Profile Deviance for Parameters in a Generalized Nonlinear Model
For one or more parameters in a generalized nonlinear model, profile the deviance over a range of values about the fitted estimate.
## S3 method for class 'gnm'
profile(fitted, which = ofInterest(fitted), alpha = 0.05, maxsteps = 10,
stepsize = NULL, trace = FALSE, ...)fitted |
an object of class |
which |
(optional) either a numeric vector of indices or a
character vector of names, specifying the parameters over which the
deviance is to be profiled. If |
alpha |
the significance level of the z statistic, indicating the range that the profile must cover (see details). |
maxsteps |
the maximum number of steps to take either side of the fitted parameter. |
stepsize |
(optional) a numeric vector of length two, specifying
the size of steps to take when profiling down and up respectively,
or a single number specifying the step size in both directions. If
|
trace |
logical, indicating whether profiling should be traced. |
... |
further arguments. |
This is a method for the generic function profile in the
base package.
For a given parameter, the deviance is profiled by constraining that parameter to certain values either side of its estimate in the fitted model and refitting the model.
For each updated model, the following "z statistic" is computed
z(theta) = (theta - theta.hat) * sqrt((D_theta - D_theta.hat)/delta)
where theta is the constrained value of the parameter; theta.hat is the original fitted value; D_theta is the deviance when the parameter is equal to theta, and delta is the dispersion parameter.
When the deviance is quadratic in theta, z will be
linear in theta. Therefore departures from quadratic
behaviour can easily be identified by plotting z against
theta using plot.profile.gnm.
confint.profile.gnm estimates confidence intervals for the
parameters by interpolating the deviance profiles and identifying the
parameter values at which z is equal to the relevant percentage points
of the normal distribution. The alpha argument to
profile.gnm specifies the significance level of z which must be
covered by the profile. In particular, the profiling in a given
direction will stop when maxsteps is reached or two steps have
been taken in which
z(theta) > (theta - theta.hat) * z_{(1 - alpha)/2}
By default, the stepsize is
z_{(1 - alpha)/2} * s_theta.hat
where s_theta.hat is the standard error of
theta.hat. Strong asymmetry is detected and
the stepsize is adjusted accordingly, to try to ensure that the range
determined by alpha is adequately covered. profile.gnm
will also attempt to detect if the deviance is asymptotic such that
the desired significance level cannot be reached. Each profile has an
attribute "asymptote", a two-length logical vector specifying
whether an asymptote has been detected in either direction.
For unidentified parameters the profile will be NA, as such
parameters cannot be profiled.
A list of profiles, with one named component for each parameter profiled. Each profile is a data.frame: the first column, "z", contains the z statistics and the second column "par.vals" contains a matrix of parameter values, with one column for each parameter in the model.
The list has two attributes: "original.fit" containing fitted
and "summary" containing summary(fitted).
Modification of profile.glm from the MASS
package. Originally D. M. Bates and W. N. Venables, ported to R by
B. D. Ripley, adapted for "gnm" objects by Heather Turner.
Chambers, J. M. and Hastie, T. J. (1992) Statistical Models in S
set.seed(1)
### Example in which deviance is near quadratic
count <- with(voting, percentage/100 * total)
yvar <- cbind(count, voting$total - count)
classMobility <- gnm(yvar ~ -1 + Dref(origin, destination),
constrain = "delta1", family = binomial,
data = voting)
prof <- profile(classMobility, trace = TRUE)
plot(prof)
## confint similar to MLE +/- 1.96*s.e.
confint(prof, trace = TRUE)
coefData <- se(classMobility)
cbind(coefData[1] - 1.96 * coefData[2], coefData[1] + 1.96 * coefData[2])
## Not run:
### These examples take longer to run
### Another near quadratic example
RChomog <- gnm(Freq ~ origin + destination + Diag(origin, destination) +
MultHomog(origin, destination),
ofInterest = "MultHomog", constrain = "MultHomog.*1",
family = poisson, data = occupationalStatus)
prof <- profile(RChomog, trace = TRUE)
plot(prof)
## confint similar to MLE +/- 1.96*s.e.
confint(prof)
coefData <- se(RChomog)
cbind(coefData[1] - 1.96 * coefData[2], coefData[1] + 1.96 * coefData[2])
## Another near quadratic example, with more complex constraints
count <- with(voting, percentage/100 * total)
yvar <- cbind(count, voting$total - count)
classMobility <- gnm(yvar ~ -1 + Dref(origin, destination),
family = binomial, data = voting)
wts <- prop.table(exp(coef(classMobility))[1:2])
classMobility <- update(classMobility, constrain = "delta1",
constrainTo = log(wts[1]))
sum(exp(parameters(classMobility))[1:2]) #=1
prof <- profile(classMobility, trace = TRUE)
plot(prof)
## confint similar to MLE +/- 1.96*s.e.
confint(prof, trace = TRUE)
coefData <- se(classMobility)
cbind(coefData[1] - 1.96 * coefData[2], coefData[1] + 1.96 * coefData[2])
### An example showing asymptotic deviance
unidiff <- gnm(Freq ~ educ*orig + educ*dest +
Mult(Exp(educ), orig:dest),
ofInterest = "[.]educ", constrain = "[.]educ1",
family = poisson, data = yaish, subset = (dest != 7))
prof <- profile(unidiff, trace = TRUE)
plot(prof)
## clearly not quadratic for Mult1.Factor1.educ4 or Mult1.Factor1.educ5!
confint(prof)
## 2.5 % 97.5 %
## Mult(Exp(.), orig:dest).educ1 NA NA
## Mult(Exp(.), orig:dest).educ2 -0.5978901 0.1022447
## Mult(Exp(.), orig:dest).educ3 -1.4836854 -0.2362378
## Mult(Exp(.), orig:dest).educ4 -2.5792398 -0.2953420
## Mult(Exp(.), orig:dest).educ5 -Inf -0.7006889
coefData <- se(unidiff)
cbind(coefData[1] - 1.96 * coefData[2], coefData[1] + 1.96 * coefData[2])
### A far from quadratic example, also with eliminated parameters
backPainLong <- expandCategorical(backPain, "pain")
oneDimensional <- gnm(count ~ pain + Mult(pain, x1 + x2 + x3),
eliminate = id, family = "poisson",
constrain = "[.](painworse|x1)", constrainTo = c(0, 1),
data = backPainLong)
prof <- profile(oneDimensional, trace = TRUE)
plot(prof)
## not quadratic for any non-eliminated parameter
confint(prof)
coefData <- se(oneDimensional)
cbind(coefData[1] - 1.96 * coefData[2], coefData[1] + 1.96 * coefData[2])
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