# Equation of mixed model for one kernel, a wrapper of two methods This function estimates maximum-likelihood (ML/REML; resticted maximum likelihood) solutions for the following mixed model. [REMOVE_ME]$$ y = X \beta + Z u + \epsilon [REMOVE_ME_2]$$ where $\beta$ is a vector of fixed effects and $u$ is a vector of random effects with $Var[u] = K \sigma^2_u$. The residual variance is $Var[\epsilon] = I \sigma^2_e$. ```r EMM.cpp( y, X = NULL, ZETA, eigen.G = NULL, eigen.SGS = NULL, n.thres = 450, reestimation = FALSE, n.core = NA, lam.len = 4, init.range = c(1e-06, 100), init.one = 0.5, conv.param = 1e-06, count.max = 20, bounds = c(1e-06, 1e+06), tol = NULL, optimizer = "nlminb", traceInside = 0, REML = TRUE, silent = TRUE, plot.l = FALSE, SE = FALSE, return.Hinv = TRUE ) ``` ## Arguments - `y`: A $n \times 1$ vector. A vector of phenotypic values should be used. NA is allowed. - `X`: A $n \times p$ matrix. You should assign mean vector (rep(1, n)) and covariates. NA is not allowed. - `ZETA`: A list of variance (relationship) matrix (K; $m \times m$) and its design matrix (Z; $n \times m$) of random effects. You can use only one kernel matrix. For example, ZETA = list(A = list(Z = Z, K = K)) Please set names of list "Z" and "K"! - `eigen.G`: A list with - **$values**: Eigen values - **$vectors**: Eigen vectors The result of the eigen decompsition of $G = ZKZ'$. You can use "spectralG.cpp" function in RAINBOWR. If this argument is NULL, the eigen decomposition will be performed in this function. We recommend you assign the result of the eigen decomposition beforehand for time saving. - `eigen.SGS`: A list with - **$values**: Eigen values - **$vectors**: Eigen vectors The result of the eigen decompsition of $SGS$, where $S = I - X(X'X)^{-1}X'$, $G = ZKZ'$. You can use "spectralG.cpp" function in RAINBOWR. If this argument is NULL, the eigen decomposition will be performed in this function. We recommend you assign the result of the eigen decomposition beforehand for time saving. - `n.thres`: If $n >= n.thres$, perform EMM1.cpp. Else perform EMM2.cpp. - `reestimation`: If TRUE, EMM2.cpp is performed when the estimation by EMM1.cpp may not be accurate. - `n.core`: Setting n.core > 1 will enable parallel execution on a machine with multiple cores. - `lam.len`: The number of initial values you set. If this number is large, the estimation will be more accurate, but computational cost will be large. We recommend setting this value 3 <= lam.len <= 6. - `init.range`: The range of the initial parameters. For example, if lam.len = 5 and init.range = c(1e-06, 1e02), corresponding initial heritabilities will be calculated as seq(1e-06, 1 - 1e-02, length = 5), and then initial lambdas will be set. - `init.one`: The initial parameter if lam.len = 1. - `conv.param`: The convergence parameter. If the diffrence of log-likelihood by updating the parameter "lambda" is smaller than this conv.param, the iteration steps will be stopped. - `count.max`: Sometimes algorithms won't converge for some initial parameters. So if the iteration steps reache to this argument, you can stop the calculation even if algorithm doesn't converge. - `bounds`: Lower and Upper bounds of the parameter lambda. If the updated parameter goes out of this range, the parameter is reset to the value in this range. - `tol`: The tolerance for detecting linear dependencies in the columns of G = ZKZ'. Eigen vectors whose eigen values are less than "tol" argument will be omitted from results. If tol is NULL, top 'n' eigen values will be effective. - `optimizer`: The function used in the optimization process. We offer "optim", "optimx", and "nlminb" functions. - `traceInside`: Perform trace for the optimzation if traceInside >= 1, and this argument shows the frequency of reports. - `REML`: You can choose which method you will use, "REML" or "ML". If REML = TRUE, you will perform "REML", and if REML = FALSE, you will perform "ML". - `silent`: If this argument is TRUE, warning messages will be shown when estimation is not accurate. - `plot.l`: If you want to plot log-likelihood, please set plot.l = TRUE. We don't recommend plot.l = TRUE when lam.len >= 2. - `SE`: If TRUE, standard errors are calculated. - `return.Hinv`: If TRUE, the function returns the inverse of $H = ZKZ' + \lambda I$ where $\lambda = \sigma^2_e / \sigma^2_u$. This is useful for GWAS. ## Returns - **$Vu**: Estimator for $\sigma^2_u$ - **$Ve**: Estimator for $\sigma^2_e$ - **$beta**: BLUE($\beta$) - **$u**: BLUP($u$) - **$LL**: Maximized log-likelihood (full or restricted, depending on method) - **$beta.SE**: Standard error for $\beta$ (If SE = TRUE) - **$u.SE**: Standard error for $u^*-u$ (If SE = TRUE) - **$Hinv**: The inverse of $H = ZKZ' + \lambda I$ (If return.Hinv = TRUE) - **$Hinv2**: The inverse of $H2 = ZKZ'/\lambda + I$ (If return.Hinv = TRUE) - **$lambda**: Estimators for $\lambda = \sigma^2_e / \sigma^2_u$ (If $n >= n.thres$) - **$lambdas**: Lambdas for each initial values (If $n >= n.thres$) - **$reest**: If parameter estimation may not be accurate, reest = 1, else reest = 0 (If $n >= n.thres$) - **$counts**: The number of iterations until convergence for each initial values (If $n >= n.thres$) ## Description This function estimates maximum-likelihood (ML/REML; resticted maximum likelihood) solutions for the following mixed model. $$ y = X \beta + Z u + \epsilon $$ where $\beta$ is a vector of fixed effects and $u$ is a vector of random effects with $Var[u] = K \sigma^2_u$. The residual variance is $Var[\epsilon] = I \sigma^2_e$. ## Examples ```r ### Perform genomic prediction with 10-fold cross validation ### Import RAINBOWR require(RAINBOWR) ### Load example datasets data("Rice_Zhao_etal") Rice_geno_score <- Rice_Zhao_etal$genoScore Rice_geno_map <- Rice_Zhao_etal$genoMap Rice_pheno <- Rice_Zhao_etal$pheno ### View each dataset See(Rice_geno_score) See(Rice_geno_map) See(Rice_pheno) ### Select one trait for example trait.name <- "Flowering.time.at.Arkansas" y <- as.matrix(Rice_pheno[, trait.name, drop = FALSE]) ### Remove SNPs whose MAF <= 0.05 x.0 <- t(Rice_geno_score) MAF.cut.res <- MAF.cut(x.0 = x.0, map.0 = Rice_geno_map) x <- MAF.cut.res$x map <- MAF.cut.res$map ### Estimate genomic relationship matrix (GRM) K.A <- calcGRM(genoMat = x) ### Modify data modify.res <- modify.data(pheno.mat = y, geno.mat = x, return.ZETA = TRUE) pheno.mat <- modify.res$pheno.modi ZETA <- modify.res$ZETA ### Solve linear mixed effects model EMM.res <- EMM.cpp(y = pheno.mat, X = NULL, ZETA = ZETA) (Vu <- EMM.res$Vu) ### estimated genetic variance (Ve <- EMM.res$Ve) ### estimated residual variance (herit <- Vu / (Vu + Ve)) ### genomic heritability (beta <- EMM.res$beta) ### Here, this is an intercept. u <- EMM.res$u ### estimated genotypic values See(u) ### Estimate marker effects from estimated genotypic values x.modi <- modify.res$geno.modi WMat <- calcGRM(genoMat = x.modi, methodGRM = "addNOIA", returnWMat = TRUE) K.A <- ZETA$A$K if (min(eigen(K.A)$values) < 1e-08) { diag(K.A) <- diag(K.A) + 1e-06 } mrkEffectsForW <- crossprod(x = WMat, y = solve(K.A)) %*% as.matrix(u) mrkEffects <- mrkEffectsForW / mean(scale(x.modi %*% mrkEffectsForW, scale = FALSE) / u) #### Cross-validation for genomic prediction noNA <- !is.na(c(pheno.mat)) ### NA (missing) in the phenotype data phenoNoNA <- pheno.mat[noNA, , drop = FALSE] ### remove NA ZETANoNA <- ZETA ZETANoNA$A$Z <- ZETA$A$Z[noNA, ] ### remove NA nFold <- 10 ### # of folds nLine <- nrow(phenoNoNA) idCV <- sample(1:nLine %% nFold) ### assign random ids for cross-validation idCV[idCV == 0] <- nFold yPred <- rep(NA, nLine) for (noCV in 1:nFold) { yTrain <- phenoNoNA yTrain[idCV == noCV, ] <- NA ### prepare test data EMM.resCV <- EMM.cpp(y = yTrain, X = NULL, ZETA = ZETANoNA) ### prediction yTest <- EMM.resCV$beta + EMM.resCV$u ### predicted values yPred[idCV == noCV] <- (yTest[noNA])[idCV == noCV] } ### Plot the results plotRange <- range(phenoNoNA, yPred) plot(x = phenoNoNA, y = yPred,xlim = plotRange, ylim = plotRange, xlab = "Observed values", ylab = "Predicted values", main = "Results of Genomic Prediction", cex.lab = 1.5, cex.main = 1.5, cex.axis = 1.3) abline(a = 0, b = 1, col = 2, lwd = 2, lty = 2) R2 <- cor(x = phenoNoNA[, 1], y = yPred) ^ 2 text(x = plotRange[2] - 10, y = plotRange[1] + 10, paste0("R2 = ", round(R2, 3)), cex = 1.5) ``` ## References Kang, H.M. et al. (2008) Efficient Control of Population Structure in Model Organism Association Mapping. Genetics. 178(3): 1709-1723. Zhou, X. and Stephens, M. (2012) Genome-wide efficient mixed-model analysis for association studies. Nat Genet. 44(7): 821-824.

EMM.cpp function