negbinomial {VGAM}R Documentation

Negative Binomial Distribution Family Function

Description

Maximum likelihood estimation of the two parameters of a negative binomial distribution.

Usage

negbinomial(lmu = "loge", lk = "loge",
            emu =list(), ek=list(),
            ik = NULL, nsimEIM=100,
            cutoff = 0.995, Maxiter=5000, 
            deviance.arg = FALSE, method.init=1,
            shrinkage.init=0.95, zero = -2)

Arguments

lmu, lk Link functions applied to the mu and k parameters. See Links for more choices. Note that the k parameter is the size argument of rnbinom etc.
emu, ek List. Extra argument for each of the links. See earg in Links for general information.
ik Optional initial values for k. If failure to converge occurs try different values (and/or use method.init). For a S-column response, ik can be of length S. A value NULL means an initial value for each response is computed internally using a range of values. This argument is ignored if used within cqo; see the iKvector argument of qrrvglm.control instead.
nsimEIM This argument is used for computing the diagonal element of the expected information matrix (EIM) corresponding to k. See CommonVGAMffArguments for more information and the note below.
cutoff Used in the finite series approximation. A numeric which is close to 1 but never exactly 1. Used to specify how many terms of the infinite series for computing the second diagonal element of the EIM are actually used. The sum of the probabilites are added until they reach this value or more (but no more than Maxiter terms allowed). It is like specifying p in an imaginary function qnegbin(p).
Maxiter Used in the finite series approximation. Integer. The maximum number of terms allowed when computing the second diagonal element of the EIM. In theory, the value involves an infinite series. If this argument is too small then the value may be inaccurate.
deviance.arg Logical. If TRUE, the deviance function is attached to the object. Under ordinary circumstances, it should be left alone because it really assumes the index parameter is at the maximum likelihood estimate. Consequently, one cannot use that criterion to minimize within the IRLS algorithm. It should be set TRUE only when used with cqo under the fast algorithm.
method.init An integer with value 1 or 2 or 3 which specifies the initialization method for the mu parameter. If failure to converge occurs try another value and/or else specify a value for shrinkage.init and/or else specify a value for ik.
shrinkage.init How much shrinkage is used when initializing mu. The value must be between 0 and 1 inclusive, and a value of 0 means the individual response values are used, and a value of 1 means the median or mean is used. This argument is used in conjunction with method.init. If convergence failure occurs try setting this argument to 1.
zero Integer valued vector, usually assigned -2 or 2 if used at all. Specifies which of the two linear/additive predictors are modelled as an intercept only. By default, the k parameter (after lk is applied) is modelled as a single unknown number that is estimated. It can be modelled as a function of the explanatory variables by setting zero=NULL. A negative value means that the value is recycled, so setting -2 means all k are intercept-only.

Details

The negative binomial distribution can be motivated in several ways, e.g., as a Poisson distribution with a mean that is gamma distributed. There are several common parametrizations of the negative binomial distribution. The one used here uses the mean mu and an index parameter k, both which are positive. Specifically, the density of a random variable Y is

f(y;mu,k) = C_{y}^{y + k - 1} [mu/(mu+k)]^y [k/(k+mu)]^k

where y=0,1,2,..., and mu > 0 and k > 0. Note that the dispersion parameter is 1/k, so that as k approaches infinity the negative binomial distribution approaches a Poisson distribution. The response has variance Var(Y)=mu*(1+mu/k). When fitted, the fitted.values slot of the object contains the estimated value of the mu parameter, i.e., of the mean E(Y).

The negative binomial distribution can be coerced into the classical GLM framework, with one of the parameters being of interest and the other treated as a nuisance/scale parameter (and implemented in the MASS library). This VGAM family function negbinomial treats both parameters on the same footing, and estimates them both by full maximum likelihood estimation. Simulated Fisher scoring is employed as the default (see the nsimEIM argument).

The parameters mu and k are independent (diagonal EIM), and the confidence region for k is extremely skewed so that its standard error is often of no practical use. The parameter 1/k has been used as a measure of aggregation.

This VGAM function handles multivariate responses, so that a matrix can be used as the response. The number of columns is the number of species, say, and setting zero=-2 means that all species have a k equalling a (different) intercept only.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Warning

The Poisson model corresponds to k equalling infinity. If the data is Poisson or close to Poisson, numerical problems will occur. Possibly choosing a log-log link may help in such cases, otherwise use poissonff.

This function is fragile; the maximum likelihood estimate of the index parameter is fraught (see Lawless, 1987). In general, the quasipoissonff is more robust than this function. Assigning values to the ik argument may lead to a local solution, and smaller values are preferred over large values when using this argument.

Yet to do: write a family function which uses the methods of moments estimator for k.

Note

Suppose the response is called ymat. The diagonal element of the expected information matrix (EIM) for parameter k involves an infinite series; consequently simulated Fisher scoring (see nsimEIM) is the default. This algorithm should definitely be used if max(ymat) is large, e.g., max(ymat) > 300 or there are any outliers in ymat. A second algorithm involving a finite series approximation can be invoked by setting nsimEIM = NULL. Then the arguments Maxiter and cutoff are pertinent.

Regardless of the algorithm used, convergence problems may occur, especially when the response has large outliers or is large in magnitude. If convergence failure occurs, try using arguments (in recommended decreasing order) nsimEIM, shrinkage.init, method.init, Maxiter, cutoff, ik, zero.

This function can be used by the fast algorithm in cqo, however, setting EqualTolerances=TRUE and ITolerances=FALSE is recommended.

In the first example below (Bliss and Fisher, 1953), from each of 6 McIntosh apple trees in an orchard that had been sprayed, 25 leaves were randomly selected. On each of the leaves, the number of adult female European red mites were counted.

Author(s)

Thomas W. Yee

References

Lawless, J. F. (1987) Negative binomial and mixed Poisson regression. The Canadian Journal of Statistics 15, 209–225.

Hilbe, J. M. (2007) Negative Binomial Regression. Cambridge: Cambridge University Press.

Bliss, C. and Fisher, R. A. (1953) Fitting the negative binomial distribution to biological data. Biometrics 9, 174–200.

See Also

quasipoissonff, poissonff, cao, cqo, zinegbinomial, posnegbinomial, invbinomial, rnbinom, nbolf.

Examples

# Example 1: apple tree data
y = 0:7
w = c(70, 38, 17, 10, 9, 3, 2, 1)
fit = vglm(y ~ 1, negbinomial, weights=w)
summary(fit)
coef(fit, matrix=TRUE)
Coef(fit)

# Example 2: simulated data with multivariate response
x = runif(n <- 500)
y1 = rnbinom(n, mu=exp(3+x), size=exp(1)) # k is size
y2 = rnbinom(n, mu=exp(2-x), size=exp(0))
fit = vglm(cbind(y1,y2) ~ x, negbinomial, trace=TRUE)
coef(fit, matrix=TRUE)

# Example 3: large counts so definitely use the nsimEIM argument
x = runif(n <- 500)
y = rnbinom(n, mu=exp(12+x), size=exp(1)) # k is size
range(y)  # Large counts
fit = vglm(y ~ x, negbinomial(nsimEIM=100), trace=TRUE)
coef(fit, matrix=TRUE)

[Package VGAM version 0.7-9 Index]