mgcv (version 1.6-0)

negbin: GAM negative binomial family

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), with or without a known $\theta$ parameter. Two approaches to estimating the theta parameter are available:

• If `performance iteration' is used for smoothing parameter estimation (seegam), then smoothing parameters are chosen by GCV andthetais 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, and smoothing parameters are chosen by UBRE/AIC (with scale parameter set to 1) then a value ofthetais searched for which minimizes the AIC of the model. Alternatively If (RE)ML is used for smoothing parameter estimation then a value ofthetais searched for which maximizes the (restricted) likelihood.

The second option is much slower than the first, but the first can sometimes fail to converge. To use the first option, set the optimizer argument of gam to "perf".

Usage

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

Arguments

theta

Either i) a single value known value of theta, ii) two values of theta specifying the endpoints of an interval over which to search for theta or iii) an array of values of theta, specifying the set of theta values to search. (iii) is only available with

link

The link function: one of "log", "identity" or "sqrt"

Value

• 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).

WARNINGS

gamm does not support theta estimation

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

Details

If a single value of theta is supplied then it is always taken as the known fixed value, and estimation of smoothing paramaters is then by UBRE/AIC. 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. If theta is any other array of numbers then they are taken as the discrete set of values of theta over which to search for theta. The latter option only works with AIC based outer iteration, if performance iteration is used then an array will only be used to define a search range.

If performance iteration is used (see gam argument optimizer) then the method of estimation is to choose $\hat \theta$ 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 $\hat \theta$ 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.< p="">

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.

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 ...
b <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(3),data=dat)
plot(b,pages=1)
print(b)

## 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)

## unknown theta via outer iteration and AIC search...
b2<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(c(1,10)),
        data=dat)
plot(b2,pages=1)
print(b2)

## Same again all by  REML...
b2a <- gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(c(1,10)),
        data=dat,method="REML")
plot(b2a,pages=1)
print(b2a)


## how to retrieve Theta...
b2a$family$getTheta()

## unknown theta via outer iteration and AIC search
## over a discrete set of values...
b3<-gam(y~s(x0)+s(x1)+s(x2)+s(x3),family=negbin(2:10/2),
        data=dat)
plot(b3,pages=1)
print(b3)

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