Generalized Poisson Regression (GP-2 Parameterization)

Estimation of the two-parameter generalized Poisson distribution (GP-2 parameterization) which has the variance as a cubic function of the mean.

Usage

genpoisson2(lmeanpar = "loglink", ldisppar = "loglink",
    parallel = FALSE, zero = "disppar",
    vfl = FALSE, oparallel = FALSE,
    imeanpar = NULL, idisppar = NULL, imethod = c(1, 1),
    ishrinkage = 0.95, gdisppar = exp(1:5))

Arguments

lmeanpar, ldisppar

Parameter link functions for \(\mu\) and \(\alpha\). They are called the mean and dispersion parameters respectively. See Links for more choices. In theory the \(\alpha\) parameter might be allowed to be negative to handle underdispersion but this is not supported. All parameters are positive, therefore the defaults are the log link.

imeanpar, idisppar

Optional initial values for \(\mu\) and \(\alpha\). The default is to choose values internally.

vfl, oparallel

Argument oparallel is similar to parallel but uses rbind(1, -1) instead. If vfl = TRUE then oparallel should be assigned a formula having terms comprising \(\eta_1=\log \mu\), and then the other terms in the main formula are for \(\eta_2=\log \alpha\) . See CommonVGAMffArguments for information.

imethod

See CommonVGAMffArguments for information. The argument is recycled to length 2, and the first value corresponds to \(\mu\), etc.

ishrinkage, zero

See CommonVGAMffArguments for information.

gdisppar, parallel

See CommonVGAMffArguments for information. Argument gdisppar is similar to gsigma there and is currently used only if imethod[2] = 2.

Details

This is a variant of the generalized Poisson distribution (GPD) and called GP-2 by some writers such as Yang, et al. (2009). Compared to the original GP-0 (see genpoisson0) the GP-2 has \(\theta = \mu / (1 + \alpha \mu)\) and \(\lambda = \alpha \mu / (1 + \alpha \mu)\) so that the variance is \(\mu (1 + \alpha \mu)^2\). The first linear predictor by default is \(\eta_1 = \log \mu\) so that the GP-2 is more suitable for regression than the GP-0.

This family function can handle only overdispersion relative to the Poisson. An ordinary Poisson distribution corresponds to \(\alpha = 0\). The mean (returned as the fitted values) is \(E(Y) = \mu\).

Value

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

References

Letac, G. and Mora, M. (1990). Natural real exponential familes with cubic variance functions. Annals of Statistics 18, 1–37.

Warning

See genpoisson0 for warnings relevant here, e.g., it is a good idea to monitor convergence because of equidispersion and underdispersion.

Author

T. W. Yee.

Examples

gdata <- data.frame(x2 = runif(nn <- 500))
gdata <- transform(gdata, y1 = rgenpois2(nn, exp(2 + x2),
                               loglink(-1, inverse = TRUE)))
gfit2 <- vglm(y1 ~ x2, genpoisson2, gdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1729.6661
#> Iteration 2: loglikelihood = -1728.2174
#> Iteration 3: loglikelihood = -1728.1782
#> Iteration 4: loglikelihood = -1728.1774
#> Iteration 5: loglikelihood = -1728.1774
#> Iteration 6: loglikelihood = -1728.1774
coef(gfit2, matrix = TRUE)
#>             loglink(meanpar) loglink(disppar)
#> (Intercept)        2.0166132        -1.051532
#> x2                 0.9414887         0.000000
summary(gfit2)
#> 
#> Call:
#> vglm(formula = y1 ~ x2, family = genpoisson2, data = gdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1   2.0166     0.1229  16.415  < 2e-16 ***
#> (Intercept):2  -1.0515     0.0513 -20.500  < 2e-16 ***
#> x2              0.9415     0.2346   4.014 5.97e-05 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(meanpar), loglink(disppar)
#> 
#> Log-likelihood: -1728.177 on 997 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 6 
#>