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I have data from two different groups of research participants. One group of participants took my survey in the fall semester while the second group took my survey in the spring semester. I've come up with what I think is a good confirmatory factor analysis model based on the fall data and I want to see if that model holds in the spring data. I believe this is called a "multiple group" analysis. How can I perform this analysis using AMOS?
This FAQ assumes that you know how to run and interpret a single group analysis using AMOS. If you do not, see our online AMOS tutorial.
This FAQ also assumes that you know how to use AMOS to perform nested model comparisons and that you understand the assumptions and principles underlying nested model comparisons. See AMOS FAQ #6: Nested Model Comparisons for details.
Multiple group analysis in structural equation modeling is very useful because it allows you to compare multiple samples across the same measurement instrument or multiple population groups (e.g., males vs. females) for any identified structural equation model. AMOS allows you to test whether your groups meet the assumption that they are equal by examining whether different sets of path coefficients are invariant. In other words, you will be testing whether path coefficients in your model are equal for your groups. You can test the equalities of variables' variances, means, and intercepts, as well as the covariances between variables, and the equalities of path coefficients across two or more groups.
Before you begin testing invariance across groups, you should assess carefully your overall sample size and the equality of sample sizes across groups. Since the multiple group analysis estimates more parameters than a single group analysis, you will need proportionally more cases for a multiple group analysis to ensure stable parameter estimates and replicable results. For instance, if you had 150 cases for a single group analysis, you would want at least 300 cases for an analysis that used two groups. Furthermore, your analysis should ideally have equal numbers of cases in each group. Little is known about the impact of sharply unequal group sizes on results obtained from a multiple group SEM analysis, except that larger groups will exert more influence on the results than smaller groups. This property of multiple group analysis is not especially problematic if the group sizes mirror the proportion of individuals' group membership in the population from which the sample was drawn. On the other hand, if the sample sizes are not proportional to population sizes, errors of inference may be more likely to occur.
Different assumptions of group equality can be tested and they are often tested in a particular order (Bollen, 1989). For illustrative purposes, this example will consider one assumption of group equality. An example of sequential equality constraints in the confirmatory factor analysis context can be found in Bollen (1989); the example is discussed at length (with illustrative computer output) in LISREL FAQ #7: Comparing Groups using LISREL.
Consider example 20-2r from the AMOS program example set. This example illustrates a confirmatory factor analysis model in which two types of learning achievement, F1 (visual learning) and F2 (verbal learning) are each indicated by three measures. Are the factor structures the same for boys and girls? You can use a multiple group analysis in AMOS to address this question. This example program and the accompanying database is available with all versions of AMOS, including the student version.
Suppose you want to test the equality of the factor loadings for two separate groups of school children, girls and boys. To fit the models in AMOS, first draw the model for a single group and fit it for that group's sample data to ensure that the model is properly identified and that no minimization or other unexpected problems arise during the model fitting process. Then fit the same model to the second group's data. Assuming that both models converged correctly and no unusual problems were encountered during the model fitting process, you are now ready to perform the multiple group analysis. Follow these steps to perform the analysis:
1. Select Manage Groups... from the Model Fit menu. Name the first group Girls. Next, click on the New button to add a second group to the analysis. Name this group Boys. Click the New button successively to add additional groups as needed. AMOS 4.0 will allow you to consider up to 16 groups per analysis. Click the Close button when you are done creating and naming the multiple groups.
2. Each newly created group is represented by its own path diagram. Select the first group's path diagram by clicking on the group's label on the left-hand side of the path diagram window. The snapshot shown below features the Girls group highlighted.
From the Tools menu, choose Macro and then select Name Parameters. Check the relevant parameters you want AMOS to name in your path diagram. If you want to test the equality of factor loadings or path coefficients, choose the Regression Weights check box. If you want to compare variances of the groups, select the Variances check box. If you have any covariances represented by double-headed arrows in your path diagram and you want to compare their equality across the groups, select the Covariances check box. If you are explicitly modeling means or intercepts, you may wish to select those check boxes as well. Once you have selected the Regression Weights, Variances, and Covariances check boxes, the Name Parameters macro dialog box will look like this:
Select the second group's path diagram, Boys, and go through the same parameter naming process with one notable exception: For the second group, change the prefix for covariances from C to Ca, the prefix for variances from W to Wa, and the prefix for variances from V to Va. This alteration of the parameter naming process ensures that the second group will have different parameter names for each path coefficient, variance, and covariance when it is compared to the first group.
3. Select File, Data Files... to launch the Data Files dialog box. For each group, specify the relevant data file name. For this example, choose the Grant_fem SPSS database for the girls' group; choose the Grant_mal SPSS database for the boys' group. If the groups' information appears in separate data files for each group, as is the case in this example, you need to locate and specify the relevant data file for each group in the analysis by selecting the group, clicking on the File Name button, and locating the appropriate data file. Repeat this process for each group in the analysis. When you finish assigning data files to the groups, the Data Files dialog box will look like this:
If you have a single database that contains multiple groups, AMOS can also select groups of cases from a larger data file based upon a known grouping variable through the use of the Grouping Variable and Group Value buttons: Select the Grouping Variable button to identify the relevant grouping variable within a database and then use the Group Value button to select which value of the grouping variable represents the group of interest. Repeat this process for each group in the analysis.
4. Double-click on the Your Model label shown on the left side of the path diagram window (see the first figure shown above in section 2). This action launches the Manage Models window, shown below (you can also reach the Manage Models window from the Model Fit menu).
5. Rename the original model to be more meaningful. In this example, unrestricted loadings is a good choice because the starting model allows different factor loadings for boys and girls. You are now ready to define a second model that imposes a set of equality constraints on the unrestriced loadings model such that the unstandardized factor loadings are equal across boys' and girls' groups. To set up this model, first type the name of the original model, unrestricted loadings, on the first row of the Parameter Constraints section of the Manage Models dialog box. Referring to the unrestricted loadings model here lets AMOS know that you want to impose the constraints that follow subject to the assumptions or constraints already implied by the unrestricted loadings model.
Next, identify the four pairs relevant factor loadings of interest: w1 through w4 in the girls group, and wa1 through wa4 in the boys group. By double-clicking on w1 and then double-clicking on wa1, AMOS will insert the appropriate equality constraint in the Parameter Constraints section of the Manage Models dialog box. Similarly, you would perform the same operation for w2 and wa2, w3 and w3a, and w4 and wa4. When you finish, your Manage Models window looks like this:
Click the Close button to dismiss the Manage Models window. You are now ready to run the analysis.
Hint: The original AMOS example program enabled boostrapping. The example will run more quickly with bootstrapping disabled. To disable bootstrapping, select View/Set, then Analysis Properties.... Click on the Bootstrap tab, then uncheck the Perform bootstrap check box. For more information about bootstrapping, see AMOS FAQ #7: Handling non-normal data using AMOS.
After you have run the analysis, you can examine the model fit statistics for each model side by side. This information appears below:
These results show that both models fit the data exceptionally well. Interestingly, the more restricted equal loadings model fits the data better than the original model in which the factor loadings are allowed to vary across the girls and boys.
AMOS outputs the model comparison tests below the global fit statistics:
The results from this model comparison (Chi-square = 1.812 with 4 DF, p = .77) suggest that imposing the additional restrictions of four equal factor loadings across the two sexes of school children did not result in a statistically significant worsening of overall model fit. Notice that the nested model comparison heading mentions that the nested tests assume the baseline model is true. Be sure to begin the multiple group comparison process with a well-fitting default or starting model.
Once you have selected a final model that fits the data well, you may interpret the parameter estimates for the model. In the preceding figure, you can select the appropriate model from the left-hand side of the results table window and you can also select the group results to examine. For instance, you could select the Equal Loadings model and then examine the girls' and boys' regression weights and variances, as well as the covariance between the two factors. The path diagram mirrors this information.
Notice that while the unstandardized regression weights (factor loadings) will be the same for girls and boys in this model, the covariance between the factors and the residual variance estimates will vary across the two groups because you have not imposed any equality constraints on these quantities. You might choose to follow up the analysis above by specifying a third model that imposes further equality constraints on the factor covariance or the residual variance estimates, or both. See Bollen (1989) and LISREL FAQ #7: Comparing Groups using LISREL for more details on this approach to test factorial invariance.
For more information about muliple group SEM see the following references:
Arbuckle, J., & Wothke, W. (1999). AMOS 4.0 User's Guide. Chicago: Smallwaters Corporation, Inc.
Bollen, K.A. (1989). Structural equations with latent variables. New York: John Wiley & Sons.
Jaccard, J. & Wan, C. K. (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage Publications.
Joreskog, K.G., & Sorbom, D. (1993). LISREL 8 user’s reference guide. Chicago: Scientific Software International, Inc.
If you have further questions, send E-mail to stats@ssc.utexas.edu.