quickle: Penalized Quasi-Likelihood for Aster Models
In aster: Aster Models

Description Usage Arguments Details Value Note Examples

Evaluates the objective function for approximate maximum likelihood for an aster model with random effects. Uses Laplace approximation to integrate out the random effects analytically. The “quasi” in the title is a misnomer in the context of aster models but the acronym PQL for this procedure is well-established in the generalized linear mixed models literature.

1	quickle(alphanu, bee, fixed, random, obj, y, origin, zwz, deriv = 0)

`alphanu`	the parameter vector value where the function is evaluated, a numeric vector, see details.
`bee`	the random effects vector that is used as the starting point for the inner optimization, which maximizes the penalized log likelihood to find the optimal random effects vector matching `alphanu`.
`fixed`	the model matrix for fixed effects. The number of rows is `nrow(obj$data)`. The number of columns is the number of fixed effects.
`random`	the model matrix or matrices for random effects. The number of rows is `nrow(obj$data)`. The number of columns is the number of random effects in a group. Either a matrix or a list each element of which is a matrix.
`obj`	aster model object, the result of a call to `aster`.
`y`	response vector. May be omitted, in which case `obj$x` is used. If supplied, must be a matrix of the same dimensions as `obj$x`.
`origin`	origin of aster model. May be omitted, in which case default origin (see `aster`) is used. If supplied, must be a matrix of the same dimensions `obj$x`.
`zwz`	A possible value of t(Z) W Z, where Z is the model matrix for all random effects and W is the variance matrix of the response. See details. Typically constructed by the function `makezwz`.
`deriv`	Number of derivatives wanted, zero, one, or two.

Define

p(alpha, b, nu) = m(a + M alpha + Z b) + t(b) D^(- 1) b / 2 + log det[t(Z) W Z D + I] / 2

where m is minus the log likelihood function of a saturated aster model, where a is a known vector (the offset vector in the terminology of glm but the origin in the terminology of aster), where M is a known matrix, the model matrix for fixed effects (the argument fixed of this function), where Z is a known matrix, the model matrix for random effects (either the argument random of this function if it is a matrix or Reduce(cbind, random) if random is a list of matrices), where D is a diagonal matrix whose diagonal is the vector rep(nu, times = nrand) where nrand is sapply(random, ncol) when random is a list of matrices and ncol(random) when random is a matrix, where W is an arbitrary symmetric positive semidefinite matrix (t(Z) W Z is the argument zwz of this function), and where I is the identity matrix. Note that D is a function of nu although the notation does not explicitly indicate this.

The argument alphanu of this function is the concatenation of the parameter vectors alpha and ν. The argument bee of this function is a possible value of b. The length of alpha is the column dimension of M. The length of b is the column dimension of Z. The length of ν is the length of the argument random of this function if it is a list and is one otherwise.

Let bstar denote the minimizer of p(alpha, b, nu) considered as a function of b for fixed alpha and nu, so bstar is a function of alpha and nu. This function evaluates

q(alpha, nu) = p(alpha, bstar, nu)

and its gradient vector and Hessian matrix (if requested). Note that bstar is a function of alpha and nu although the notation does not explicitly indicate this.

a list with some of the following components: value, gradient, hessian, alpha, bee, nu. The first three are the requested derivatives. The second three are the corresponding parameter values: alpha and nu are the corresponding parts of the argument alphanu, the value of bee is the result of the inner optimization (bstar in the notation in details), not the argument bee of this function.

Not intended for use by naive users. Use summary.reaster, which calls it.

data(radish)

pred <- c(0,1,2)
fam <- c(1,3,2)

rout <- reaster(resp ~ varb + fit : (Site * Region),
    list(block = ~ 0 + fit : Block, pop = ~ 0 + fit : Pop),
    pred, fam, varb, id, root, data = radish)

alpha.mle <- rout$alpha
bee.mle <- rout$b
nu.mle <- rout$sigma^2
zwz.mle <- rout$zwz
obj <- rout$obj
fixed <- rout$fixed
random <- rout$random
alphanu.mle <- c(alpha.mle, nu.mle)

qout <- quickle(alphanu.mle, bee.mle, fixed, random, obj,
    zwz = zwz.mle, deriv = 2)