| quickle {aster} | R Documentation |
Penalized Quasi-Likelihood for Aster Models
Description
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.
Usage
quickle(alphanu, bee, fixed, random, obj, y, origin, zwz, deriv = 0)
Arguments
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 |
fixed |
the model matrix for fixed effects. The number of rows
is |
random |
the model matrix or matrices for random effects.
The number of rows is |
obj |
aster model object, the result of a call to |
y |
response vector. May be omitted, in which case |
origin |
origin of aster model. May be omitted, in which case
default origin (see |
zwz |
A possible value of |
deriv |
Number of derivatives wanted, zero, one, or two. |
Details
Define
p(\alpha, b, \nu) = m(a + M \alpha + Z b) + {\textstyle \frac{1}{2}} b^T D^{- 1} b + {\textstyle \frac{1}{2}} \log \det[Z^T W Z D + I]
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
(Z^T 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 \nu.
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 \nu is the length of the argument random
of this function if it is a list and is one otherwise.
Let b^* denote the minimizer
of p(\alpha, b, \nu) considered as a function of
b for fixed \alpha and \nu, so b^*
is a function of \alpha and \nu.
This function evaluates
q(\alpha, \nu) = p(\alpha, b^*, \nu)
and its gradient vector and Hessian matrix (if requested).
Note that b^* is a function of \alpha
and \nu although the notation does not explicitly indicate this.
Value
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 (b^* in the notation in details),
not the argument bee of this function.
Note
Not intended for use by naive users. Use summary.reaster,
which calls it.
Examples
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)