We evaluated variation in developmental time among genotypes by means of a four-way analysis of variance (ANOVA) according to the mixed model: Y = μ + L + S + G+ R (L ×S× G) + L × G + L × S + G × S + L × G × S + E, where L, G and S are the fixed cross-classified effects of line (second chromosome substitution line), genotype (Canton-S B or m) and sex, respectively.
Gene expression heritability analysis was performed in the four HapMap populations combined, and also in each individual HapMap population using a simplified linear mixed model y ij = μ + u i + e ij, where u ~ N 0, ) and e ~ N 0, ).
The adjusted phenotypes were fitted using a linear mixed model Y = μ + M + g + E, where μ was the population mean, M was the fixed effect for marker effect, and g was a polygenic term with its covariance among lines determined by the genomic relationship matrix (Huang et al. 2014).
For each variant, we fitted the following linear mixed model: y = μ + x β + u + ε, u ∼ M V N (0, σ u 2 K ), ε ∼ M V N (0, σ e 2 I ), and tested the null hypothesis H0: β = 0 vs the alternative H1: β ≠ 0. Here, y is the n by 1 vector of gene expression levels for the n individuals in the sample.
Association analyses were performed on best linear unbiased estimations (BLUEs) of genotypes' performance for a given trait, inferred from the following linear mixed model: y i j l m r c = m e a n + g e n o t y p e i + l o c a t i o n j + y e a r (l o c a t i o n ) j l + r e p. (l o c a t i o n ) j m + (year × rep.
A linear mixed model y i j k = μ + C O 2 i + G j + D k + e i j k was fitted to the data, with yijk as the normalized-transformed gene expression, μ as the group mean, CO2i as the effect of the CO2 level, Gj as the effect of ith gill, Dk as the effect of jth day, and eijk as the sample effect (random error).
Consider the linear mixed model (7) y = X β + Zu + e with y the vector of observations for q traits, β, u and e vectors of fixed effects, random effects and residuals, and X and Z the design matrices pertaining to β and u.
The linear mixed model was y = μ + Xβ + Ζu = e where y is a vector of phenotypes (DTDs for bulls and TDs for cows.
The FA animal mixed model was: Y hij = μ + η h + β h AG E h + FER T hi + A C j + A S hj + e hij, where A C j is the random genetic effect of the latent common factor across environments for animal j and A S hj is the random genetic effect specific to environment h for animal j.
To remove the dependency within the longitudinal data, each of the three intermediate risk factors was summarized into two summary measures, a random intercept and a random slope, using the following mixed effect model: y i 0 was the measurement of risk factor y taken at baseline for patient i; i.e. the time point before medication was given.
Phenotypic data from the IBM density trial were analyzed using SAS PROC MIXED version 9.2 (SAS Institute) with the following mixed linear model: Y i j k = μ + d i + R (D ) j + G k + G D i k + ε i j k [1]where Yijk is the response variable of the kth genotype (G) in the jth replicate (R) nested in the ith density (D).
