R: Profile Deviance for Parameters in a Generalized Nonlinear...

profile.gnm {gnm} | R Documentation

## 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 class 'gnm'
    profile(fitted, which = ofInterest(fitted), alpha = 0.05, maxsteps = 10,
                stepsize = NULL, trace = FALSE, ...)

### 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.

### 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 - 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.

### 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)`.

### Author(s)

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.

### References

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

### See Also

`confint.gnm`, `gnm`, `profile.glm`, `ofInterest`

### Examples

    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])

    ## End(Not run)

[Package gnm version 1.1-2 Index]