Table 3.
Results Simulation study 2: false positive rates under the extended univariate moderation model and the full bivariate moderation model when the covariance between T and M is not moderated
| Drop β a | Drop β c | Drop β e | ||||
|---|---|---|---|---|---|---|
| % hits | nsim | % hits | nsim | % hits | nsim | |
| rM,T .24 via A | ||||||
| Ext univariate | .04 | 2000 | .04 | 2000 | .04 | 1998 |
| Full bivariate | .01 | 2000 | .01 | 2000 | .04 | 2000 |
| p < .001 | p < .001 | p = .88 | ||||
| rM,T .24 via C | ||||||
| Ext univariate | .04 | 2000 | .04 | 2000 | .05 | 2000 |
| Full bivariate | .01 | 1999 | .01 | 1998 | .04 | 2000 |
| p < .001 | p < .001 | p = .08 | ||||
| rM,T .24 via E | ||||||
| Ext univariate | .05 | 1998 | .04 | 1999 | .05 | 1999 |
| Full bivariate | .01 | 1996 | .01 | 1996 | .05 | 2000 |
| p < .001 | p < .001 | p = .65 | ||||
| rM,T = .24 via A, C and E in equal proportions | ||||||
| Ext univariate | .04 | 2000 | .05 | 2000 | .05 | 2000 |
| Full bivariate | .02 | 2000 | .01 | 2000 | .05 | 2000 |
| p < .005 | p < .001 | p = .43 | ||||
| rM,T = .62 via A, C and E in equal proportions | ||||||
| Ext univariate | .05 | 1998 | .05 | 2000 | .05 | 2000 |
| Full bivariate | .02 | 2000 | .02 | 2000 | .05 | 2000 |
| p < .001 | p < .001 | p = .94 | ||||
| rM1,M2 = 0 (fully E), rM,T = .24 | ||||||
| Ext univariate | .05 | 1999 | .05 | 2000 | .04 | 1999 |
| rM1,M2 = 1 (fully C), rM,T = .24 | ||||||
| Ext univariate | .03 | 2000 | .03 | 2000 | .05 | 1999 |
Note: β a, β c, and β e denote the moderation parameters on the variance components unique to T (see Fig. 1). nsim denotes the number of data sets (out of 2000) for which the extended univariate moderation model and the full bivariate moderation model converged without problems. % hits denotes the percentage (of the nsim converged models) that the likelihood-ratio test was smaller than the critical value 3.84 (i.e., significant given α = .05). % hits outside the .04–.06 range should be considered incorrect (i.e., significantly too low or too high). The p-values concern p-values of the binomial test for comparing two proportions, used to test whether the number of hits under the extended univariate moderation model is significantly different from the number of hits under the full bivariate moderation model. Note that the full bivariate moderation model was not used to analyze data generated according to the final two settings (rM1,M2 = 0 and rM1,M2 = 1) because one would never use a bivariate model for data with such a variance–covariance structure