| eql {EQL} | R Documentation |
The Extended Quasi-Likelihood Function
Description
Computes the Extended Quasi Likelihood (EQL) function for a given set of variance functions from a particular variance family.
Usage
eql(formula, param.space, family = powerVarianceFamily(),
phi.method = c("pearson", "mean.dev"), include.model = TRUE,
smooth.grid = 10, do.smooth = dim(family) == 1,
verbose = 1, ...)
## S3 method for class 'eql'
plot(x, do.points = (dim(x) == 1 && sum(!x$is.smoothed) <= 20),
do.ci = TRUE, alpha = 0.95, do.bw = TRUE,
show.max = TRUE, ...)
Arguments
formula |
an object of class |
param.space |
a list of parameters for which the EQL value should be evaluated. If provided as a named list, the names must equal the names of the parameters defined by the variance family. |
family |
an object of class |
phi.method |
a character string giving the name of the method
used to estimate the dispersion parameter
|
include.model |
logical. If |
x |
an object of class |
do.smooth, smooth.grid |
|
verbose |
the amount of feedback requested: ‘0’ or |
... |
further arguments to be passed to the |
do.points, show.max |
logical. If |
do.ci, alpha |
|
do.bw |
logical. If |
Details
The EQL function as defined by Nelder and Pregibon (see ‘References’) is given by:
Q^+_\theta(y,\mu) = -\frac{1}{2} \log [2\pi\phi V_\theta(y)] -
\frac{1}{2\phi} D_\theta(y,\mu),
where D_\theta() and V_\theta() denote the
deviance function and the variance function, respectively, determined by
the particular choice of the variance family.
The goal is to maximize the EQL function over \mu
and the not necessarily one-dimensional space of parameters
\theta. The function eql takes a particular finite
set of candidate parameters and computes the corresponding
EQL value for each of these parameters and returns the
maximum EQL value for the given set. That implies that the
function is only capable of capturing local maxima. If the maximum occurs
at the boundary of the set, the set of parameters may be badly chosen
and one should consider a larger set with the found maximum as an
interior point.
The plot function is an important tool to investigate the
structure of the EQL function. Confidence intervals and
confidence ellipsoids give an idea of plausible parameter values
for the variance function. The contour plot
used for two-dimensional variance families is generated using the
package lattice, which in turn relies on so called
trellis plots. Hence, for two-dimensional families the
plot function does not only generate the plot, but also
returns the plot object to allow for further modifications of the
plot. This is not true for one-dimensional variance models, which are
plotted using the R standard graphical engine.
For large parameter sets the computation may take a long time. If no feedback is chosen, the function seems to be hung up, because the function does not provide any textual feedback while computing. Hence, a minimal feedback (including a progress bar) should be chosen to have an idea of the remaining calculation time.
An explicitely given deviance function speeds up calculation. A rather large amount of the total calculation time is used to determine the numerical values of the integral in the deviance function.
Value
eql returns an object of class eql, which contains the
following components:
eql |
a numerical vector with the computed eql values for the given set of parameter values. For one-dimensional variance families (i.e. those families with only one parameter), a smoothing operation can be performed to obtain intermediate values. |
param |
a |
eql.max |
the maximum value of the eql function in the considered range. |
param.max |
a |
dim |
an integer value giving the dimension of the parameters in the underlying variance family. |
smooth |
a logical value indicating whether a smoothing operation was performed. |
is.smoothed |
a vector of logical values of the same length as
|
smooth.grid |
an integer value giving the number of points used
in the smoothing process or |
model |
if |
Note
The EQL for variance functions with V_\theta(0)=0
becomes infinite. Hence, if there are exact zeros in the data, one
should provide a variance family, which do not equate to zero at the
origin. Nelder and Pregibon propose some adjustment of V(y)
at the origin, which leads to a modified variance function.
The predefined families powerVarianceFamily and
extBinomialVarianceFamily are, however, not capable of
dealing with exact zeros, for there is no general mechanism to modify
the variance function for all possible values of the particular variance
family.
The confidence interval for one-dimensional variance families is not calculated exactly, but depends on the amount of EQL values available. Hence, if one is interested in a confidence interval, one should allow for smoothing.
The function eql does not use a direct maximization routine, but
rather do a simple maximation over a finite set. Hence, all obtained
values including confidence intervals and confidence ellipsoids have a
“local flavour” and should not be regarded as global solutions.
The confidence bounds are determined rather empirically and do heavily depend on the amount of parameter values under consideration.
Author(s)
Thorn Thaler
References
Nelder, J.A. and Pregibon, D. (1987). An extended quasi-likelihood function. Biometrika, 74, 221–232.
See Also
Examples
## Power Variance Family
# Data from Box and Cox (1964)
x <- (-1:1)
y <- c(674,370,292,338,266,210,170,118,90,1414,1198,634,1022,620,438,
442,332,220,3636,3184,2000,1568,1070,566,1140,884,360)
yarn.raw <- data.frame(expand.grid(x3=x, x2=x, x1=x), cycles=y)
yarn <- data.frame(x1=yarn.raw$x1, x2=yarn.raw$x2, x3=yarn.raw$x3,
cycles=yarn.raw$cycles)
attach(yarn)
ps.power <- list(theta=seq(1, 4, length = 20))
eq.power <- eql(cycles~x1+x2+x3, param.space=ps.power,
family=powerVarianceFamily("log"), smooth.grid=500)
plot(eq.power)
## Not run:
## Extended Binomial Variance Family
# Data from McCullagh & Nelder: GLM, p. 329
# (zeros replaced by 'NA')
site <- rep(1:9, each=10)
variety <- rep(1:10, 9)
resp <- c(0.05,NA,NA,0.10,0.25,0.05,0.50,1.30,1.50,1.50,
NA,0.05,0.05,0.30,0.75,0.30,3,7.50,1,12.70,1.25,1.25,
2.50,16.60,2.50,2.50,NA,20,37.50,26.25,2.50,0.50,0.01,
3,2.50,0.01,25,55,5,40,5.50,1,6,1.10,2.50,8,16.50,
29.50,20,43.50,1,5,5,5,5,5,10,5,50,75,5,0.10,5,5,
50,10,50,25,50,75,5,10,5,5,25,75,50,75,75,75,17.50,
25,42.50,50,37.50,95,62.50,95,95,95) / 100
ps.binomial <- list(seq(1, 2.2, length=32), seq(1, 3, length=32))
eq.binomial <- eql(resp~site*variety, param.space=ps.binomial,
family=extBinomialVarianceFamily())
plot(eq.binomial)
## End(Not run)