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Example 20
fail for models that fit poorly. If some way could be found to successfully fit Model 2
to these 19 samples—for example, with hand-picked start values or a superior
algorithm—it seems likely that the discrepancies would be large. According to this line
of reasoning, discarding bootstrap samples for which estimation failed would lead to a
downward bias in the mean discrepancy. Thus, you should be concerned by estimation
failures during bootstrapping, primarily when they occur for the model with the lowest
mean discrepancy.
In this example, the lowest mean discrepancy (26.57) occurs for Model 2R,
confirming the model choice based on the BCC, AIC, and CAIC criteria. The
differences among the mean discrepancies are large compared to their standard errors.
Since all models were fitted to the same bootstrap samples (except for samples where
Model 2 was not successfully fitted), you would expect to find positive correlations
across bootstrap samples between discrepancies for similar models. Unfortunately,
Amos does not report those correlations. Calculating the correlations by hand shows
that they are close to 1, so that standard errors for the differences between means in the
table are, on the whole, even smaller than the standard errors of the means.
Summary
The bootstrap can be a practical aid in model selection for analysis of moment
structures. The Linhart and Zucchini (1986) approach uses the expected discrepancy
between implied and population moments as the basis for model comparisons. The
method is conceptually simple and easy to apply. It does not employ any arbitrary
magic number such as a significance level. Of course, the theoretical appropriateness
of competing models and the reasonableness of their associated parameter estimates
are not taken into account by the bootstrap procedure and need to be given appropriate
weight at some other stage in the model evaluation process.
Modeling in VB.NET
Visual Basic programs for this example are in the files Ex20-1.vb, Ex20-2.vb, Ex20-
2r.vb, Ex20-ind.vb, and Ex20-sat.vb.