Here we have focused on computations using the usual relatedness matrix, computed from all SNPs genome-wide, whose rank, r, is typically equal to the number of individuals n. However, as noted by Lippert et al8, if a lower-rank relatedness matrix is used then this reduces computing time (computational complexity of the singular value decomposition can scale with nr2) and in some cases memory requirements (e.g. Lippert et al.8 suggest using a relatedness matrix based on only a few thousand SNPs; this has the nice property that required singular value decompositions can be done without computing the n by n relatedness matrix itself). Using the usual full-rank relatedness matrix, our current implementation of GEMMA can handle approximately 23,000 individuals on a machine with 64 Gb memory (in double precision); using a lower-rank relatedness matrix, much larger problems could be tackled. However, we note that changing the relatedness matrix can produce much larger changes in p values than, for example, the differences between EMMAX and exact calculations (e.g. Supplementary Fig. 2), and for both the HMDP and WTCCC data using a lower-rank
