N1poisson: Linear Model and Poisson Mixed Data Type Family Function

Description

Estimate the four parameters of the (bivariate) \(N_1\)--Poisson copula mixed data type model by maximum likelihood estimation.

Usage

N1poisson(lmean = "identitylink", lsd = "loglink",
    lvar = "loglink", llambda = "loglink", lapar = "rhobitlink",
    zero = c(if (var.arg) "var" else "sd", "apar"),
    doff = 5, nnodes = 20, copula = "gaussian",
    var.arg = FALSE, imethod = 1, isd = NULL,
    ilambda = NULL, iapar = NULL)

Arguments

Value

An object of class "vglmff"

(see vglmff-class). The object is used by modelling functions such as vglm

and vgam.

Details

The bivariate response comprises \(Y_1\) from a linear model having parameters mean and sd for \(\mu_1\) and \(\sigma_1\), and the Poisson count \(Y_2\) having parameter lambda for its mean \(\lambda_2\). The joint probability density/mass function is \(P(y_1, Y_2 = y_2) = \phi_1(y_1; \mu_1, \sigma_1) \exp(-h^{-1}(\Delta)) [h^{-1}(\Delta)]^{y_2} / y_2!\) where \(\Delta\) adjusts \(\lambda_2\) according to the association parameter \(\alpha\). The quantity \(\Delta\) is \(\Phi((\Phi^{-1}(h(\lambda_2)) - \alpha Z_1) / \sqrt{1 - \alpha^2})\) where \(h\) maps \(\lambda_2\) onto the unit interval. The quantity \(Z_1\) is \((Y_1-\mu_1) / \sigma_1\). Thus there is an underlying bivariate normal distribution, and a copula is used to bring the two marginal distributions together. Here, \(-1 < \alpha < 1\), and \(\Phi\) is the cumulative distribution function pnorm of a standard univariate normal.

The first marginal distribution is a normal distribution for the linear model. The second column of the response must have nonnegative integer values. When \(\alpha = 0\) then \(\Delta=\Delta^*\). Together, this family function combines uninormal and poissonff. If the response are correlated then a more efficient joint analysis should result.

The second marginal distribution allows for overdispersion relative to an ordinary Poisson distribution---a property due to \(\alpha\).

This VGAM family function cannot handle multiple responses. Only a two-column matrix is allowed. The two-column fitted value matrix has columns \(\mu_1\) and \(\lambda_2\).

Examples

Run this code

apar <- rhobitlink(0.3, inverse = TRUE)
nn <- 1000; mymu <- 1; sdev <- exp(1)
lambda <- loglink(1, inverse = TRUE)
mat <- rN1pois(nn, mymu, sdev, lambda, apar)
npdata <- data.frame(y1 = mat[, 1], y2 = mat[, 2])
with(npdata, var(y2) / mean(y2))  # Overdispersion
fit1 <- vglm(cbind(y1, y2) ~ 1, N1poisson,
             npdata, trace = TRUE)
coef(fit1, matrix = TRUE)
Coef(fit1)
head(fitted(fit1))
summary(fit1)
confint(fit1)