varComprob: Fitting variance component models using robust procedures

Description

varComprob and varComprob.fit fit linear mixed-effect models where the marginal variance-covariance matrix is linear in known positive semidefinite matrices. varComprob uses a formula interface, whereas varComprob.fit is the underlying working horse.

Usage

varComprob(fixed, data, random, groups, varcov, weights, subset,
  family = stats::gaussian("identity"), na.action, offset,
  control = varComprob.control(...), doFit = TRUE,
  normalizeTrace = FALSE, contrasts = NULL,
  model = TRUE, X = TRUE, Y = TRUE, K = TRUE, ...)
varComprob.fit(Y, X, V, control = varComprob.control(), ...)

Value

The value of any of the two functions is a list with class

varComprob or varComprob.fit respectively. The following elements are present in both

call: the actual call.
beta: a named numeric vector of parameter estimates. These are the estimated of the fixed effect parameters.
vcov.beta: estimated variance-covariance matrix of the estimated fixed effects parameters.
eta: a named numeric vector of parameter estimates. These are the estimated variance components.
vcov.eta: estimated variance-covariance matrix of the estimated random effects variance parameters.
gamma: a named numeric vector of parameter estimates. These are the estimated ratio of each variance component relative to the error variance.
vcov.gamma: estimated variance-covariance matrix of the estimated ratio of random effects variance parameters with respect the estimated error variance.
eta0: the estimated error variance.
resid: residuals in matrix form $p \times n$.
weights: final weights of the iterative fixed point equations.
dotweights: another type of weight. See references for details.
Sigma: an estimates of the variance-covariance marginal matrix.
scales: the scales in case of composite robust procedures otherwise NULL.
scale: the scale in case of classical robust procedures otherwise NULL.
min: the minimum attains by the goal function.
scale0:
initial.values: a list with the following components: beta: initial value for the fixed parameters; gamma: initial value for the ratio of each variance component relative to the error variance; eta0: initial value for the errot variance; scales: initial value for the scales in case of composite robust methods otherwise is not available; scale: initial value for the scale in case of classic robust methods otherwise is not available.
iterations: number of iterations.
control: the control argument.
method: the robust method used to perform the estimation.

The function varComprob returns also

fixed: the same as beta.
parms: the same as gamma.
sigma2: the same as eta0.
nobs: the number of observations.
na.action: the na.action used in the model frame.
offset: the same as input.
contrasts: the contrast used in the fixed-effect design matrix.
random.labels: the labels used to differentiate random effects. Note that the error variance is not included here. It is safe to check the number of variance components specified by the model by checking the length of random.labels.

Arguments

fixed: A two-sided formula specifying the fixed effects of the model. This is the same as in stats::lm.
data: An optional data frame, list or environment containing the variables in the model. If not found in data, the variables are taken from environment(fixed), typically the environment from which varComprob is called. This is the same as the data argument of stats::lm.
random: This argument is not yet used.
groups: a numeric matrix with two columns which is used to group appropriately the observations according to subject. See details below and the example.
varcov: An list of symmetric positive semidefinite matrices. The weighted sum of these matrices represent the contribution of random effects to the marginal variance of the response variable, with unknown weights representing variance components.
weights: This argument is not yet used.
subset: An optional vector specifying a subset of observations to be used in the fitting process.
family: The same as the family argument of stats::glm. However, only gaussian('identity') is supported currently.
na.action: The same as in stats::lm.
offset: The same as in stats::glm. These offsets are assumed as known fixed effects.
control: Usually an object from calling varComprob.control.
doFit: This argument is not yet used.
normalizeTrace: A logical scalar, indicating whether the individual variance-covariance matrices should be normalized such that variance components are on the same scale.
contrasts: The same as in stats::lm.
model: A logical scalar, indicating whether the model frame will be included in the result.
X: For varComprob, this is a logical scalar, indicating whether the fixed-effect design matrix should be included in the result. For varComprob.fit this is the optional numeric array containing the fixed effect design matrix for the model, see details below. If X is missing or an array with zero dimension, it is assumed that Y has zero mean.
Y: For varComprob, this is a logical scalar, indicating whether the response variable should be included in the result. For varComprob.fit, this is a numeric matrix of response variables. See details below.
K: This is a logical scalar, indicating whether the list of variance-covariance matrices should be included in the result. These matrices are those specified by varcov. See details below.
V: This is a numeric array containing the variance-covariance matrices. See details below.
...: When control is given, this is ignored. Otherwise, these arguments are passed to varComprob.control and the result will be used as the control argument.

Author

Claudio Agostinelli and Victor J. Yohai

Details

The variance component model is of form $$\mathbf{y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{e}$$ where $\mathbf{e}$ is multivariate normally distributed with mean zero and variance-covariance matrix $\mathbf{\Sigma}$ being $$\mathbf{\Sigma} = \sum_{j=1}^R \sigma_j^2 \mathbf{K}_j + \sigma_e^2 \mathbf{I}$$ in which $\mathbf{K}_j$ are known positive semidefinite matrices and $\mathbf{I}$ is the identity matrix.

In the varComprob formula interface, the $\mathbf{X}$ matrix and response variable are specified by the fixed argument. The varcov argument specifies each variance-covariance matrix in a list.

In varComprob.fit, let $p$ the number of observations, $n$ the number of independent replicates and $k$ the number of regressors. Then Y must be a matrix of dimension $p \times n$, X must be an array of dimension $p \times n \times k$ and V must be an array of dimension $p \times p \times R$, where $R$ is the number of variance-covariance matrices.

The model fitting process is performed by a robust procedures. See references for more details.

See varComprob.control for arguments controlling the modeling fitting.

References

C. Agostinelli and V.J. Yohai (2014) Composite Robust Estimators for Linear Mixed Models. arXiv:1407.2176.

M.P. Victoria-Feser and Copt (2006) High Breakdown Inference in the Mixed Linear Model. Journal of American Statistical Association, 101, 292-300.

Examples

Run this code

  if (!require(nlme))
    stop()
  data(Orthodont)

  z1 <- rep(1, 4)
  z2 <- c(8,10,12,14)
  K <- list()
  K[[1]] <- tcrossprod(z1,z1) ## Int
  K[[2]] <- tcrossprod(z1,z2) + tcrossprod(z2,z1) ## Int:age
  K[[3]] <- tcrossprod(z2,z2) ## age
  names(K) <- c("Int", "Int:age", "age")
  p <- 4
  n <- 27
  groups <- cbind(rep(1:p, each=n), rep((1:n), p))

  if (FALSE) {

  ## Composite S
  OrthodontCompositeS <- varComprob(distance ~ age*Sex, groups = groups,
      data = Orthodont, varcov = K,
      control=varComprob.control(method="compositeS", lower=c(0,-Inf,0)))
  }

  ## Composite Tau
  OrthodontCompositeTau <- varComprob(distance ~ age*Sex, groups = groups,
      data = Orthodont, varcov = K,
      control=varComprob.control(lower=c(0,-Inf,0)))

  if (FALSE) {

  summary(OrthodontCompositeTau)

  ## Classic S
  OrthodontS <- varComprob(distance ~ age*Sex, groups = groups,
      data = Orthodont, varcov = K,
      control=varComprob.control(lower=c(0,-Inf,0),
      method="S", psi="rocke"))

  summary(OrthodontS)
  }
  if (FALSE) {

  if (!require(WWGbook))
    stop()
  if (!require(nlme))
    stop()
  data(autism)
  autism <- autism[complete.cases(autism),]
  completi <- table(autism$childid)==5
  completi <- names(completi[completi])
  indici <- as.vector(unlist(sapply(completi,
              function(x) which(autism$childid==x))))
  ind <- rep(FALSE, nrow(autism))
  ind[indici] <- TRUE
  autism <- subset(autism, subset=ind) ## complete cases 41
  attach(autism)
  sicdegp.f <- factor(sicdegp)
  age.f <- factor(age)
  age.2 <- age - 2
  sicdegp2 <- sicdegp
  sicdegp2[sicdegp == 3] <- 0
  sicdegp2[sicdegp == 2] <- 2 
  sicdegp2[sicdegp == 1] <- 1
  sicdegp2.f <- factor(sicdegp2)
  autism.updated <- subset(data.frame(autism, 
                    sicdegp2.f, age.2), !is.na(vsae))
  autism.grouped <- groupedData(vsae ~ age.2 | childid, 
                    data=autism.updated, order.groups = FALSE)
  p <- 5
  n <- 41
  z1 <- rep(1, p)
  z2 <- c(0, 1, 3, 7, 11)
  z3 <- z2^2
  K <- list()
  K[[1]] <- tcrossprod(z1,z1)
  K[[2]] <- tcrossprod(z2,z2)
  K[[3]] <- tcrossprod(z3,z3)
  K[[4]] <- tcrossprod(z1,z2) + tcrossprod(z2,z1)
  K[[5]] <- tcrossprod(z1,z3) + tcrossprod(z3,z1)
  K[[6]] <- tcrossprod(z3,z2) + tcrossprod(z2,z3)
  names(K) <- c("Int", "age", "age2", "Int:age", "Int:age2", "age:age2")

  groups <- cbind(rep(1:p, each=n), rep((1:n), p))

  ## Composite Tau
  AutismCompositeTau <- varComprob(vsae ~ age.2 + I(age.2^2)
    + sicdegp2.f + age.2:sicdegp2.f + I(age.2^2):sicdegp2.f, 
    groups = groups,
    data = autism.grouped, varcov = K,
    control=varComprob.control(
    lower=c(0.01,0.01,0.01,-Inf,-Inf,-Inf)))

  summary(AutismCompositeTau)

  ## Classic S
  AutismS <- varComprob(vsae ~ age.2 + I(age.2^2)
    + sicdegp2.f + age.2:sicdegp2.f + I(age.2^2):sicdegp2.f, 
    groups = groups,
    data = autism.grouped, varcov = K,
    control=varComprob.control(
    method="S", psi="rocke", cov.init="covOGK",
    lower=c(0.01,0.01,0.01,-Inf,-Inf,-Inf)))
  summary(AutismS)
  }

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