Table 4.

Runtimes and time complexity for the 13,500 WTCCC dataset

Algorithm	Time	Time complexity
One variance component model
SKAT (Wu et al., 2011)	0.03 s	$O(N{k}_1^2)$
FaST-LMM-set score	0.03 s	$O(N{k}_1^2)$
FaST-LMM-Set LR test	0.04 s	$O(N{k}_1^2)$
Two variance component model full rank background kernel
FaST-LMM-set score	2 s	$O(N^2{k}_1)$
FaST-LMM-set LR test	1.6 h	$O(N^2{k}_1)$
LMM-Set LR test (before improvement)	150 h	$O(N^3)$
Two variance component model low rank background kernel
FaST-LMM-set score	See text	$O((N+k_g)k_1^2$)
LMM-set score (before improvement)	See text	$O(N^2{k}_1)$
FaST-LMM-set LR test	See text	$O(N{(k_g+k_1)}^2)$

Runtimes on a single core and time complexities for various linear set tests, both without a background kernel (one variance component model) and with (two variance component model) after applying our improvements with exceptions noted. The time reported is the time per test averaged over 13 850 tests from the WTCC1 type 1 diabetes dataset. Runtimes and complexities for the two-variance full rank cases exclude the O(N³) computations shared across all tests and done upfront (2 s, when amortized over the 13 850 tests). The logistic score model had approximately the same timing as the linear score, and so here we report only the linear score. For the LR test, the time includes the 10 permutations that are required. Regarding the notation for time complexity, $k_g$ and $k_1$ refer to the size of the background and test components, respectively.