penmlogl: Penalized Minus Log Likelihood for Aster Models

In aster: Aster Models

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.

See Also

For an example using this function see the example for pickle.

aster documentation built on May 29, 2024, 2:09 a.m.