genpoisson0: Generalized Poisson Regression (Original Parameterization)

In VGAM: Vector Generalized Linear and Additive Models

Generalized Poisson Regression (Original Parameterization)

Description

Estimation of the two-parameter generalized Poisson distribution (original parameterization).

Usage

genpoisson0(ltheta = "loglink", llambda = "logitlink",
            itheta = NULL, ilambda = NULL, imethod = c(1, 1),
            ishrinkage = 0.95, glambda = ppoints(5),
            parallel = FALSE, zero = "lambda")

Arguments

ltheta, llambda	Parameter link functions for \theta and \lambda. See Links for more choices. In theory the \lambda parameter is allowed to be negative to handle underdispersion, however this is no longer supported, hence 0 < \lambda < 1. The \theta parameter is positive, therefore the default is the log link.
itheta, ilambda	Optional initial values for \lambda and \theta. The default is to choose values internally.
imethod	See CommonVGAMffArguments for information. Each value is an integer 1 or 2 or 3 which specifies the initialization method for each of the parameters. If failure to converge occurs try another value and/or else specify a value for ilambda and/or itheta. The argument is recycled to length 2, and the first value corresponds to theta, etc.
ishrinkage, zero	See CommonVGAMffArguments for information.
glambda, parallel	See CommonVGAMffArguments for information. Argument glambda is similar to gsigma there and is currently used only if imethod[2] = 1.

Details

The generalized Poisson distribution (GPD) was proposed by Consul and Jain (1973), and it has PMF

f(y)=\theta(\theta+\lambda y)^{y-1} \exp(-\theta-\lambda y) / y!

for 0 < \theta and y = 0,1,2,\ldots. Theoretically, \max(-1,-\theta/m) \leq \lambda \leq 1 where m (\geq 4) is the greatest positive integer satisfying \theta + m\lambda > 0 when \lambda < 0 [and then Pr(Y=y) = 0 for y > m]. However, there are problems with a negative \lambda such as it not being normalized, so this family function restricts \lambda to (0, 1).

This original parameterization is called the GP-0 by VGAM, partly because there are two other common parameterizations called the GP-1 and GP-2 (see Yang et al. (2009)), genpoisson1 and genpoisson2) that are more suitable for regression. However, genpoisson() has been simplified to genpoisson0 by only handling positive parameters, hence only overdispersion relative to the Poisson is accommodated. Some of the reasons for this are described in Scollnik (1998), e.g., the probabilities do not sum to unity when lambda is negative. To simply things, VGAM 1.1-4 and later will only handle positive lambda.

An ordinary Poisson distribution corresponds to \lambda = 0. The mean (returned as the fitted values) is E(Y) = \theta / (1 - \lambda) and the variance is \theta / (1 - \lambda)^3 so that the variance is proportional to the mean, just like the NB-1 and quasi-Poisson.

For more information see Consul and Famoye (2006) for a summary and Consul (1989) for more details.

Value

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

Warning

Although this family function is far less fragile compared to what used to be called genpoisson() it is still a good idea to monitor convergence because equidispersion may result in numerical problems; try poissonff instead. And underdispersed data will definitely result in numerical problems and warnings; try quasipoisson instead.

Note

This family function replaces genpoisson(), and some of the major changes are: (i) the swapping of the linear predictors; (ii) the change from rhobitlink to logitlink in llambda to reflect the no longer handling of underdispersion; (iii) proper Fisher scoring is implemented to give improved convergence.

Notationally, and in the literature too, don't get confused because theta (and not lambda) here really matches more closely with lambda of dpois.

This family function handles multiple responses. This distribution is potentially useful for dispersion modelling. Convergence and numerical problems may occur when lambda becomes very close to 0 or 1.

Author(s)

T. W. Yee. Easton Huch derived the EIM and it has been implemented in the weights slot.

References

Consul, P. C. and Jain, G. C. (1973). A generalization of the Poisson distribution. Technometrics, 15, 791–799.

Consul, P. C. and Famoye, F. (2006). Lagrangian Probability Distributions, Boston, USA: Birkhauser.

Jorgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall.

Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Applications. New York, USA: Marcel Dekker.

Yang, Z., Hardin, J. W., Addy, C. L. (2009). A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model. J. Statist. Plann. Infer., 139, 1514–1521.

Yee, T. W. (2020). On generalized Poisson regression. In preparation.

See Also

Genpois0, genpoisson1, genpoisson2, poissonff, negbinomial, Poisson, quasipoisson.

Examples

gdata <- data.frame(x2 = runif(nn <- 500))
gdata <- transform(gdata, y1 = rgenpois0(nn, theta = exp(2 + x2),
                                         logitlink(1, inverse = TRUE)))
gfit0 <- vglm(y1 ~ x2, genpoisson0, data = gdata, trace = TRUE)
coef(gfit0, matrix = TRUE)
summary(gfit0)

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.