R: GAM negative binomial families

negbin {mgcv} | R Documentation

## GAM negative binomial families

### Description

The `gam` modelling function is designed to be able to use the `negbin` family (a modification of MASS library `negative.binomial` family by Venables and Ripley), or the `nb` function designed for integrated estimation of parameter `theta`. θ is the parameter such that var(y) = μ + μ^2/θ, where μ = E(y).

Two approaches to estimating `theta` are available (with `gam` only):

- With `negbin` then if ‘performance iteration’ is used for smoothing parameter estimation (see `gam`), then smoothing parameters are chosen by GCV and `theta` is chosen in order to ensure that the Pearson estimate of the scale parameter is as close as possible to 1, the value that the scale parameter should have.

- If ‘outer iteration’ is used for smoothing parameter selection with the `nb` family then `theta` is estimated alongside the smoothing parameters by ML or REML.

To use the first option, set the `optimizer` argument of `gam` to `"perf"` (it can sometimes fail to converge).

### Usage

    negbin(theta = stop("'theta' must be specified"), link = "log")
    nb(theta = NULL, link = "log")

### Arguments

`theta` | Either i) a single value known value of theta or ii) two values of theta specifying the endpoints of an interval over which to search for theta (this is an option only for `negbin`). For `nb` then a positive supplied `theta` is treated as a fixed known parameter, otherwise it is estimated (the absolute value of a negative `theta` is taken as a starting value).
`link` | The link function: one of `"log"`, `"identity"` or `"sqrt"`

### Details

`nb` allows estimation of the `theta` parameter alongside the model smoothing parameters, but is only useable with `gam` (not `bam` or `gam`).

For `negbin`, if a single value of `theta` is supplied then it is always taken as the known fixed value and this is useable with `bam` and `gamm`. If `theta` is two numbers (`theta[2]>theta[1]`) then they are taken as specifying the range of values over which to search for the optimal theta. This option should only be used with performance iteration estimation (see `gam` argument `optimizer`), in which case the method of estimation is to choose theta so that the GCV (Pearson) estimate of the scale parameter is one (since the scale parameter is one for the negative binomial). In this case theta estimation is nested within the IRLS loop used for GAM fitting. After each call to fit an iteratively weighted additive model to the IRLS pseudodata, the theta estimate is updated. This is done by conditioning on all components of the current GCV/Pearson estimator of the scale parameter except theta and then searching for the theta which equates this conditional estimator to one. The search is a simple bisection search after an initial crude line search to bracket one. The search will terminate at the upper boundary of the search region is a Poisson fit would have yielded an estimated scale parameter <1.

The following `negbin` based approaches are now deprecated:

If outer iteration is used then theta is estimated by searching for the value yielding the lowest AIC. The search is either over the supplied array of values, or is a grid search over the supplied range, followed by a golden section search. A full fit is required for each trial theta, so the process is slow, but speed is enhanced by making the changes in theta as small as possible, from one step to the next, and using the previous smothing parameter and fitted values to start the new fit.

In a simulation test based on 800 replicates of the first example data, given below, the GCV based (performance iteration) method yielded models with, on avergage 6% better MSE performance than the AIC based (outer iteration) method. `theta` had a 0.86 correlation coefficient between the two methods. `theta` estimates averaged 3.36 with a standard deviation of 0.44 for the AIC based method and 3.22 with a standard deviation of 0.43 for the GCV based method. However the GCV based method is less computationally reliable, failing in around 4% of replicates.

### Value

For `negbin` an object inheriting from class `family`, with additional elements

`dvar` | the function giving the first derivative of the variance function w.r.t. `mu`.
`d2var` | the function giving the second derivative of the variance function w.r.t. `mu`.
`getTheta` | A function for retrieving the value(s) of theta. This also useful for retriving the estimate of `theta` after fitting (see example).

For `nb` an object inheriting from class `extended.family`.

### WARNINGS

`gamm` and `bam` do not support `theta` estimation

The negative binomial functions from the MASS library are no longer supported.

### Author(s)

Simon N. Wood simon.wood@r-project.org modified from Venables and Ripley's `negative.binomial` family.

### References

Venables, B. and B.R. Ripley (2002) Modern Applied Statistics in S, Springer.

### Examples

    library(mgcv)
    set.seed(3)
    n<-400
    dat <- gamSim(1,n=n)
    g <- exp(dat$f/5)

    ## negative binomial data...
    dat$y <- rnbinom(g,size=3,mu=g)
    ## known theta fit ...
    b0 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(3),data=dat)
    plot(b0,pages=1)
    print(b0)

    ## same with theta estimation...
    b <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=nb(),data=dat)
    plot(b,pages=1)
    print(b)
    b$family$getTheta(TRUE) ## extract final theta estimate

    ## unknown theta via performance iteration...
    b1 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(c(1,10)),
              optimizer="perf",data=dat)
    plot(b1,pages=1)
    print(b1)

    ## another example...
    set.seed(1)
    f <- dat$f
    f <- f - min(f)+5;g <- f^2/10
    dat$y <- rnbinom(g,size=3,mu=g)
    b2 <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=nb(link="sqrt"),
             data=dat,method="REML")
    plot(b2,pages=1)
    print(b2)
    rm(dat)

[Package mgcv version 1.8-3 Index]