EM3.general: Equation of mixed model for multi-kernel including using...

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

Equation of mixed model for multi-kernel including using other packages (with other packages, much faster than EM3.cpp)

Description

This function solves the following multi-kernel linear mixed effects model using MMEst function in 'MM4LMM' package, lmm.aireml or lmm.diago functions in 'gaston' package, or EM3.cpp function in 'RAINBOWR' package.

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

where Var[y] = \sum _{l=1} ^ {L} Z _ {l} K _ {l} Z _ {l}' \sigma _ {l} ^ 2 + I \sigma _ {e} ^ {2}.

Usage

EM3.general(
  y,
  X0 = NULL,
  ZETA,
  eigen.G = NULL,
  package = "gaston",
  tol = NULL,
  n.core = 1,
  optimizer = "nlminb",
  REML = TRUE,
  pred = TRUE,
  return.u.always = TRUE,
  return.u.each = TRUE,
  return.Hinv = TRUE,
  recheck.RAINBOWR = TRUE,
  var.ratio.range = c(1e-09, 1e+07)
)

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"!
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.
package	Package name to be used in this function. We only offer the following three packages: "RAINBOWR", "MM4LMM" and "gaston". Default package is 'gaston'.
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. (‘n.core' will be replaced by 1 for 'package = ’gaston'')
optimizer	The function used in the optimization process. We offer "optim", "optimx", and "nlminb" functions. This argument is only valid when ‘package = ’RAINBOWR''.
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	When using the "gaston" package with missing values or using the "MM4LMM" package (with/without missings), computing BLUP will take some time in addition to solving the mixed-effects model. You can choose whether BLUP ('u'; u) will be returned or not.
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' when using packages other than 'RAINBOWR'.
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}. It will take some time in addition to solving the mixed-effects model when using packages other than 'RAINBOWR'.
recheck.RAINBOWR	When you use the package other than 'RAINBOWR' and the ratio of variance components is out of the range of 'var.ratio.range', the function will solve the mixed-effects model again with 'RAINBOWR' package, if 'recheck.RAINBOWR = TRUE'.
var.ratio.range	The range of variance components to check that the results by the package other than RAINBOWR is correct or not when 'recheck.RAINBOWR = TRUE'.

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

$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.

Johnson, D. L., & Thompson, R. (1995). Restricted maximum likelihood estimation of variance components for univariate animal models using sparse matrix techniques and average information. Journal of dairy science, 78(2), 449-456.

Hunter, D. R., & Lange, K. (2004). A tutorial on MM algorithms. The American Statistician, 58(1), 30-37.

Zhou, H., Hu, L., Zhou, J., & Lange, K. (2015). MM algorithms for variance components models. arXiv preprint arXiv:1509.07426.

Gilmour, A. R., Thompson, R., & Cullis, B. R. (1995), Average information REML: an efficient algorithm for variance parameter estimation in linear mixed models, Biometrics, 1440-1450.

See Also

MMEst, lmm.aireml, lmm.diago

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
  
  
  ### Estimate additive genomic relationship matrix (GRM) & epistatic relationship matrix
  K.A <- calcGRM(genoMat = x) 
  K.AA <- K.A * K.A   ### additive x additive epistatic effects
  
  
  ### 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),
               AA = list(Z = Z, K = K.AA))
  
  
  ### Solve multi-kernel linear mixed effects model using gaston package (2 random efects)
  EM3.gaston.res <- EM3.general(y = pheno.mat, X0 = NULL, ZETA = ZETA,
                                package = "gaston", return.u.always = TRUE,
                                pred = TRUE, return.u.each = TRUE,
                                return.Hinv = TRUE)
  (Vu <- EM3.gaston.res$Vu)   ### estimated genetic variance
  (Ve <- EM3.gaston.res$Ve)   ### estimated residual variance
  (weights <- EM3.gaston.res$weights)   ### estimated proportion of two genetic variances
  (herit <- Vu * weights / (Vu + Ve))   ### genomic heritability (additive, additive x additive)
  
  (beta <- EM3.gaston.res$beta)   ### Here, this is an intercept.
  u.each <- EM3.gaston.res$u.each   ### estimated genotypic values (additive, additive x additive)
  See(u.each)
  
  
  ### Perform genomic prediction with 10-fold cross validation using gaston package (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 <- rep(NA, nLine)
  
  for (noCV in 1:nFold) {
    print(paste0("Fold: ", noCV))
    yTrain <- phenoNoNA
    yTrain[idCV == noCV, ] <- NA   ### prepare test data
    
    EM3.gaston.resCV <- EM3.general(y = yTrain, X0 = NULL, ZETA = ZETANoNA,
                                    package = "gaston", return.u.always = TRUE,
                                    pred = TRUE, return.u.each = TRUE,
                                    return.Hinv = TRUE)   ### prediction
    yTest <-  EM3.gaston.resCV$y.pred     ### predicted values
    
    yPred[idCV == noCV] <- yTest[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 (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)

RAINBOWR documentation built on Sept. 12, 2023, 9:08 a.m.