EM3.linker.cpp: Equation of mixed model for multi-kernel (fast, for limited...
In KosukeHamazaki/RAINBOW: Genome-Wide Association Study with SNP-Set Methods

Description Usage Arguments Value References Examples

This function solves multi-kernel mixed model using fastlmm.snpset approach (Lippert et al., 2014). This function can be used only when the kernels other than genomic relationship matrix are linear kernels.

EM3.linker.cpp(
  y0,
  X0 = NULL,
  ZETA = NULL,
  Zs0 = NULL,
  Ws0,
  Gammas0 = lapply(Ws0, function(x) diag(ncol(x))),
  gammas.diag = TRUE,
  X.fix = TRUE,
  eigen.SGS = NULL,
  eigen.G = NULL,
  tol = NULL,
  bounds = c(1e-06, 1e+06),
  optimizer = "nlminb",
  traceInside = 0,
  n.thres = 450,
  spectral.method = NULL,
  REML = TRUE,
  pred = TRUE
)

`y0`	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 (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"!
`Zs0`	A list of design matrices (Z; n \times m matrix) for Ws. For example, Zs0 = list(A.part = Z.A.part, D.part = Z.D.part)
`Ws0`	A list of low rank matrices (W; m \times k matrix). This forms linear kernel K = W Γ W'. For example, Ws0 = list(A.part = W.A, D.part = W.D)
`Gammas0`	A list of matrices for weighting SNPs (Gamma; k \times k matrix). This forms linear kernel K = W Γ W'. For example, if there is no weighting, Gammas0 = lapply(Ws0, function(x) diag(ncol(x)))
`gammas.diag`	If each Gamma is the diagonal matrix, please set this argument TRUE. The calculationtime can be saved.
`X.fix`	If you repeat this function and when X0 is fixed during iterations, please set this argument TRUE.
`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 RAINBOW. 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.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 RAINBOW. 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.
`bounds`	Lower and upper bounds for weights.
`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.
`n.thres`	If n >= n.thres, perform EMM1.cpp. Else perform EMM2.cpp.
`spectral.method`	The method of spectral decomposition. In this function, "eigen" : eigen decomposition and "cholesky" : cholesky and singular value decomposition are offered. If this argument is NULL, either method will be chosen accorsing to the dimension of Z and X.
`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.

$y.pred: The fitting values of y y = Xβ + Zu
$Vu: Estimator for σ^2_u, all of the genetic variance
$Ve: Estimator for σ^2_e
$beta: BLUE(β)
$u: BLUP(u)
$weights: The proportion of each genetic variance (corresponding to each kernel of ZETA) to Vu
$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' + λ I

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.

Lippert, C. et al. (2014) Greater power and computational efficiency for kernel-based association testing of sets of genetic variants. Bioinformatics. 30(22): 3206-3214.

  ### Import RAINBOW
  require(RAINBOW)
  
  ### 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 additive genomic relationship matrix (GRM)
  K.A <- calcGRM(genoMat = x)
  
  
  ### 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))
  
  
  ### Including the additional linear kernel for chromosome 12
  chrNo <- 12
  W.A <- x[, map$chr == chrNo]    ### marker genotype data of chromosome 12
  
  Zs0 <- list(A.part = Z)
  Ws0 <- list(A.part = W.A)       ### This will be regarded as linear kernel
  ### for the variance-covariance matrix of another random effects.
  
  
  ### Solve multi-kernel linear mixed effects model (2 random efects)
  EM3.linker.res <- EM3.linker.cpp(y0 = pheno.mat, X0 = NULL, ZETA = ZETA,
                                   Zs0 = Zs0, Ws0 = Ws0)
  (Vu <- EM3.linker.res$Vu)   ### estimated genetic variance
  (Ve <- EM3.linker.res$Ve)   ### estimated residual variance
  (weights <- EM3.linker.res$weights)   ### estimated proportion of two genetic variances
  (herit <- Vu * weights / (Vu + Ve))   ### genomic heritability (all chromosomes, chromosome 12)
  
  (beta <- EM3.linker.res$beta)   ### Here, this is an intercept.
  u <- EM3.linker.res$u   ### estimated genotypic values (all chromosomes, chromosome 12)
  See(u)