LMM—the gold standard for comparison

Using LASSO to analyze the bin data is our new contribution. However, we need to compare the new method with alternative methods to demonstrate the advantages of our method over existing methods. The LMM is the routine method for GWAS and has been the gold standard for comparison. It automatically controls population structure effects (Q) and polygenic effect via a genomic relationship (kinship) matrix (K). Therefore, it is called the Q + K model (Yu et al. 2006). Let y be an n × 1 vector of phenotypic values for a trait of interest, the LMM describing y is

where X is a design matrix capturing all non-genetic (fixed) effects, e.g., systematic environmental effects, population structural effects, and so on, β are the fixed effects, W_j is a vector of numerical codes for the jth marker for j = 1,…,M (M is the total number of markers, not the number of bins), μ_j is the effect of marker j on the trait (treated as a fixed effect), ξ is a vector of polygenic (random) effects with an assumed N(0, Kϕ²) distribution where ϕ² is the polygenic variance and K is a marker inferred kinship matrix (defined later), e is a vector of residual errors with an assumed N(0, Iσ²) distribution, where σ² is the residual error variance. The marker inferred kinship matrix is defined as

where d (a normalization factor) is the average value of the diagonal elements of the unnormalized matrix K. The restricted maximum likelihood (REML) method was used to estimate parameters {ϕ², σ²} and the best linear unbiased estimates (BLUE) of the fixed effects, β and μ_j, were obtained via mixed model equations. The Wald test for H₀:μ_j = 0 is

Under the null hypothesis, W_j follows approximately a Chi-square distribution with one degree of freedom. The corresponding p value was calculated using