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 relatedness matrix seems to compromise the ability of the LMM to control for sample structure (Supplementary Table 1). Thus choice of relatedness matrix could affect statistical efficiency (both power, and correct control of type I error due to stratification or relatedness) as well as computational efficiency. Interestingly, statistical and computational considerations may not necessarily conflict: for example,7 suggest that use of compressed MLM, which yields a lower-rank relatedness matrix by clustering individuals, can both reduce computation and increase power compared with the full-rank matrix. The general question of which low-rank relatedness matrices produce the best combination of computational and statistical performance seems to be an interesting avenue for further study.
