Estimate the baseline model (best fitting model) for each group seperately to find the best fitting model for each group, the best fitting models might be different for each other. Tests are conducted for each group seperately. Even the baseline models are different for each groups, we can still continue the following procedures. The baseline model represents one that is most parsimonious, as well as statistically best fitting and substantively most meaningful.
Test the factorial structure simultaneously across groups, parameters are estimated for all groups at the same time, the fit of this simultaneously estiamted model can provide the baseline value against which all subsequently specified models are compared. This multigroup analysis yields only one set of fit statistics for overall model fit.
- Each group has its own data file.
- A model structure needs only to be drawn for the first group. By default, all other groups will have the same structure. ps, the structure comes from the basline model (best fitting model in the previous stage)
- The fit fo this simultaneously estiamted model provides the baseline value against which all subsequently specified models are compared. This multigroup analysis yields only one set of fit statistics or overall model fit. Given chi-square statistics are summative, the overall chi-square value for the multipgroup model should equal the sum of the chi-square values obtained when the baseline model is tested seperately for each group (without any cross-group constraints imposed). (df of the multiple group is also the sum of the df of all different groups--not very sure about this)
- Analyze--manage groups--new (give group name to each group, eg, sample1, sample2,sample3)--data file (identify data file for each group)--analyze (calculate estimate)----this step provide overall fit index for the one model (simultaneously estimated multiple groups)--this is M1
- all factor loadings, all factor variances, and all factor covariances constrained equal (in general, testing for the quality of error variances across grups is considered to be excessively stringent) --- open the hypothesized model Amos input file---right click the mouse (Object Properties)-- give the paremeter we want to constrain equal across groups label (each parameter to be held equal across groups is given a label; any parameter that is unlabeled will be freely estimated. how to name the lable is arbitrary, depending on my mood; one indicator of each factor initial was set at 1 to give metric to the factor, we let it remain as 1 and don't give it any label, given this parameter, ie., the path from the indicator and the factor is alredy constrained to equal 1, its value will be constant across groups) -- check "all groups" in the object properties box--this is M2
- If M1 (no equality constraints, more free parameters) has chi-square value=2243.21, df=495; M2 (has equality constraints, more parsimonious) has chi-square value= 2344.75, df= 545; the chi-square difference is 101.54 and df difference 50; -- statistically significant--M2 has a worse fit which is significant, ie, M2 is significantly worse than M1, so we choos M1-- some equality constraints do not hold across the group--so we need to further investigate which parameter is not equal across the groups
- If M2 is not significantly worse than M1, since M2 is more parsimonious, we choose M2--indicate that parameters are equal across groups. --- a significant chi-square difference indicates non-invariance, a significant chi-square difference indicates non-invariance (ie, non-equivalence)
- Example, we have sample A and sample B. we first specify a mdoel with an underlying three-factor structure (irrespective of factor loading pattern) across two samples. After simultaneously testing the model for the two groups, we get M1: chi-square (330)=1004.37--this model means testing the number of underlying factor (or underlying three-factor structure). Second, we put constraints on factor loadings across the two groups, we get M2: chi-square (347)=1025.81. Compare M1 with M2, we get chi-square difference (17,df)=21.44, which is not significant. M2 (more parsimonious) although has a worse fit, but is not significantly worse, thus we choose M2, indicating the equivalent of item loadings. The item loadings (found to be invariant) were cumulatively constrained equal across groups---maintaining equality constraints on the factor loadings parameters across groups, we specify M3 (constraining all factor covariances to be group-invariant+factor loading invariance). Comparing M2 with M3 yields a chi-squrare difference (6)= 3, insignificant--indicate support for M3--- from Byrne (1993) the Maslach burnout inventory: testing for factorial validity and invariance across elementary, intermediate and secondary teachers
- Example, M1 is the multigroup model without any constraints, M1 tests for the equivalence of an underlying three-factor structure. M1: chi-squre=604.18, df= 373, CFI=0.92. The CFI is high, so we can say that M1 indicate equivalent three-factor structure. M2 constrains the pattern of factor loadings to be equal across two groups. M2: chi-squre=641.25, df=390. Compare M1 with M2 yield chi-square difference 37.07, df difference 17. This is a statistically significant difference in model fit. Thus, substantiating rejection of the "hypothesis that item measurements were equivalent across two groups".--ie, factor loadings are different across two groups
- Marsh (1994) the invariance of factor variances and of hte uniqueness (error variance) associated with measured variables is typically less substantively relevant.
- Byrne, B.M. (1993). The Maslach Burnout Inventory: Testing for factorial validity and invariance across elementary, intermediate and secondary teachers. Journal of Occupational and Organizational Psychology, 66: 197-212.
- Byrne, B.M. (1994) Testing for the factorial validity, replication, and invariance of a measuring instrument: a paradigmatic application based on the Maslach burnout inventory. Multivariate Behavioral Research, 29 (3), 289--311
- Byrne, B.M. (2004). Testing for multigroup invariance using AMOS graphics: a road less traveled. Structrual Equation Modeling, 11(2): 272-300
- Byrne, B.M.(2001) Structural Equation Modeling with AMOS: basic concepts, applications, and programming. Lawrence Erlbaum.
- Marsh,H.W. (1994) Confirmatory factor analysis models of factorial invariance: a multifaceted approach. Structrual Equation Modeling, 1(1): 5-34--gender, age, gender multiply age (gender differences in structure that varies with age)
- Torkzadeh, G., Koufteros, X., Pflughoeft,K. (2003), Confirmatory analysis of computer self-efficacy. Structural Equation Modeling, 10(2), 263-275--across male vs. famale