Multi-sample structural equation models with mean structures, with special emphasis on assessing measurement invariance in cross-national research

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1 1 Multi-sample structural equation models with mean structures, with special emphasis on assessin measurement invariance in cross-national research Measurement invariance measurement invariance: whether or t, under different conditions of observin and studyin phemena, measurement operations yield measures of the same attribute (Horn and McArdle 1992); frequent lack of concern for, or inappropriate examination of, measurement invariance (esp. in cross-national research); this is problematic because in the absence of measurement invariance across-roup (e.., crossnational) comparisons may be meaninless;

2 2 Measurement model Let x be a p x 1 vector of observed variables in roup, an m x 1 vector of latent variables, a p x 1 vector of unique factors (errors of measurement), a p x 1 vector of item intercepts, and a p x m matrix of factor loadins. Then x = + + The means part of the model is iven by = + and the covariance part is iven by = ' + Model identification covariance part: the latent constructs have to be assined a scale in which they are measured; this is done by choosin a marker item and settin its loadin to one; means part: two possibilities the intercepts of the marker items are fixed to zero (which equates the means of the latent constructs to the means of their marker variables, m = ); the vector of latent means is set to zero in the reference roup and one intercept per factor is constrained to be equal across roups; althouh these constraints help with identifyin the model, in eneral they are t sufficient for meaninful across-roup comparisons;

3 3 Types of invariance confiural invariance: the pattern of salient and nsalient loadins is the same across different roups; metric invariance: the scale metrics are the same across roups; 1 = 2 =... = G scalar invariance: in addition to the scale metrics, the item intercepts are the same across roups; 1 = 2 =... = G Types of invariance (cont d) factor (co)variance invariance: factor covariances and factor variances are the same across roups; 1 = 2 =... = G error variance invariance: error variances are the same across roups; 1 = 2 =... = G

4 4 Linkin the types of invariance required to the research objective Confiural invariance Metric invariance Scalar invariance Explorin the basic structure of the construct across roups Examinin structural relationships with other constructs across roups Conductin acrossroup comparisons of means Partial measurement invariance for identification purposes, one item per factor has to have invariant loadins and intercepts (marker item); the marker item has to be chosen carefully; at least one other invariance constraint on the loadins/ intercepts is necessary to ascertain whether the marker item satisfies metric/scalar invariance; Cheun and Rensvold (1999) proposed the factor-ratio test in which all possible pairs of items are tested for metric/scalar invariance and sets of invariant items are identified; an alternative is to start with the fully invariant model of a iven type and relax invariance constraints based on sinificant modification indices, chanes in alternative fit indices, and expected parameter chanes;

5 5 Assessin metric invariance across different roups choose a marker item for each factor (e.., based on reliability or other considerations; if it later turns out that the marker item does t satisfy metric or scalar invariance, a different marker item may have to be chosen) compare the confiural with the full metric invariance model; use Bonferroni-adjusted modification indices, chanes in alternative fit indices, and expected parameter chanes to free loadins that are t invariant; the final model should fit as well as the confiural model; cross-validate the model, if possible; Assessin scalar invariance across different roups compare the final (partial) metric invariance model with the full (or initial) scalar invariance model; use Bonferroni-adjusted modification indices, chanes in fit indices, and expected parameter chanes to free item intercepts that are t invariant; the final model should fit as well as the final (partial) metric invariance model; cross-validate the model, if possible;

6 6 Fiure 1: Proposed Procedure for Assessin Measurement Invariance in Cross-National Consumer Research pool covariance matrices and mean vectors equality of and? equality of? equality of? the factorial structure of the construct is similar across countries full confiural invariance? continue testin for confiurally invariant factors partial confiuralinvariance? the factorial structure of the construct differs across countries full metric invariance? free loadin w/ the larest modification index partial metric invariance? the latent means can be meaninfully compared across countries full scalar invariance? free intercept w/ the larest modification index partial scalar invariance? (some) factor covariances are identical across countries full factor covariance invariance? free covariance w/ the larest modification index partial factor covariance invariance? (some) factor correlations are identical across countries full factor variance invariance? free variance w/ the larest modification index partial factor variance invariance? (invariant) items are equally reliable across countries full error variance invariance? free error w/ the larest modification index partial error variance invariance? Note: If the researcher is t interested in comparin means across countries, tests of scalar invariance can be omitted and the analysis proceeds from assessin metric invariance to investiatin factor covariance invariance.

7 7 Illustration: 393 Austrian and 1181 U.S. respondents completed the Satisfaction with Life Scale (SWLS; Diener et al. 1985), which is a well-kwn instrument used to assess the conitive component of subjective well-bein. The scale consists of the followin five items: (1) In most ways my life is close to my ideal. (2) The conditions of my life are excellent. (3) I am satisfied with my life. (4) So far I have otten the important thins I want in life. (5) If I could live my life over, I would chane almost thin. Respondents indicated their areement or disareement with these statements usin the followin five-point scale: 1 = stronly disaree, 2 = disaree, 3 = neither aree r disaree, 4 = aree, and 5 = stronly aree. Perform an analysis of measurement invariance on the SWLS and test whether Austrian or American respondents are more satisfied with their lives (if possible). Multi-sample structural equation models A one-factor model for life satisfaction j 11 LS l 2 l 3 l 4 l ls1 ls2 ls3 ls4 ls

8 8 MODEL COMPARISONS FOR LIFE SATISFACTION (AUSTRIA VS. U.S.) 2 value df RMSEA CAIC CFI TLI Confiural invariance Full metric invariance Final partial metric invariance Initial (and final) partial scalar invariance Mean invariance Factor loadins Item intercepts AUT US AUT US ls ls ls ls ls Latent means AUT: 3.91 US: 3.26

9 9 DATA usa; INFILE 'd:\m554\multrou\ls-usa.dat'; INPUT ls1 ls2 ls3 ls4 ls5; DATA aut; INFILE 'd:\m554\multrou\ls-aut.dat'; INPUT ls1 ls2 ls3 ls4 ls5; title 'initial and final partial scalar invariance (ls5, ls4)'; run; PROC CALIS modification; roup 1 / data=usa label="usa"; roup 2 / data=aut label="aut"; model 1 / roup=1; PATH ls1 <--- LS = L11, ls2 <--- LS = L21, ls3 <--- LS = 1.0, ls4 <--- LS = L411, ls5 <--- LS = L511; MEAN ls1 = t1, ls2 = t2, ls3 = 0., ls4 = t41, ls5 = t51, LS = k1; PVAR LS = ph111, ls1 = th111, ls2 = th221, ls3 = th331, ls4 = th441, ls5 = th551;

10 10 model 2 / roup=2; PATH ls1 <--- LS = L11, ls2 <--- LS = L21, ls3 <--- LS = 1.0, ls4 <--- LS = L412, ls5 <--- LS = L512; MEAN ls1 = t1, ls2 = t2, ls3 = 0., ls4 = t42, ls5 = t52, LS = k2; PVAR LS = ph112, ls1 = th112, ls2 = th222, ls3 = th332, ls4 = th442, ls5 = th552; run;

11 11 --Data Records Group Label Data Set N Read N Used N Obs Model Type 1 USA WORK.USA Model 1 PATH 2 AUT WORK.AUT Model 2 PATH Group 1. USA Simple Statistics Variable Mean Std Dev ls ls ls ls ls Group 2. AUT Simple Statistics Variable Mean Std Dev ls ls ls ls ls

12 12 Fit Summary Modelin Info Number of Observations 1574 Number of Variables 5 Number of Moments 40 Number of Parameters 26 Number of Active Constraints 0 Baseline Model Function Value Baseline Model Chi-Square Baseline Model Chi-Square DF 20 Pr > Baseline Model Chi-Square <.0001 Absolute Index Fit Function Chi-Square Chi-Square DF 14 Pr > Chi-Square <.0001 Z-Test of Wilson & Hilferty Hoelter Critical N 605 Root Mean Square Residual (RMR) Standardized RMR (SRMR) Goodness of Fit Index (GFI) Parsimony Index Adjusted GFI (AGFI) Parsimonious GFI RMSEA Estimate RMSEA Lower 90% Confidence Limit RMSEA Upper 90% Confidence Limit Probability of Close Fit Akaike Information Criterion Bozdoan CAIC Schwarz Baian Criterion McDonald Centrality Incremental Index Bentler Comparative Fit Index Bentler-Bonett NFI Bentler-Bonett Non-rmed Index Bollen Normed Index Rho Bollen Non-rmed Index Delta James et al. Parsimonious NFI

13 13 Fit Comparison Amon Groups Overall USA Modelin Info Number of Observations Number of Variables 5 5 Number of Moments Number of Parameters Number of Active Constraints 0 0 Baseline Model Function Value Baseline Model Chi-Square Baseline Model Chi-Square DF Fit Index Fit Function Percent Contribution to Chi-Square Root Mean Square Residual (RMR) Standardized RMR (SRMR) Goodness of Fit Index (GFI) Bentler-Bonett NFI Fit Comparison Amon Groups AUT Modelin Info Number of Observations 393 Number of Variables 5 Number of Moments 20 Number of Parameters 15 Number of Active Constraints 0 Baseline Model Function Value Baseline Model Chi-Square Baseline Model Chi-Square DF 10 Fit Index Fit Function Percent Contribution to Chi-Square 30 Root Mean Square Residual (RMR) Standardized RMR (SRMR) Goodness of Fit Index (GFI) Bentler-Bonett NFI

14 14 Model 1. PATH List Standard Path Parameter Estimate Error t Value ls1 <=== LS L ls2 <=== LS L ls3 <=== LS ls4 <=== LS L ls5 <=== LS L Model 1. Variance Parameters Variance Standard Type Variable Parameter Estimate Error t Value Exoeus LS ph Error ls1 th ls2 th ls3 th ls4 th ls5 th Model 1. Means and Intercepts Standard Type Variable Parameter Estimate Error t Value Intercept ls1 t ls2 t ls3 0 ls4 t ls5 t Mean LS k Model 1. Squared Multiple Correlations Error Total Variable Variance Variance R-Square ls ls ls ls ls

15 15 Model 2. PATH List Standard Path Parameter Estimate Error t Value ls1 <=== LS L ls2 <=== LS L ls3 <=== LS ls4 <=== LS L ls5 <=== LS L Model 2. Variance Parameters Variance Standard Type Variable Parameter Estimate Error t Value Exoeus LS ph Error ls1 th ls2 th ls3 th ls4 th ls5 th Model 2. Means and Intercepts Standard Type Variable Parameter Estimate Error t Value Intercept ls1 t ls2 t ls3 0 ls4 t ls5 t Mean LS k Model 2. Squared Multiple Correlations Error Total Variable Variance Variance R-Square ls ls ls ls ls

16 16 Model 1. Rank Order of the 10 Larest LM Stat for Path Relations Parm To From LM Stat Pr > ChiSq Chane ls4 ls < ls5 ls ls4 ls ls5 ls ls2 ls ls4 ls ls2 ls LS ls ls5 ls ls3 ls Model 2. Rank Order of the 10 Larest LM Stat for Path Relations Parm To From LM Stat Pr > ChiSq Chane ls5 ls ls4 ls ls4 ls ls5 ls LS ls ls5 ls LS ls ls3 ls LS ls ls5 ls Larane Multiplier Statistics for Releasin Equality Constraints -----Released Parameter Chanes----- Oriinal Released Parm Model Type Var1 Var2 LM Stat Pr > ChiSq Parm Parm L11 1 DV_IV ls1 LS DV_IV ls1 LS L21 1 DV_IV ls2 LS DV_IV ls2 LS t1 1 INTERCEPT ls INTERCEPT ls t2 1 INTERCEPT ls INTERCEPT ls

17 17 What to do when the number of items and/or factors varies across roups? Unequal number of observed variables introduce imainary observed variables; specify their intercepts and loadins as zero and error variances as one; adjust derees of freedom; Unequal number of latent variables introduce imainary observed and latent variables; specify latent variables to have zero means, unit variance, and relationships with other constructs; add imainary observed variables and adjust derees of freedom; y 1 = *(x 1 ) + 1 Var(x 1 ) Var( 1 ) b r y 2 = *(x 2 + ) + 2 Var(x 2 ) Var( ) Var( 2 ) b r

18 18 The full multi-sample LISREL model with mean structures y x y x y x with E Baozzi, Baumartner, and Pieters (1998) 0.31 (7.4) positive anticipated emotions.14 (3.6).16 (4.4) dietin volitions dietin behaviors.20 (7.4).36 (6.8) d -.07 (-.6).54 (4.9) b.61 (7.7) a,b.29 (3.1) c,d oal achievement positive oal-outcome emotions neative anticipated emotions.07 (1.9).16 (4.0) exercisin volitions.24 (8.7) exercisin behaviors.08 (.9).56 (3.7) a -.18 (-2.3) a,c -.46 (-8.7) b,d neative oal-outcome emotions 0.11 (3.4).33 (5.3) b a men wantin to lose weiht b women wantin to lose weiht c men wantin to maintain their weiht d women wantin to maintain their weiht 2 (110)= RMSEA=.030 CFI=.94 TLI=.92

19 19 How to test mediated paths across roups (moderated mediation) Example: Test whether the effect of positive and neative anticipated emotions on actual dietin is mediated by dietin volitions, separately for male and female respondents, and whether the manitude of the indirect effect differs by ender. DATA males; INFILE 'd:\m554\multrou\wloss-males.dat'; INPUT sexm posem neem bidiet biexer diet exer oalach; DATA females; INFILE 'd:\m554\multrou\wloss-females.dat'; INPUT sexm posem neem bidiet biexer diet exer oalach; title 'Moderated mediation'; run; PROC CALIS modification; roup 1 / data=males label="males"; roup 2 / data=females label="females"; model 1 / roup=1; PATH bidiet <--- posem neem = a111 a121, diet <--- bidiet = be211; PVAR posem = ph111, neem = ph221, bidiet = psi111, diet = psi221; PCOV posem neem = ph211; model 2 / roup=2; PATH bidiet <--- posem neem = a112 a122, diet <--- bidiet = be212; PVAR posem = ph112, neem = ph222, bidiet = psi112, diet = psi222; PCOV posem neem = ph212;

20 20 run; parameters ie11 ie21 ie12 ie22 diffie1 diffie2; ie11=a111*be211; ie21=a121*be211; ie12=a112*be212; ie22=a122*be212; diffie1=a111*be211-a112*be212; diffie2=a121*be211-a122*be212; Type Additional Parameters Parameter Estimate Standard Error t Value Dependent ie ie ie ie diffie diffie