profile.gnm: Profile Deviance for Parameters in a Generalized Nonlinear Model

Description

For one or more parameters in a generalized nonlinear model, profile the deviance over a range of values about the fitted estimate.

Usage

# S3 method for gnm
profile(fitted, which = ofInterest(fitted), alpha = 0.05, maxsteps = 10,
            stepsize = NULL, trace = FALSE, ...)

Value

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).

Arguments

fitted: an object of class "gnm".
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 NULL, the deviance is profiled over all parameters.
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 NULL, the step sizes will be determined automatically.
trace: logical, indicating whether profiling should be traced.
...: further arguments.

Author

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.

Details

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 $$z(\theta) = (\theta - \hat{\theta}) * \sqrt{\frac{D_{theta} - D_{\hat{theta}}}{\delta}}$$ where $\theta$ is the constrained value of the parameter; $\hat{\theta}$ 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 $$z(\theta) > (\theta - \hat{\theta}) * z_{(1 - \alpha)/2}$$

By default, the stepsize is $$z_{(1 - \alpha)/2} * s_{\hat{\theta}}$$ where $s_{\hat{\theta}}$ is the standard error of $\hat{\theta}$. 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.

References

Chambers, J. M. and Hastie, T. J. (1992) Statistical Models in S

Examples

Run this code

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])

if (FALSE) {
### 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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