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I want to compare the equality of factor loadings with a confirmatory factor analysis. How can I do this using AMOS?
This FAQ assumes that you understand the assumptions of structural equation models (SEM) and can specify and test SEMs using AMOS. If not, see our AMOS tutorial.
You can use a method known as nested model comparisons to address your question. Note that you can also compare path coefficients (regression weights), means, intercepts, and variances within path analytic and structural equation models using this approach.
Nested model comparisons work by imposing a constraint or set of multiple constraints on a starting or less retricted model to obtain a more restricted final model. Loehlin (1997) provides a nice discussion, with illustrated diagrams, of models that may be considered nested versus those that are not nested. The method described here is only appropriate for comparing nested models. Non-nested models may be compared descriptively by examining descriptive model fit statistics such as Akaike's Information Criterion (AIC) or the Bayesian Information Criterion (BIC); other things being equal, models with smaller AIC and BIC values fit better. Also, while the nested model comparisons provide a powerful tool to test competing structural equation (and related) models, the technique can be overused. As is the case with any test statistic, employing too many nested model comparisons on the same set of data can result in increased type 1 error for any given test (i.e., obtaining a seemingly significant hypothesis test that is not reflective of an underlying population difference but is instead due to chance alone). Jaccard & Wan (1996) address this issue in the context of nested model comparisons; they offer a suggested remedy, documented in General FAQ #29: Adjusted Bonferroni Comparisons.
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.This example program and the accompanying database is available in the Examples directory within the AMOS directory with all versions of AMOS, including the student version.
Suppose you wanted to test the equality of the visperc factor loading and the paragraph factor loading, as well as the cubes factor loading and the lozenges factor loading. Follow the steps shown below to obtain the appropriate nested model comparison.
The first step is to name the parameters you want to constrain. In this example, you can name the visperc factor loading by double-clicking on the single-headed arrow pointing from F1 to the visperc observed variable. This action launches the Object Properties window. Click on the Parameters tab and enter a suitable name in the dialog box labeled Regression weight. For instance, you might call this factor loading visperc-loading. The Object Properties dialog box will then resemble the following figure.
Close the Object Properties dialog box. Next, assign similar names to the other three unconstrained factor loadings using the same method. Naming the factor loadings allows you to refer to them when you establish equality constraints as part of the nested model comparison, as shown below. Note: If you have a large number of parameters to name, AMOS has a macro program that will automatically name parameters. See AMOS FAQ #3: Multiple Group Analysis for an example of how to use the AMOS parameter naming macro.
Next, double-click on the section of the AMOS diagram window labeled XX: Your Model. This action launches the Manage Models window. You can also reach this window by selecting Manage Models... from the Model Fit menu item.
Change the name of Your Model to Unrestricted Loadings. 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, click the New button to create a new model. Assign the model a meaningful name, e.g., Equal Loadings. In the first section of the Parameter Constraints segment of the window, type Unrestricted Loadings. Next, double-click on the visperc-loading weight. Then double-click on the paragraph-loading weight. AMOS automatically inserts the two weights separated by an equal sign into the Parameter Constraints area of the Manage Models window. It is also possible to manually type the constraint(s) in the window. For example, try typing cubes-loading = sentence-loading in the window. When you are done, your Manage Models window should look like this:
Click the Close button to close the window, and then save your work by choosing Save As from the File menu and saving your new model file under a unique file name. Run the model and examine the overall fit measures.
Hint: The original AMOS example program enabled bootstrapping. 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.
AMOS produces the fit statistics table for the original model that does not assume equal factor loadings and it also produces a second set of fit statistics for the more restricted model that assumes equal factor loadings. You can visually compare the inferential and descriptive fit statistics for each model. Below the fit measures, AMOS displays a section of output titled Model Comparisons. Select this output.
Although the overall fit statistics show that both models are acceptable, the nested model comparison that assesses the worsening of overall fit due to imposing the two restrictions on the original model shows a statistically significant chi-square value of 12.795 with two degrees of freedom, resulting in a probability value of .002. This finding suggests that the parsimony you achieve with the more restricted equal factor loadings model comes at too high a cost: the fact that the two models differ indicates that constraining the parameters in the Unrestricted Loadings model to obtain the Equal Loadings model results in a substantial worsening of overall model fit. Therefore, in most circumstances you would reject the equal factor loadings model in favor of the original model.
Unstandardized and standardized regression coefficients for both models are available in the AMOS output. To examine the parameter estimates for a particular model, select the model's name from the list of models shown on the left-hand side of the AMOS tabular output (shown immediately above). Then click on the Estimates output section. Notice in this example that the unstandardized path coefficients from F1 to W1 and F2 to W4 are equal, as are the unstandardized path coefficients from F1 to W2, and F2 to W3. This is as it should be because the Equal Loadings model contains these constraints. By contrast, the unstandardized regression coefficients for the first model are free to vary across the factors.
With AMOS 4.0 you can also perform model comparisons where you fix a parameter or set of parameters to a specific value. Zero is the most common value used, but other numeric values are permissable. At the time of this writing, AMOS does not support addition, subtraction, multiplication, or exponentiation in establishing model equality constraints.
For more information about nested model comparisons in the conext of SEM, see the following references:
Arbuckle, J., & Wothke, W. (1999). AMOS 4.0 User's Guide. Chicago: Smallwaters Corporation, Inc.
Jaccard, J. & Wan, C. K. (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage Publications.
Loehlin, J. C. (1997). Latent variable models. Mahwah, NJ: Lawrence Erlbaum.
If you have further questions, send E-mail to stats@ssc.utexas.edu.