BackgroundDyadic data collection has become common in many areas of psychological and broader social science research. Due to the interactive nature of dyads, researchers are unable to assume that observations are independent, a key assumption of most classical statistical methodologies. Structural equation modeling allows researchers to properly model dependency in dyadic analyses. The Actor-Partner Interdependence Model (APIM) is specifically oriented towards modeling dependency in dyadic data. This KUant Guide provides instruction on how to model APIM's with both distinguishable (e.g. Mother-Child) and indistinguishable (e.g. Friend A-Friend B) dyads as well as LISREL and MPLUS code for how to do so. For thorough discussion of the Actor-Partner Interdependence Model, see Cook & Kenny (International Journal of Behavioral Development, 2005, Vol 29 (2), 101-109).

All analyses require that the dyad be the unit of analysis in the dataset (i.e., data for both members of the dyad appear on the same line in the dataset).

The APIMThe simplest generic actor partner interdependence model can be seen below. Person A’s constructs (1 & 3) appear in the top row and Person B’s equivalent constructs (2 & 4) appear in the bottom row. Within-person regression lines are called ‘actor effects’ and are represented by a or a’. Between-person regression lines are called ‘partner effects’ and are represented by p or p’. Slight variations must be made to the model depending on whether dyads are distinguishable or not.

Guide No.KUANT 027.1

www.crmda.ku.edu

KUant GuidesDyadic Analyses in SEM:Modeling Latent Actor-Partner Interdependence Models using LISREL and Mplus

Stump, K.N., Howard, W.J., & Gallagher, M.W. (2012)

APIM with Distinguishable DyadsDyads are distinguishable if each member of the dyad has a specific role in the relationship (e.g., parent-child; husband-wife). Two adjustments must be made to the generic APIM when dyads are distinguishable. All LISREL and MPLUS syntax are available in Appendix A.

1) Account for dependency in the dyad by estimating correlated residual variances. Indicators 1, 2, 3, 7, 8, and 9 for Person A are the same as indicators 4, 5, 6, 10, 11, and 12, respectively, for Person B.

LISREL syntax:

FR TE(4,1) TE(5,2) TE(6,3) TE(10,7) TE(11,8) TE(12,9)

MPlus syntax:

IND1 WITH IND4; !Free the residual variance among indicator 4 and indicator 1IND2 WITH IND5; !Free the residual variance among indicator 5 and indicator 2IND3 WITH IND6; !Free the residual variance among indicator 6 and indicator 3IND7 WITH IND10;!Free the residual variance among indicator 10, indicator 7IND8 WITH IND11;!Free the residual variance among indicator 11, indicator 8IND9 WITH IND12;!Free the residual variance among indicator 12, indicator 9

2) To ensure measurement invariance among members of the dyad, equate respective factor loadings among constructs.

LISREL syntax:

EQ LY(1,1) LY(4,2) EQ LY(2,1) LY(5,2) EQ LY(3,1) LY(6,2) EQ LY(7,3) LY(10,4) EQ LY(8,3) LY(11,4) EQ LY(9,3) LY(12,4)

MPlus syntax:

IV_1 BY IND1* (L1)IND2 (L2)IND3 (L3);

IV_2 BY IND4* (L1)!EQUATE ALL LOADINGS WITH (L1), i.e., LY(4,2) and LY(1,1) IND5 (L2)!EQUATEALL LOADINGS WITH (L2), i.e., LY(5,2) and LY(2,1) IND6 (L3);!EQUATE ALL LOADINGS WITH (L3), i.e., LY(6,2) and LY(3,1)

DV_1 BY IND7* (L4) IND8 (L5)IND9 (L6);

DV_2 BY IND10* (L4)!EQUATE ALL LOADINGS WITH (L4), i.e., LY(10,4) and LY(7,3)IND11 (L5)!EQUATE ALL LOADINGS WITH (L5), i.e., LY(11,4) and LY(8,3)IND12 (L6);!EQUATE ALL LOADINGS WITH (L6),i.e., LY(12,4) and LY(9,3)

APIM with Indistinguishable Dyads:Dyads are indistinguishable if members’ roles are interchangeable (i.e., if a person’s placement as Person A or Person B is arbitrary). For extensive discussion regarding APIM in SEM with interchangeable dyads, see Olsen and Kenny (2006). Adjustments must be made to the generic APIM when dyads are indistinguishable. All LISREL and MPLUS syntax are available in Appendix B.

To account for the arbitrary placement of a dyadmember as Person A or Person B,

1) Equate corresponding factor loadings.

LISREL syntax:

EQ LY(1,1) LY(4,2)EQ LY(2,1) LY(5,2)EQ LY(3,1) LY(6,2)EQ LY(7,3) LY(10,4)EQ LY(8,3) LY(11,4)EQ LY(9,3) LY(12,4)

MPlus syntax:

IV_1 BY IND1* (L1)IND2 (L2)IND3 (L3);

IV_2 BY IND4* (L1)!EQUATE ALL LOADINGS WITH (L1), i.e., LY(4,2) and LY(1,1)IND5 (L2)!EQUATE ALL LOADINGS WITH (L2), i.e., LY(5,2) and LY(2,1)IND6 (L3);!EQUATE ALL LOADINGS WITH (L3), i.e., LY(6,2) and LY(3,1)

DV_1 BY IND7* (L4) IND8 (L5)IND9 (L6);

DV_2 BY IND10* (L4)!EQUATE ALL LOADINGS WITH (L4), i.e., LY(10,4) and LY(7,3)IND11 (L5)!EQUATE ALL LOADINGS WITH (L5), i.e., LY(11,4) and LY(8,3)IND12 (L6);!EQUATE ALL LOADINGS WITH (L6), i.e., LY(12,4) and LY(9,3)

2) Equate corresponding residual variances.

LISREL syntax:

EQ TE(1,1) TE(4,4)EQ TE(2,2) TE(5,5)EQ TE(3,3) TE(6,6)EQ TE(7,7) TE(10,10)EQ TE(8,8) TE(11,11)EQ TE(9,9) TE(12,12)

MPlus syntax:

IND1 IND4 (T1)!EQUATE ALL RESIDUAL VARIANCES WITH (T1), i.e., TE(1,1) and TE(4,4)IND2 IND5 (T2)!EQUATE ALL RESIDUAL VARIANCES WITH (T2), i.e., TE(2,2) and TE(5,5)IND3 IND6 (T3)!EQUATE ALL RESIDUAL VARIANCES WITH (T3), i.e., TE(3,3) and TE(6,6)

IND7 IND10 (T4)!EQUATE ALL RESIDUAL VARIANCES WITH (T4), i.e., TE(7,7) and TE(10,10)IND8 IND11 (T5)!EQUATE ALL RESIDUAL VARIANCES WITH (T5), i.e., TE(8,8) and TE(11,11)IND9 IND12 (T6)!EQUATE ALL RESIDUAL VARIANCES WITH (T6), i.e., TE(9,9) and TE(12,12)

3) Equate within-person (actor) regression paths.

LISREL syntax:

EQ BE(3,1) BE(4,2) !ACTOR EFFECTS

MPlus syntax:

DV_1 ON IV_1 (B1);!ACTOR EFFECTSDV_2 ON IV_2 (B1);

4) Equate between-person (partner) regression paths.

LISREL syntax:

EQ BE(3,2) BE(4,1) !PARTNER EFFECTS

MPlus syntax:

DV_2 ON IV_1 (BE2);!PARTNER EFFECTSDV_1 ON IV_2 (BE2);

Appendix A: Code for APIM with Distinguishable DyadsLISREL Syntax

DA NG=1 NI=12 MA=CM ME=ML NO=85LAIND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12RA FI=DYADS.PSFMO NY=12 NE=4 LY=FU,FI PS=SY,FI TE=SY,FI BE=FU,FIFI PS(1,1) PS(2,2) PS(3,3) PS(4,4) !SET ALL LATENT VARIANCES TO 1.0VA 1.0 PS(1,1) PS(2,2) PS(3,3) PS(4,4)

FR PS(2,1) PS(4,3) !FREE COVARIANCES (ICC’S)

FR LY(1,1) LY(2,1) LY(3,1) EQ LY(1,1) LY(4,2) !EQUATE CORRESPONDING FACTOR LOADINGS FOR IVEQ LY(2,1) LY(5,2)EQ LY(3,1) LY(6,2)

FR LY(7,3) LY(8,3) LY(9,3) EQ LY(7,3) LY(10,4) !EQUATE CORRESPONDING FACTOR LOADINGS FOR DVEQ LY(8,3) LY(11,4) EQ LY(9,3) LY(12,4)

FR TE(1,1) TE(2,2) TE(3,3) TE(4,4) TE(5,5) TE(6,6) TE(7,7) TE(8,8) TE(9,9) TE(10,10) TE(11,11) TE(12,12)

FR TE(4,1) TE(5,2) TE(6,3) TE(10,7) TE(11,8) TE(12,9) !CORRELATE RESIDUAL VARIANCES

FR BE(3,1) BE(4,2) !ACTOR EFFECTSFR BE(3,2) BE(4,1) !PARTNER EFFECTS

ST .3 ALL

LEIV_1 IV_2 DV_1 DV_2

PDOU AD=OFF RS SC SO ND=3

MPlus SyntaxTITLE: Dyadic Data - Distinguishable example

DATA: FILE IS dyads.dat;

VARIABLE: NAMES ARE IND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12;

USEVARIABLES ARE IND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12;

MODEL:!INDEPENDENT VARIABLE FOR MEMBER 1 IV_1 BY IND1* (L1) !NOTE: the asterix (*) freely estimates the first LY loadingIND2 (L2)IND3 (L3); IV_1@1.0; !SET LATENT VARIANCE OF IV_1 TO 1.0

!INDEPENDENT VARIABLE FOR MEMBER 2 IV_2 BY IND4* (L1)!EQUATE INDICATOR 4 TO INDICATOR 1IND5 (L2) !EQUATE INDICATOR 5 TO INDICATOR 2IND6 (L3); !EQUATE INDICATOR 6 TO INDICATOR 3 IV_2@1.0; !SET LATENT VARIANCE OF IV_2 TO 1.0

!DEPENDENT VARIABLE FOR FRIEND MEMBER 1 DV_1 BY IND7* (L4) IND8 (L5)IND9 (L6); DV_1@1.0; !SET LATENT VARIANCE OF DV_1 TO 1.0

!DEPENDENT VARIABLE FOR FRIEND MEMBER 2 DV_2 BY IND10* (L4)!EQUATE INDICATOR 10 TO INDICATOR 7IND11 (L5) !EQUATE INDICATOR 11 TO INDICATOR 8IND12 (L6); !EQUATE INDICATOR 12 TO INDICATOR 9 DV_2@1.0; !SET LATENT VARIANCE OF DV_2 TO 1.0

!correlated residual terms IND1 WITH IND4; IND2 WITH IND5; IND3 WITH IND6; IND7 WITH IND10; IND8 WITH IND11; IND9 WITH IND12;

!free residual covariances IV_1 WITH IV_2; DV_1 WITH DV_2;

!BETAS (i.e., REGRESSION PATHS) DV_1 ON IV_1; !ACTOR EFFECT DV_2 ON IV_1; !PARTNER EFFECT DV_1 ON IV_2; !PARTNER EFFECT DV_2 ON IV_2; !ACTOR EFFECT

OUTPUT: TECH1 STANDARDIZED;

Appendix B: Code for APIM with Indistinguishable DyadsLISREL Syntax

DA NG=1 NI=12 MA=CM ME=ML NO=85LA0IND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12

RA FI=DYADS.PSF

MO NY=12 NE=4 LY=FU,FI PS=SY,FI TE=SY,FI BE=FU,FI

FI PS(1,1) PS(2,2) PS(3,3) PS(4,4) !SET ALL LATENT VARIANCES EQUAL TO 1.0VA 1.0 PS(1,1) PS(2,2) PS(3,3) PS(4,4)

FR PS(2,1) PS(4,3) !FREE COVARIANCES (ICC’S)

FR LY(1,1) LY(2,1) LY(3,1) EQ LY(1,1) LY(4,2) !EQUATE CORRESPONDING FACTOR LOADINGS FOR IVEQ LY(2,1) LY(5,2)EQ LY(3,1) LY(6,2)

FR LY(7,3) LY(8,3) LY(9,3) EQ LY(7,3) LY(10,4) !EQUATE CORRESPONDING FACTOR LOADINGS FOR DVEQ LY(8,3) LY(11,4) EQ LY(9,3) LY(12,4)

FR TE(1,1) TE(2,2) TE(3,3)EQ TE(1,1) TE(4,4) !EQUATE CORRESPONDING RESIDUAL VARIANCES FOR IVEQ TE(2,2) TE(5,5) EQ TE(3,3) TE(6,6)

FR TE(7,7) TE(8,8) TE(9,9)EQ TE(7,7) TE(10,10) !EQUATE CORRESPONDING RESIDUAL VARIANCES FOR DVEQ TE(8,8) TE(11,11)EQ TE(9,9) TE(12,12)

FR BE(3,1) EQ BE(3,1) BE(4,2) !EQUATE ACTOR EFFECTSFR BE(3,2) EQ BE(3,2) BE(4,1) !EQUATE PARTNER EFFECTS

ST .3 ALL

LEIV_1 IV_2 DV_1 DV_2

PDOU AD=OFF RS SC SO ND=3

MPlus Syntax

TITLE: Dyadic Data - Indistinguishable example

DATA: FILE IS dyads.dat;

VARIABLE: NAMES ARE IND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12; USEVARIABLES ARE IND1 IND2 IND3 IND4 IND5 IND6 IND7 IND8 IND9 IND10 IND11 IND12;

MODEL: !INDEPENDENT VARIABLE FOR MEMBER 1 IV_1 BY IND1* (L1) !NOTE: the asterix (*) freely estimates the first LYIND2 (L2) ! loadingbecause it is fixed to 1.0 by defaultIND3 (L3); IV_1@1.0; !SET LATENT VARIANCE OF IV_1 TO 1.0

!INDEPENDENT VARIABLE FOR MEMBER 2 IV_2 BY IND4* (L1) !EQUATE INDICATOR 4 TO INDICATOR 1IND5 (L2) !EQUATE INDICATOR 5 TO INDICATOR 2IND6 (L3); !EQUATE INDICATOR 6 TO INDICATOR 3IV_2@1.0; !SET LATENT VARIANCE OF IV_2 TO 1.0

!DEPENDENT VARIABLE FOR FRIEND MEMBER 1 DV_1 BY IND7* (L4) IND8 (L5)IND9 (L6); DV_1@1.0; !SET LATENT VARIANCE OF DV_1 TO 1.0

!DEPENDENT VARIABLE FOR FRIEND MEMBER 2 DV_2 BY IND10* (L4) !EQUATE INDICATOR 10 TO INDICATOR 7IND11 (L5) !EQUATE INDICATOR 11 TO INDICATOR 8IND12 (L6); !EQUATE INDICATOR 12 TO INDICATOR 9 DV_2@1.0; !SET LATENT VARIANCE OF DV_2 TO 1.0

IND1 WITH IND4; !ALLOW CORRELATED RESIDUAL TERMS IND2 WITH IND5; IND3 WITH IND6; IND7 WITH IND10; IND8 WITH IND11; IND9 WITH IND12;

IV_1 WITH IV_2;!FREE LATENT VARIANCES DV_1 WITH DV_2;

!BETAS (i.e., REGRESSION PATHS) DV_1 ON IV_1 (BE1); !ACTOR EFFECT DV_2 ON IV_1 (BE1); !PARTNER EFFECT DV_1 ON IV_2 (BE2); !PARTNER EFFECT DV_2 ON IV_2 (BE2); !ACTOR EFFECT

!EQUATE CORRESPONDING RESIDUAL VARIANCES IND1 IND4 (T1) IND2 IND5 (T2) IND3 IND6 (T3) IND7 IND10 (T4) IND8 IND11 (T5) IND9 IND12 (T6)

OUTPUT: TECH1 STANDARDIZED;