Boosting powers by combining spatial econometrics with dyadic analysis and SEM. Racial/ethnic differences in life expectancy across the US states

Methodological Innovation - Modern Modeling Methods, Storrs, CT, June 26-28 2023

Emil Coman^P, Adrian-Gabriel Enescu², Peter Chen³, Sandro Steinbach⁴

^P: Health Disparities Institute, UConn School of Medicine; ²: Univ. of Brasov, Romania, Economics, ³: Uconn Geography; ⁴: Uconn Agricultural Economics; coman@uchc.edu

Slides at tinyurl.com/mmmdyadic

What would your department say?

2

1. Challenges of spatial data and analytics and solutions

2. Applying path analysis to spatial data: naïve/a-spatial vs. spatial modeling

3. A modern spatial twist of a statistical fix for auto-correlated data – dyadic non-independence

4. Dyadic analysis and Gonzalez and Griffin approach

5. ‘Auto’-correlations and interference: causality

6. Future extensions: 1-to-many relations; spatial factor analysis

General plan

3

Why spatial analytics is needed

Spatial perspectives in family health research https://academic.oup.com/fampra/article/39/3/556/6463006

4

Intuition for minimum Moran’s I

“all the variation is within classes [neighbors of red squares], with the result that there is no variation between class (i.e., each class sum equals [the same #]).”

​

Haggard, E. A. (1958). Intraclass correlation and the analysis of variance

Intuition for Maximum Moran’s I

“there is no variation between the scores in any of the [classes [neighbors of red squares]; rather all the variation is between the [classes [the same #]).”

​

Haggard, E. A. (1958). Intraclass correlation and the analysis of variance

Networks spatial structure

NY

CT

NH

RI

Lines from the actual *.gal weights file in GeoDa for CT:

CT 3

NY MA RI

MA

VT

ME

A classic regression Y_i = α. + β.·X_i + ε_i would become for spatially connected/nonindependent data e.g.:

Y_CT = ρ·(1/3·Y_MA + 1/3·Y_NY + 1/3·Y_RI) + α. + β.·X_CT + ε_CT,

which says that MA, NY, and RI are neighbors of CT

Y_ME = ρ·(1·Y_NH) + α. + β.·X_ME + ε_ME,

which says that only NH is a US state neighbor of ME

Y_MA = ρ·(1/5·Y_CT + 1/5·Y_NY + 1/5·Y_NH+ 1/5·Y_RI+ 1/5·Y_VT) + α. + β.·X_MA + ε_MA,

which says that CT, NY, NH, RI and VT and RI are neighbors of NY

Y_RI = ρ·(1/2·Y_CT + 1/2·Y_MA) + α. + β.·X_RI + ε_RI,

which says that CT and MA are neighbors of RI

etc., 45 more times

​

Self & Other

From naïve/a-spatial to spatial regression

Spelling out the ‘auto’-correlation meanings

min I possible 2: I = -0.402

By-hand method: started to the NW corner, and walked down and right, to ensure YELLOW states are surrounded by PURPLE ones

Small’s possible patterns - 2

I = -.007, z = 1.819, p = .018

Large I’s possible patterns

I = +.652, z = 6.82, p < .001

https://geodacenter.github.io/workbook/5a_global_auto/lab5a.html ; #of neighbors Moran’s Is .245, pseudo p = .007

15

%Minority._s

Income._s

LifeExp._s

+.38^.002_LEx.I

-.18^.006_%M.LEx

Univariate and bivariate Moran’s Is (##_{2nd .1st})

+.54^.001

+.43^.001

+.38^.002

-.07^.169_I.%M

-.02^.368_%M.I

+.35^.001_I.LEx

-.27^{. 001}_LEx.%M

. pcorr lfex20stcdc pminrty15_19st inck1519st (obs=49)

Partial and semipartial correlations of LIFEEXPECTANCY with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

%Minority | -0.4185 -0.3030 0.1751 0.0918 0.0031

INCOME | 0.7475 0.7400 0.5587 0.5475 0.0000

​

Partial and semipartial correlations of %Minority with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

INCOME | 0.4475 0.4430 0.2003 0.1963 0.0014

LIFEEXPECTANCY -0.4185 -0.4079 0.1751 0.1664 0.0031

​

Partial and semipartial correlations of INCOME with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

%Minority | 0.4475 0.3241 0.2003 0.1050 0.0014

LIFEEXPECTANCY |0.7475 0.7286 0.5587 0.5308 0.0000

​

​

​

​

%Minority._s

Income._s

LifeExp._s

(##_{Outcome.Predictor})

* The partial correlation between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 if the other x's did not vary.

* The semipartial correlation, also called part correlation, between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 after the effects of all other x's are removed from x1 but not from y.

Sample naïve = a-spatial PARTIAL correlations (3^rd variable is the 3^rd)

NON-Directional!

+.75*

+.45*

-.42*

17

%Minority._s

Income._s

LifeExp._s

+.32*_I.%M

-.30*_LEx.%M

Sample naïve = a-spatial SEMI-PARTIAL correlations

Directional

+.44*_%M.I

+.74*_LEx.I

-.41*_%M.LEx

+.73*_I.LEx

Partial and semipartial correlations of lfex20stcdc with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

pminrty15~t | 0.0739 0.0525 0.0055 0.0028 0.6176

lglfex20 | 0.6988 0.6917 0.4883 0.4785 0.0000

Partial and semipartial correlations of lfex20stcdc with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

inck1519st | 0.5536 0.3931 0.3064 0.1545 0.0000

lglfex20 | 0.5770 0.4178 0.3330 0.1746 0.0000

Partial and semipartial correlations of pminrty15_19st with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

inck1519st | 0.3511 0.2512 0.1233 0.0631 0.0144

lgpmin1519 | 0.7262 0.7079 0.5274 0.5011 0.0000

Partial and semipartial correlations of pminrty15_19st with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

lfex20stcdc | 0.0877 0.0627 0.0077 0.0039 0.5535

lgpmin1519 | 0.6939 0.6869 0.4814 0.4718 0.0000

​

Partial and semipartial correlations of inck1519st with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

pminrty15~t | 0.3305 0.2789 0.1092 0.0778 0.0218

lginck19st | 0.5764 0.5618 0.3322 0.3156 0.0000

​

Partial and semipartial correlations of inck1519st with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

lfex20stcdc | 0.5798 0.4894 0.3362 0.2395 0.0000

lginck19st | 0.3133 0.2269 0.0982 0.0515 0.0301

[COMPARE SAME COLORS]

​

​

%Minority._s

Income._s

LifeExp._s

* The partial correlation between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 if the other x's did not vary.

* The semipartial correlation, also called part correlation, between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 after the effects of all other x's are removed from x1 but not from y.

(##_{Outcome<-Predictor})

​

Sample naïve = a-spatial PARTIAL correlations

NON-Directional!

+.75*

+.45*

-.42*

19

%Minority._s

Income._s

LifeExp._s

+.74_LEx<-I

-.30_LEx<-%M

Sample naïve = a-spatial PARTIAL correlations

Directional (3^rd variable is the ‘other’)

+.32_I<-%M

+.73_I<-LEx

-.41_%M<-LEx

+.44_%M<-I

20

%Minority._s

Income._s

LifeExp._s

Spatial SEMI-PARTIAL correlations – Directional

(3^rd variable is the ‘DV lag’)

%Minority._s

Income._s

LifeExp._s

Sample naïve = a-spatial SEMI-PARTIAL correlations

Directional

(3^rd variable is the 3rd)

​

sLAG Inc_s

*

sLAG Inc_s

*

sLAG Inc_s

*

+.39_LEx<-I

+.25_%M<-I

+.05_LEx<-%M

+.74_LEx<-I

+.32_I<-%M

-.30_LEx<-%M

+.73_I<-LEx

+.44_%M<-I

-.41_%M<-LEx

+.49_I<-LEx

+.28_I<-%M

+.06_%M<-LEx

* The partial correlation between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 if the other x's did not vary (bidirectional)

* The semipartial correlation, also called part correlation, between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 after the effects of all other x's are removed from x1 but not from y (directional)

%Minority_s

Income_s

LifeExp._s

+.160^.001

d_P=-.037^.001

-.312^.001

Standardized coefficients from

X + M + X*M + -> Y

X + -> M

​

Total (spatial) effect

TE = -.061^.892

Mediated Interaction (spatial) effect

INT_Med = -.001^.152

Hence i_T = i_P = i_BK & d_T = d_P

Controlled Direct Effect = -.037^.001

_{ContrDE = cPrime + ReferenceInteraction *Interaction*MIntercept}

​

​

TE = i_T + d_P = d_T + i_P

INT_Med = i_T - i_P = d_T – d_P

Direct Pure = d_P

Direct Total = d_T

Indirect Pure = i_P

Indirect Total = i_T

​

​

(p values in superscripts)

i_T =-.023^.252

+.154^.109

Naïve/a-spatial ‘causal’ mediation model

USEVARIABLES ARE

CovY1 CovM1 M Y X XbyM ;! CovM1 can add; all new variables, not in data, but are defined below

define: ! we define X, M, Y here so anyone can replace only this upfront define section

X = YOUR_X - XMean ;

MODEL:

Y on X (cPrime) !cPrime, the original mediation a-b-c-c' labeling

M (b) !b, the original mediation a-b-c-c' labeling

XbyM (Interact) !each label HAS to be entered in separate rows for each predictor

CovY1 ; !lagXbyM

M on X (a)

CovM1; !a, the original mediation a-b-c-c' labeling

!can add COVARIATES specific to M: CovM1 ;

[M] (mInt); !define M intercept for use below in ContrDE & INTref

MODEL CONSTRAINT: ! Additional effects from VanderWeele NEW (DirPure IndTot TotEff IndPure DirTot INTmed ContrDE INTref PrAttrIn PrElim );

!INTref & ContrDE are NOT fixed parameters, but varying with mInt=Mediator intercept

DirTot = cPrime + Interact*mInt + Interact*a; ! TOTAL DIRECT EFFECT as in 3. M&A

DirPure = cPrime + Interact*mInt; ! PURE DIRECT EFFECT as in 3. M&A

!dp = ContrDE + INTref; !as in 2. VanderWeele: alternative estimation

IndTot = a*b + Interact*a; ! TOTAL INDIRECT EFFECT as in 3. M&A

IndPure = a*b ; ! PURE INDIRECT EFFECT = B&K effect as in 0. Baron & Kenny

TotEff = IndTot+DirPure; !TOTAL EFFECT

INTmed = TotEff - IndPure - DirPure; ! MEDIATED INTERACTION as in 2. VanderWeele / ContrDE = cPrime + INTref*Interact*mInt; ! CONTROLLED DIRECT EFFECT as in 2. VanderWeele /added EC FT JF/

!ContrDE = dp - INTref; ! CONTROLLED DIRECT EFFECT as in 2. VanderWeele: alternative estimation; is estimated at a particular M value, here at the M intercept

!one can replace its intercept m0 used here with specific M values of interest;

INTref = DirPure - ContrDE ; ! REFERENCE INTERACTION as in 2. VanderWeele /added EC FT JF/

PrAttrIn = TotEff - IndPure - ContrDE ;! Proportion Attributable to Interaction as in 2. VanderWeele /added EC FT JF/

PrElim = IndPure + INTmed + INTref ; ! Portion Eliminated as in 2. VanderWeele

! PrElim = TE - ContrDE ; !as in 2. VanderWeele: alternative estimation /added EC FT JF/

OUTPUT: SAMP TECH1 TECH4 standardized CINTERVAL

Coman, E. N., Thoemmes, F., & Fifield, J. (2017). Commentary: Causal Effects in Mediation Modeling: An Introduction with Applications to Latent Variables. Frontiers in Psychology, 8(151). https://www.frontiersin.org/articles/10.3389/fpsyg.2017.00151/full

Mplus ‘causal’ mediation ‘by hand’ + VanderWeele

%Minority_s

Income_s

LifeExp._s

Standardized coefficients from

X + M + X*M + Y_Sp.Lag -> Y

X + M_Sp.Lag -> M

​

Total (spatial) effect

TE = -.047^.878

i_P = +.023^.023

Mediated Interaction (spatial) effect

INT_Med = +.051^.058

Hence i_T > i_P = i_BK & d_T > d_P

Controlled Direct Effect = -.019^.082

_{ContrDE = cPrime + ReferenceInteraction *Interaction*MIntercept}

​

​

TE = i_T + d_P = d_T + i_P

INT_Med = i_T - i_P = d_T – d_P

Direct Pure = d_P

Direct Total = d_T

Indirect Pure = i_P

Indirect Total = i_T

​

​

(p values in superscripts)

sLAG Income_s

sLAG Debt_s.

*

*

-.317^.001

Spatial ‘causal’ mediation model

+.160^.001

d_P=-.019^.950

-.312^.001

i_T =-.028^.152

Wide = Horizontal

Long = Vertical

Wide Dyadic = Horizontal Dyadic

Long Dyadic = Vertical Dyadic

24

Gonzalez Griffin dyadic approach expanded -spatial

+++

Turn the data from n=49 to self+others -> Moran’s I becomes a Pearson correlation in the N = 98 data.

Merge them

Gonzalez Griffin dyadic approach expanded -spatial

* 1. Overall within-partner XY correlation rxy.c = rx'y'.c

. pcorr pmin lfex20 self1oth2(obs=98)Partial and semipartial correlations of pmin with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

lfex20 | -0.2127 -0.2126 0.0453 0.0452 0.0364

self1oth2 | -0.0309 -0.0302 0.0010 0.0009 0.7639

​

. * 2. X pairwise intraclass correlation rxx'.c

. pcorr pmin pmino self1oth2(obs=98)Partial and semipartial correlations of pmin with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

pmino | 0.6216 0.6214 0.3864 0.3861 0.0000

self1oth2 | -0.0623 -0.0489 0.0039 0.0024 0.5444

​

. * 3. Y pairwise intraclass correlation ryy'.c

. pcorr lfexp lfex20o self1oth2(obs=98)Partial and semipartial correlations of lfexp with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

lfex20o | 0.6809 0.6809 0.4636 0.4636 0.0000

self1oth2 | -0.0003 -0.0002 0.0000 0.0000 0.9975

​

. * 4. XY cross-intraclass correlation rxy’.c = rx'y.c

. pcorr pmin lfex20o self1oth2(obs=98)Partial and semipartial correlations of pmin with

Partial Semipartial Partial Semipartial Significance

Variable | corr. corr. corr.^2 corr.^2 value

------------+-----------------------------------------------------------------

lfex20o | -0.2611 -0.2610 0.0682 0.0681 0.0098

self1oth2 | -0.0312 -0.0301 0.0010 0.0009 0.7616

​

Gonzalez & Griffin correlation decomposition

Gonzalez, R., & Griffin, D. (2000). On the statistics of interdependence: Treating dyadic data with respect https://drive.google.com/file/d/1z3I7NjKL7RKAhKNhsOfTAvajLkPuESsh/view?usp=share_link In W. Ickes & S. Duck (Eds.),

The Social Psychology of Personal Relationships (pp. 271-301).

27

Figure 9.1 Figure 9.1 All possible pairwise correlations between variables X, Y, and their corresponding "reverse codes".

X pairwise

intraclass

correlation

Y pairwise

intraclass

correlation

Overall within-partner

XY correlation

XY cross-intraclass correlation

Note that in this framework r_xy' = r_x'y and

r_xy = r_x'y'

Gonzalez & Griffin - distinguishable case

Gonzalez, R., & Griffin, D. (2000). On the statistics of interdependence: Treating dyadic data with respect https://drive.google.com/file/d/1z3I7NjKL7RKAhKNhsOfTAvajLkPuESsh/view?usp=share_link In W. Ickes & S. Duck (Eds.),

The Social Psychology of Personal Relationships (pp. 271-301).

28

Figure 9.2 All possible pairwise correlations between X, Y, and their corresponding "reverse codes" in the distinguishable case. Variable C has been partialled out from all correlations.

[partial correlations; C is 1 for ‘self’ and 2 for ‘other’]

X pairwise

intraclass

correlation

Y pairwise

intraclass

correlation

Overall within-partner

XY correlation

XY cross-intraclass correlation

‘Others’= Neighbors’ Average

%Minority_s

Spatial Lag

‘Others’= Neighbors’ Average

LifeExp._s

Spatial Lag

‘SELF’

%Minority_s

‘‘SELF’

LifeExp._s

+.21^.036

@ +.21^.036

+.62^.001

+.68^.001

Gonzalez & Griffin semipartial correlations

-.26^.010

Y pairwise

intraclass

correlation

X pairwise

intraclass

correlation

Overall province-neighbors

XY correlation

Overall province-neighbors

XY correlation

XY cross-intraclass correlation

p values are irrelevant: the N = 49 is the population of provinces, there are no other contiguous US States out there!

29

r_xy.c

r_x'y’.c

r_xx’.c

r_xy'.c

r_yy’.c

r_xy'.c = r_x’y.c by data design: confirmed

%Minority._s

Income._s

LifeExp._s

%Minority._s

Income._s

LifeExp._s

Spatial perspectives in family health research

Kenny chapter

Moran’s Is &

‘Spatial’ correlations

Model implied Correlation from Nonrecursive with IVs

+.71

+.29

-.11

Correlations between all pairs of residuals were NS, so were all set @0

‘Self-’Other’ intraclass correlations r_xx’ &

Partial pairwise correlations r_xx’.c

​

30

+.79*

-.06^NS

+.12^NS

+.54*

+.43*

+.38*

+.70*

+.70*

+.54*

%Minority._s

Income._s

LifeExp._s

%Minority._s

Income._s

LifeExp._s

Spatial perspectives in family health research

Kenny chapter

‘Spatial’ correlations

Model implied Correlation from Nonrecursive with IVs

+.69

+.22

-.14

Standardized semi-partial correlations:

from 3 bivariate pairwise models to separate out within-partner from cross-intraclass

correlations in the distinguishable ‘dyadic’ case

= Overall within-partner XY correlation r_xy.c = r_x'y'.c

31

+.72^.001

-.21^.036

+.12^.231

‘Others’= Neighbors’ Average

%Minority_s

Spatial Lag

‘SELF’

%Minority_s

‘‘SELF’

LifeExp._s

Actor Partner model (G&G)

32

“An actor effect, which represents the extent to which a dyad member's (the "actor") standing on variable X determines that actor's standing on variable Y, and a partner effect, which represents the extent to which the partner's standing on X determines the actor's standing on Y.” “Following the structural model illustrated in Figure 9.5 leads to the interpretation of the (semi-partial) pairwise rxy as the "actor correlation“ and the (semi-partial) pairwise rxy' as the "partner correlation".

Gonzalez, R., & Griffin, D. (2000). On the statistics of interdependence: Treating dyadic data with respect https://drive.google.com/file/d/1z3I7NjKL7RKAhKNhsOfTAvajLkPuESsh/view?usp=share_link In W. Ickes & S. Duck (Eds.),

The Social Psychology of Personal Relationships (pp. 271-301).

-.08^.506

-.21^.091

“If the individuals in the block share no common causes of A or Y , as in the DAG in Figure 4, then Ci suffices to block the backdoor paths from Ai to Yi

and from Aj to Yi and, therefore, exchangeability for the effect of A on Yi holds conditional on Ci.

That is, Yi(ai, aj ) ∐ A |Ci for all i.” [1]:565 [subscripts upgraded for clarity]

Interference and causal issues

1. Ogburn, E. L., & VanderWeele, T. J. (2014). Causal Diagrams for Interference. Statistical Science, 29(4), 559-578.

FIG. 4.

“The principles of covariate control in the presence of interference are straightforward: like in the case of no interference, they follow from the fact that all backdoor paths from treatment to outcome must be blocked by a measured set of covariates.

However, without taking the time to draw the operative causal DAG with interference it is easy to make mistakes, like controlling only for individual-level covariates when block-level covariates are necessary to identify the causal effect of interest.” [1]:565

“If Cj is a direct cause of Yi for i = j , as in Figure 5(a), then exchangeability for the effect of A on Yi necessitates block- and not just individual-level covariates.

Even if each individual’s treatment is randomized conditional on his own covariates (this corresponds to the absence of arrows Cj to Ai for i = j on the DAG), there is still a backdoor path from Aj to Yi via Cj and, somewhat counterintuitively, it is necessary to control for Cj in addition to Ci in any model for the effect of A on Yi.

On the other hand, if Cj directly affects Ai but not Yi for i = j , as in Figure 5(b), then for exchangeability for the effect of A on Yi it suffices to condition only on Ci .” [1]:565

[subscripts upgraded for clarity]

Interference and causal issues

1. Ogburn, E. L., & VanderWeele, T. J. (2014). Causal Diagrams for Interference. Statistical Science, 29(4), 559-578.

FIG. 5.a.

FIG. 5.b.

Direct interference

“if individual i receives treatment and individual j does

not, individual j may be nevertheless be exposed to the

treatment of individual i”

​

Interference by contagion

Via the first individual’s outcome - It does not represent a direct causal pathway from the exposed individual to another individual’s outcome, but rather a pathway mediated by the outcome of the exposed individual.

​

Allocational interference

Treatment in this setting allocates individuals to groups; through interactions within a group individuals’ characteristics may affect one another.

“An example that often arises in the social science literature is the allocation of children to schools or of children to classrooms within schools”

[1]:565

Spatial interference: interference by contagion?

1. Ogburn, E. L., & VanderWeele, T. J. (2014). Causal Diagrams for Interference. Statistical Science, 29(4), 559-578.

X_neighbors(i)

X_i

Y_neighbors(i)

Y_i

?

effect

Gonzalez & Griffin extensions possible

Gonzalez, R., & Griffin, D. (2000). On the statistics of interdependence: Treating dyadic data with respect https://drive.google.com/file/d/1z3I7NjKL7RKAhKNhsOfTAvajLkPuESsh/view?usp=share_link In W. Ickes & S. Duck (Eds.),

The Social Psychology of Personal Relationships (pp. 271-301).

36

Figure 9.3 A latent variable model separating individual-level (unique) and dyad-level (shared) effects

Maybe? Gonzalez Griffin approach expanded to 1-to-many

+++

Turn the data from n=49 to self+others -> Moran’s I becomes a Pearson correlation in the N = 218 data. N = 267 has G&G similar uses.

Merge 2!

Merge 1- Moran I equivalent

C1: The spatial data structure lends itself to dyadic thinking and modeling.

​

C2: Spatial non-independence can be modeled with the Gonzalez & Griffin ‘trick’ to handle dyadically linked data.

​

C3: Dyadic analogies can be more than mere coincidences: more modeling options are available.

​

Conclusions

38

Extras

%Minority._s

Income._s

LifeExp._s

%Minority._s

Income._s

LifeExp._s

Spatial perspectives in family health research

Kenny chapter

Sample naïve = a-spatial correlations

+.69^<.001

+.22^.123

-.14^.333

Correlations between all pairs of residuals were NS, so were all set @0

Standardized semi-partial correlations:

from 3 bivariate pairwise models to separate out within-partner from cross-intraclass

correlations in the distinguishable ‘dyadic’ case

40

-.??^.???

-.??^.???

-.??^.???

* The partial correlation between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 if the other x's did not vary (bidirectional) – adding the lag of the outcome makes it kind of directional.

* The semipartial correlation, also called part correlation, between y and x1 is an attempt to estimate the correlation that would be observed between y and x1 after the effects of all other x's are removed from x1 but not from y (directional)

41

%Minority_s

Income._s

LifeExp._s

+.55*_LEx<-I

+.33*_I<-%M

Spatial PARTIAL correlations - Directional

(2^nd predictor is the ‘DV spatial lag’)

sLAG Inc_s

*

sLAG Inc_s

*

sLAG Inc_s

*

%Minority_s

Income._s

LifeExp._s

+.75

+.45

-.42

Sample naïve = a-spatial PARTIAL correlations

NON-Directional!

+.07_LEx<-%M

+.58*_I<-LEx

+.35*_%M<-I

+.09_%M<-LEx

Maximum within-class (‘auto’) correlation, and minimum (-1)

“in Example A there is no variation between the scores in any of the rows[…[]; rather all the variation is between the row averages [sums] (i.e., between the classes), namely 77.9, 78.1, ... , 78.6. “

Haggard, E. A. (1958). Intraclass correlation and the analysis of variance

https://drive.google.com/file/d/15sqL7oOhYtLar-iUScwg6r7PG0sB2l8_/view?usp=share_link

“In Example B all the variation is within the rows or classes, with the result that there is no variation between classes (i.e., each class average [sum] equals 78.3).’