Performance and limitations of reference models for linear mixed model GWAS.

The time complexity of each algorithm is approximate, assuming a model with only a single marker effect and no other fixed effects. Here, l is used to index full-rank random effects; are variance component parameters; n is the number of observations; and p is number of markers to test. Denote t1…t7 to represent the number of iterations needed for convergence, which is expected to vary among methods (particularly for the iterations of grid search in Grid-LMM-fast), and may vary across markers. The terms g and gi are grid sizes for the Grid-LMM methods (i.e. the number of grid vertices that must be evaluated). Lastly, pi is the number of markers that need to be tested in iteration of i ∈ {1…t7} of the Grid-LMM-fast method. The rate limiting terms in common GWAS applications (where p ≫ n) are in bold. “Method Type” describes the estimation of . “Exact” means effectively exact, up to machine precision and subject to possible convergence of algorithms to local maxima. “Null” means estimation of parameters under the null model with no marker effects. References list additional methods that are approximately equivalent to the given model classes.