Table 3. Empirical positive rates of all tests under all significance assessment procedures in all scenarios based on 10,000 simulations, 1,000 permutations each to estimate empirical null distributions (residperm and locusperm), and a nominal false positive rate (FPR) of $\alpha =\text{0.05}$. Entries in column 1 and 5 through all rows, columns 3 and 7 in the top third, and columns 2 and 6 in the middle third represent empirical FPR. Where the empirical FPR is within one standard error of the nominal FPR of 0.05, it is written in normal font. Where it is overly conservative, it is underlined. Where it is anti-conservative, it is in boldface. The entries in the rest of the table represent power. Given the sample size of 10,000, the standard error for the values in this table are all between 0.005 and 0.01. Generally, values near 0.05 have a standard error near 0.005 and values near 0.5 have a standard error near 0.01. All standard errors are listed in Table S2 and plotted visually in Figure 1, Figure 2, Figure 3, and Figure 4.

		BVH absent				BVH present
test	procedure	null	mQTL	vQTL	mvQTL	null	mQTL	vQTL	mvQTL
SLM	standard	0.052	0.717	0.054	0.502	0.052	0.706	0.052	0.506
	RINT	0.051	0.712	0.051	0.486	0.053	0.719	0.049	0.512
	residperm	0.050	0.710	0.052	0.494	0.050	0.700	0.052	0.498
	locusperm	0.049	0.709	0.052	0.497	0.051	0.699	0.051	0.499
CaoM	standard	0.053	0.717	0.051	0.510	0.054	0.702	0.049	0.520
	RINT	0.052	0.713	0.048	0.496	0.054	0.717	0.049	0.529
	residperm	0.051	0.709	0.048	0.501	0.051	0.698	0.048	0.509
	locusperm	0.049	0.704	0.047	0.496	0.050	0.694	0.046	0.506
DGLMM	standard	0.057	0.714	0.055	0.510	0.059	0.836	0.057	0.654
	RINT	0.055	0.713	0.055	0.499	0.057	0.834	0.056	0.648
	residperm	0.049	0.693	0.048	0.490	0.052	0.822	0.050	0.631
	locusperm	0.049	0.691	0.047	0.484	0.050	0.818	0.048	0.628
Levene’s test	standard	0.045	0.048	0.655	0.452	0.050	0.045	0.566	0.388
	RINT	0.046	0.041	0.638	0.416	0.048	0.041	0.539	0.340
	residperm	0.049	0.052	0.664	0.461	0.052	0.048	0.574	0.397
	locusperm	0.047	0.051	0.665	0.461	0.052	0.049	0.576	0.399
CaoV	standard	0.054	0.053	0.738	0.536	0.135	0.131	0.736	0.568
	RINT	0.045	0.041	0.692	0.457	0.046	0.047	0.552	0.366
	residperm	0.050	0.049	0.721	0.510	0.053	0.050	0.564	0.390
	locusperm	0.049	0.047	0.717	0.505	0.051	0.047	0.561	0.384
DGLMV	standard	0.054	0.053	0.721	0.520	0.058	0.050	0.732	0.527
	RINT	0.045	0.041	0.673	0.444	0.023	0.022	0.564	0.346
	residperm	0.049	0.049	0.699	0.496	0.020	0.018	0.537	0.331
	locusperm	0.048	0.048	0.698	0.490	0.052	0.046	0.708	0.502
CaoMV	standard	0.053	0.607	0.633	0.742	0.113	0.642	0.643	0.749
	RINT	0.046	0.595	0.567	0.698	0.049	0.603	0.434	0.650
	residperm	0.049	0.597	0.606	0.726	0.054	0.516	0.503	0.632
	locusperm	0.049	0.596	0.610	0.729	0.053	0.517	0.503	0.634
DGLMMV	standard	0.056	0.606	0.621	0.734	0.060	0.746	0.631	0.811
	RINT	0.050	0.597	0.559	0.694	0.038	0.724	0.440	0.732
	residperm	0.050	0.589	0.587	0.711	0.025	0.631	0.466	0.694
	locusperm	0.050	0.586	0.585	0.713	0.054	0.735	0.596	0.790