Beyond the traditional simulation design for evaluating type 1 error rate: from ‘theoretical’ to ‘empirical’ null

When evaluating a newly developed statistical test, the first step is to check its type 1 error (TIE) control using simulations. This is often achieved by the standard simulation design S0 under the so-called ‘theoretical’ null of no association. In practice, whole-genome association analyses scan through a large number of genetic markers (Gs) for the ones associated with an outcome of interest (Y), where Y comes from an unknown alternative while the majority of Gs are not associated with Y, that is under the ‘empirical’ null. This reality can be better represented by two other simulation designs, where design S1.1 simulates Y from an alternative model based on G then evaluates its association with independently generated Gnew, while design S1.2 evaluates the association between permutated Yperm and G. More than a decade ago, Efron (2004) has noted the important distinction between the ‘theoretical’ and ‘empirical’ null in false discovery rate control. Using scale tests for variance heterogeneity and location tests of interaction effect as two examples, here we show that not all null simulation designs are equal. In examining the accuracy of a likelihood ratio test, while simulation design S0 shows the method has the correct T1E control, designs S1.1 and S1.2 suggest otherwise with empirical T1E values of 0.07 for the 0.05 nominal level. And the inflation becomes more severe at the tail and does not diminish as sample size increases. This is an important observation that calls for new practices for methods evaluation and interpretation of T1E control.