The results of Simulation study 1 as depicted in Fig. 3 are summarized in Table 2, which also includes results for some additional simulations settings. In testing β a under these three scenarios, the number of false positives (i.e., χdiff2 tests >3.84) was inflated if the correlation between T and M ran via A or C: 6.85 and 7.60%, rather than 5%, respectively. If the correlation between T and M ran via E, however, the false positive rate was even more seriously inflated: 53.23%. This inflation is clearly visible in Fig. 3 (PP-plot lower left corner). Similar results were obtained for the tests of β c: if the correlation between T and M ran via A or C, the false positive rate was 6.85 and 8.75%, respectively, while the false positive rate was 55.33% if the correlation between T and M ran via E. In testing for the significance of β e, the false positive rate was only significantly elevated if the correlation between T and M ran via E (7.50%), but not if the correlation ran via A or
