Generalized Poisson Regression (GP-1 Parameterization)

Estimation of the two-parameter generalized Poisson distribution (GP-1 parameterization) which has the variance as a linear function of the mean.

Usage

genpoisson1(lmeanpar = "loglink", ldispind = "logloglink",
     parallel = FALSE, zero = "dispind",
     vfl = FALSE, Form2 = NULL,
     imeanpar = NULL, idispind = NULL, imethod = c(1, 1),
     ishrinkage = 0.95, gdispind = exp(1:4 / 2))

Arguments

lmeanpar, ldispind

Parameter link functions for \(\mu\) and \(\varphi\). They are called the mean parameter and dispersion index respectively. See Links for more choices. In theory the \(\varphi\) parameter might be allowed to be less than unity to handle underdispersion but this is not supported. The mean is positive so its default is the log link. The dispersion index is \(> 1\) so its default is the log-log link.

vfl, Form2

If vfl = TRUE then Form2 should be assigned a formula having terms comprising \(\eta_2=\log \log \varphi\). This is similar to uninormal. See CommonVGAMffArguments for information.

imeanpar, idispind

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

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.

gdispind, parallel

See CommonVGAMffArguments for information. Argument gdispind 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 is similar to the GP-1 referred to by some writers such as Yang, et al. (2009). Compared to the original GP-0 (see genpoisson0) the GP-1 has \(\theta = \mu / \sqrt{\varphi}\) and \(\lambda = 1 - 1 / \sqrt{\varphi}\) so that the variance is \(\mu \varphi\). The first linear predictor by default is \(\eta_1 = \log \mu\) so that the GP-1 is more suitable for regression than the GP-1.

This family function can handle only overdispersion relative to the Poisson. An ordinary Poisson distribution corresponds to \(\varphi = 1\). The mean (returned as the fitted values) is \(E(Y) = \mu\). For overdispersed data, this GP parameterization is a direct competitor of the NB-1 and quasi-Poisson.

Value

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

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 = rgenpois1(nn, exp(2 + x2),
                               logloglink(-1, inverse = TRUE)))
gfit1 <- vglm(y1 ~ x2, genpoisson1, gdata, trace = TRUE)
#> Iteration 1: loglikelihood = -1429.7155
#> Iteration 2: loglikelihood = -1407.8067
#> Iteration 3: loglikelihood = -1403.5445
#> Iteration 4: loglikelihood = -1403.2269
#> Iteration 5: loglikelihood = -1403.2224
#> Iteration 6: loglikelihood = -1403.2224
#> Iteration 7: loglikelihood = -1403.2224
coef(gfit1, matrix = TRUE)
#>             loglink(meanpar) logloglink(dispind)
#> (Intercept)         2.000286           -1.275609
#> x2                  1.003630            0.000000
summary(gfit1)
#> 
#> Call:
#> vglm(formula = y1 ~ x2, family = genpoisson1, data = gdata, trace = TRUE)
#> 
#> Coefficients: 
#>               Estimate Std. Error z value Pr(>|z|)    
#> (Intercept):1  2.00029    0.03327  60.121  < 2e-16 ***
#> (Intercept):2 -1.27561    0.22975  -5.552 2.82e-08 ***
#> x2             1.00363    0.05081  19.752  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Names of linear predictors: loglink(meanpar), logloglink(dispind)
#> 
#> Log-likelihood: -1403.222 on 997 degrees of freedom
#> 
#> Number of Fisher scoring iterations: 7 
#>