A generating population with a common factor model defined by four, five, or six indicators was used to examine the null distribution of the newly derived χ2 test statistic using a set of 1,000 simulations per model. These simulations did not include individual genotypes, and were simulated solely based on a generating factor structure. For the six indicator models the standardized factor loadings in the generating population were .42, .64, .22, .59, .19, and .64. The four and five indicator models specified the same factor loadings, excluding the last, or last two loadings, respectively. Results indicated that the two-step procedure described above produced a test statistic equivalent to the χ2 statistic calculated by lavaan from the raw data (Supplementary Figure 31 and Supplementary Table 20). For a χ2 distributed test-statistic, the mean of the null sampling distribution should match the df of the test. As expected, the distribution of the test-statistic conformed to a χ2 distribution with an average approaching the df (Supplementary Figure 32). Calculated CFI values were also highly consistent with those observed using the CFI statistic provided
