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How to measure network change

Problem 2: Determining what alter-level changes underlie network-level change

Suppose the mean freq of contact with network members decreases from W1 to W2. This can be due to…

1. Ego decreasing contact with alters who were present at both W1 and W2
2. Loss of alters with whom ego had frequent contact
3. Addition of new alters with whom ego has infrequent contact

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How to measure network change

Solution: Real change or methodological artifact?

In each follow-up wave of a study…

• Have egos name their current alters
• Show them their roster from the previous wave or waves
• Have them match alters across waves
• Ask them why they didn’t name any dropped alters, and add if they report forgetting

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How to measure network change

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Measures of network change

Network-level measures that capture network turnover (pooled across two or more waves)

• N/Prop alters dropped
• N/Prop alters added
• N/Prop stable alters

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Measures of network change

N dropped or added

N unique alters pooled

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Network turnover

(Perry & Pescosolido 2012)

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How to analyze network change

If goal is to describe network-level change:

• Simple comparison of ego network characteristics over time
• E.g., Avg degree at W1 compared to avg at W2
• Measure of difference between two waves
• E.g., W2 degree – W1 degree

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How to analyze network change

Source: Cornwell et al. 2014

Comparing mean characteristics of networks across waves…

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How to analyze network change

If goal is to describe alter-level change:

• Distinguish alters dropped, maintained, or added across W1 and W2
• Can present number or percent of each
• E.g., 35% of alters dropped, 35% maintained, 30% added
• Compare characteristics of each
• E.g., 75% of maintained alters are “very close” compared to 35% of dropped alters

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How to analyze network change

Source: Cornwell et al. 2014

Comparing characteristics of stable, lost, and new alters…

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How to analyze network change

If goal is to predict network-level change

• Use two-level longitudinal multilevel models or growth (curve) models where obs over time at network level are nested in egos
• Model the effects of time and (often) time-squared
• E.g., how does network size change (non-linearly) with age
• Decompose variance into within and between effects (the latter is growth/change)
• Interact time with other predictors to test whether networks change at different rates as a function of other variables
• E.g., Does the rate of change in network size with age depend on gender?

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How to analyze network change

If goal is to predict alter-level change

• Use three-level longitudinal multilevel models or growth (curve) models where obs over time are nested in alters are nested in egos
• Model the effects of time and (often) time-squared
• E.g., how does time in the illness career affect whether an alter drops out of the network
• Interact time with other predictors
• E.g., Does the likelihood of an alter dropping as ego moves through the illness career depend on how the alter is related to ego?

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Questions?

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Three important considerations when choosing FE or RE

1. Amount of variation within clusters compared to variation across clusters

• If variation is primarily within clusters, use random effects
• Can assess using ICC, and rule of thumb is ICC < 0.50
• If variation is primarily between clusters, solution is more complicated (see next slide)
• ICC >.50 indicates a “sluggish” variable (not much within cluster variation, OR obs within cluster are highly correlated)

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Three important considerations when choosing FE or RE

2. Amount of data (obs per cluster and number of clusters) if you have a sluggish DV

• With little data, use random effects
• With lots of data, use fixed effects

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Cross-level interactions

. xtmixed support altfem##egofem c.netfem10##egofem || EGOID: altfem , mle covariance(unstructured) variance

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Mixed-effects ML regression Number of obs = 19,409

Group variable: EGOID Number of groups = 1,050

Wald chi2(5) = 103.07

Log likelihood = -24593.322 Prob > chi2 = 0.0000

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support | Coef. Std. Err. z P>|z| [95% Conf. Interval]

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1.altfem | -.0234568 .018998 -1.23 0.217 -.0606921 .0137786

1.egofem | -.0578108 .0740929 -0.78 0.435 -.2030302 .0874086

altfem#egofem |

1 1 | .191793 .0255892 7.50 0.000 .1416391 .2419468

netfem10 | -.020383 .0112081 -1.82 0.069 -.0423506 .0015845

egofem#c.netfem10 |

1 | -.0033288 .0147954 -0.22 0.822 -.0323272 .0256697

_cons | .9356164 .0481098 19.45 0.000 .841323 1.02991

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Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]

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EGOID: Unstructured |

var(altfem) | .0022581 .0066916 6.78e-06 .7518713

var(_cons) | .0358574 .0049609 .0273411 .0470265

cov(altfem,_cons) | .0067692 .0045788 -.002205 .0157434

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var(Residual) | .7093482 .007568 .6946691 .7243374

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LR test vs. linear model: chi2(3) = 385.69 Prob > chi2 = 0.0000

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Cross-level interactions

• When ego is a man, there is no significant effect of an alter being a woman (b=-0.02) on number of support functions. However, when ego is a woman, women alters are expected to provide 0.17 more support functions than men alters.
• Interaction at Level-2 is not significant

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