EM3.cov: Equation of mixed model for multi-kernel considering...

In RAINBOWR: Genome-Wide Association Study with SNP-Set Methods

Equation of mixed model for multi-kernel considering covariance structure between kernels

Description

This function solves the following multi-kernel linear mixed effects model with covairance structure.

y = X \beta + \sum _{l=1} ^ {L} Z _ {l} u _ {l} + \epsilon

where Var[y] = \sum _{i=1} ^ {L} Z _ {i} K _ {i} Z _ {i}' \sigma _ {i} ^ 2 + \sum _{i=1} ^ {L-1} \sum _{j=1} ^ {L} (Z _ {i} K _ {i j} Z _ {j}' + Z _ {j} K _ {j i} Z _ {i}') \sigma _ {i} \sigma _ {j} \rho _{i j} + I \sigma _ {e} ^ {2}.

Here, K _ {i j} and K _ {j i} are m_i \times m_j and m_j \times m_i matrices representing covariance structure between two random effects. \rho _{i j} is a correlation parameter to be estimated in addition to \sigma^2_i and \sigma_{e}^2.

Usage

EM3.cov(
  y,
  X0 = NULL,
  ZETA,
  covList,
  eigen.G = NULL,
  eigen.SGS = NULL,
  tol = NULL,
  n.core = NA,
  optimizer = "optim",
  traceInside = 0,
  nIterOptimization = NULL,
  n.thres = 450,
  REML = TRUE,
  pred = TRUE,
  return.u.always = TRUE,
  return.u.each = TRUE,
  return.Hinv = TRUE
)

Arguments

y	A n \times 1 vector. A vector of phenotypic values should be used. NA is allowed.
X0	A n \times p matrix. You should assign mean vector (rep(1, n)) and covariates. NA is not allowed.
ZETA	A list of variance matrices and its design matrices of random effects. You can use more than one kernel matrix. For example, ZETA = list(A = list(Z = Z.A, K = K.A), D = list(Z = Z.D, K = K.D)) (A for additive, D for dominance) Please set names of lists "Z" and "K"!
covList	A list of matrices representing covariance structure between paired random effects. If there are L random effects in the model, the list should contain L lists each consisting of L lists. Each \{i, j\} element of the list includes a matrix K_{ij} representing covariance structure between i-th and j-th random effects. See examples for details.
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.
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.
n.core	Setting n.core > 1 will enable parallel execution on a machine with multiple cores.
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.
nIterOptimization	Maximum number of iterations allowed. Defaults are different depending on 'optimizer'.
n.thres	If n >= n.thres, perform EMM1.cpp. Else perform EMM2.cpp.
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".
pred	If TRUE, the fitting values of y is returned.
return.u.always	If TRUE, BLUP ('u'; u) will be returned.
return.u.each	If TRUE, the function also computes each BLUP corresponding to different kernels (when solving multi-kernel mixed-effects model). It takes additional time compared to the one with 'return.u.each = FALSE'.
return.Hinv	If TRUE, H ^ {-1} = (Var[y] / \sum _{l=1} ^ {L} \sigma _ {l} ^ 2) ^ {-1} will be computed. It also returns V ^ {-1} = (Var[y]) ^ {-1}.

Value

$y.pred

The fitting values of y y = X\beta + Zu

$Vu

Estimator for \sigma^2_u, all of the genetic variance

$Ve

Estimator for \sigma^2_e

$beta

BLUE(\beta)

$u

BLUP(Sum of Zu)

$u.each

BLUP(Each u)

$weights

The proportion of each genetic variance (corresponding to each kernel of ZETA) to Vu

$rhosMat

The estimator for a matrix of correlation parameters \rho. Diagonal elements are always 0.

$LL

Maximized log-likelihood (full or restricted, depending on method)

$Vinv

The inverse of V = Vu \times ZKZ' + Ve \times I

$Hinv

The inverse of H = ZKZ' + \lambda I

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.

Examples

  ### 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


  ### Assume adjacent individuals are regarded as "neighbors"
  xAdj <- array(data = NA, dim = dim(x), dimnames = dimnames(x))
  for (i in 1:nrow(x)) {
    adjs <- (i - 1):(i + 1)
    adjs <- adjs[adjs %in% 1:nrow(x)]
    adjs <- adjs[adjs != i]
    nAdjs <- length(adjs)

    xAdj[i, ] <- x[i, , drop = FALSE] *
      apply(X = x[adjs, , drop = FALSE],
            MARGIN = 2, FUN = mean)
  }


  ### Estimate additive genomic relationship matrix (GRM)
  K.A <- tcrossprod(x) / ncol(x)
  K.Adj <- tcrossprod(xAdj) / ncol(xAdj)  # for neighbor kernel


  ### Modify data
  Z <- design.Z(pheno.labels = rownames(y),
                geno.names = rownames(K.A))  ### design matrix for random effects
  pheno.mat <- y[rownames(Z), , drop = FALSE]
  ZETA <- list(A = list(Z = Z, K = K.A),
               Adj = list(Z = Z, K = K.Adj))


  ### Prepare for covairance structure between two random effects
  K12 <- tcrossprod(x, xAdj) / sqrt(ncol(x) * ncol(xAdj))
  K21 <- tcrossprod(xAdj, x) / sqrt(ncol(x) * ncol(xAdj))

  covList <- rep(list(rep(list(NULL), 2)), 2)
  covList[[1]][[2]] <- K12
  covList[[2]][[1]] <- K21


  ### Solve multi-kernel linear mixed effects model (2 random efects)
  ### conidering covariance structure
  EM3cov.res <- EM3.cov(y = pheno.mat,
                        X0 = NULL,
                        ZETA = ZETA,
                        covList = covList)
  (Vu <- EM3cov.res$Vu)   ### estimated genetic variance
  (Ve <- EM3cov.res$Ve)   ### estimated residual variance
  (weights <- EM3cov.res$weights)   ### estimated proportion of two genetic variances
  (herit <- Vu * weights / (Vu + Ve))   ### genomic heritability (additive, neighbor)
  (rho <- EM3cov.res$rhosMat[2, 1])   ### correlation parameter

  (beta <- EM3cov.res$beta)   ### Here, this is an intercept.
  u.each <- EM3cov.res$u.each   ### estimated genotypic values (additive, neighbor)
  See(u.each)


  ### Perform genomic prediction with 10-fold cross validation (multi-kernel)
  noNA <- !is.na(c(pheno.mat))   ### NA (missing) in the phenotype data

  phenoNoNA <- pheno.mat[noNA, , drop = FALSE]   ### remove NA
  ZETANoNA <- ZETA
  ZETANoNA <- lapply(X = ZETANoNA, FUN = function (List) {
    List$Z <- List$Z[noNA, ]

    return(List)
  })   ### 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 <- yPredCov <- rep(NA, nLine)


  for (noCV in 1:nFold) {
    print(paste0("Fold: ", noCV))
    yTrain <- phenoNoNA
    yTrain[idCV == noCV, ] <- NA   ### prepare test data

    EM3.resCV <- EM3.cpp(y = yTrain, X0 = NULL,
                         ZETA = ZETANoNA)   ### prediction
    EM3cov.resCV <- EM3.cov(y = yTrain, X0 = NULL,
                            ZETA = ZETANoNA,
                            covList = covList)   ### prediction
    yTest <- EM3.resCV$y.pred     ### predicted values
    yTestCov <- EM3cov.resCV$y.pred

    yPred[idCV == noCV] <- yTest[idCV == noCV]
    yPredCov[idCV == noCV] <- yTestCov[idCV == noCV]
  }

  ### Plot the results
  plotRange <- range(phenoNoNA, yPred)
  plot(x = phenoNoNA, y = yPred,
       xlim = plotRange, ylim = plotRange,
       xlab = "Observed values", ylab = "Predicted values (EM3.cpp)",
       main = "Results of Genomic Prediction (multi-kernel)",
       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)


  plotRange <- range(phenoNoNA, yPred)
  plot(x = phenoNoNA, y = yPredCov,
       xlim = plotRange, ylim = plotRange,
       xlab = "Observed values", ylab = "Predicted values (EM3.cov)",
       main = "Results of Genomic Prediction (multi-kernel with covariance)",
       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 = yPredCov) ^ 2
  text(x = plotRange[2] - 10,
       y = plotRange[1] + 10,
       paste0("R2 = ", round(R2, 3)),
       cex = 1.5)


  plotRange <- range(yPred, yPredCov)
  plot(x = yPred, y = yPredCov,
       xlim = plotRange, ylim = plotRange,
       xlab = "Predicted values (EM3.cpp)", ylab = "Predicted values (EM3.cov)",
       main = "Comparison of Multi-Kernel 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 = yPred, y = yPredCov) ^ 2
  text(x = plotRange[2] - 10,
       y = plotRange[1] + 10,
       paste0("R2 = ", round(R2, 3)),
       cex = 1.5)

RAINBOWR documentation built on June 8, 2025, 1:54 p.m.