R: Penalized Minus Log Likelihood for Aster Models

penmlogl {aster}

R Documentation

Penalized Minus Log Likelihood for Aster Models

Description

Penalized minus log likelihood for an aster model, and its first and second derivative. The penalization allows for (approximate) random effects. These functions are called inside pickle, pickle1, pickle2, pickle3, and reaster.

Usage

penmlogl(parm, sigma, fixed, random, obj, y, origin, deriv = 2)
penmlogl2(parm, alpha, sigma, fixed, random, obj, y, origin)

Arguments

`parm`	for `penmlogl`, parameter value (vector of regression coefficients and rescaled random effects) at which we evaluate the penalized log likelihood. For `penmlogl2` the vector of rescaled random effects only (see next item).
`alpha`	the vector of fixed effects. For `penmlogl2`, the concatenation `c(alpha, parm)` is the same as `parm` that is supplied to `pemnmlogl`.
`sigma`	vector of square roots of variance components, one component for each group of random effects.
`fixed`	the model matrix for fixed effects. The number of rows is `nrow(obj$data)`. The number of columns is the number of fixed effects.
`random`	the model matrix or matrices for random effects. Each has the same number of rows as `fixed`. The number of columns is the number of random effects in a group. Either a matrix or a list of matrices.
`obj`	aster model object, the result of a call to `aster`.
`y`	response vector. May be omitted, in which case `obj$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`.
`deriv`	number of derivatives wanted. Allowed values are 0, 1, or 2.

Details

Consider an aster model with random effects and canonical parameter vector of the form

M \alpha + Z_1 b_1 + \cdots + Z_k b_k

where M and each Z_j are known matrices having the same row dimension, where \alpha is a vector of unknown parameters (the fixed effects) and each b_j is a vector of random effects that are supposed to be (marginally) independent and identically distributed mean-zero normal with variance sigma[j]^2.

These functions evaluate minus the “penalized log likelihood” for this model, which considers the random effects as parameters but adds a penalization term

b_1^2 / (2 \sigma_1^2) + \cdots + b_k^2 / (2 \sigma_k^2)

to minus the log likelihood.

To properly deal with random effects that are zero, random effects are rescaled by their standard deviation. The rescaled random effects are c_i = b_i / \sigma_i. If \sigma_i = 0, then the corresponding rescaled random effects c_i are also zero.

Value

a list containing some of the following components:

`value`	minus the penalized log likelihood.
`gradient`	minus the first derivative vector of the penalized log likelihood.
`hessian`	minus the second derivative matrix of the penalized log likelihood.
`argument`	the value of the `parm` argument for this function.
`scale`	the vector by which `parm` must be scaled to obtain the true random effects.
`mlogl.gradient`	gradient for evaluation of log likelihood; `gradient` is this plus gradient of penalty.
`mlogl.hessian`	hessian for evaluation of log likelihood; `hessian` is this plus hessian of penalty.

Note

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