VGAM (version 1.1-12)

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)

Value

An object of class "vglmff"

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

and vgam.

Arguments

lmean, lsd, lvar, llambda, lapar

Details at CommonVGAMffArguments. See Links for more link function choices. The second response is primarily controlled by the parameter \(\lambda_2\).

imethod, isd, ilambda, iapar

Initial values. Details at CommonVGAMffArguments.

zero

Details at CommonVGAMffArguments.

doff

Numeric of unit length, the denominator offset \(\delta>0\). A monotonic transformation \(\Delta^* = \lambda_2^{2/3} / (|\delta| + \lambda_2^{2/3})\) is taken to map the Poisson mean onto the unit interval. This argument is \(\delta\). The default reflects the property that the normal approximation to the Poisson work wells for \(\lambda_2 \geq 10\) or thereabouts, hence that value is mapped to the origin by qnorm. That's because 10**(2/3) is approximately 5. It is known that the \(\lambda_2\) rate parameter raised to the power of \(2/3\) is a transformation that approximates the normal density more closely.

Alternatively, delta may be assigned a single negative value. If so, then \(\Delta^* = \log(1 + \lambda_2) / [|\delta| + \log(1 + \lambda_2)]\) is used. For this, doff = -log1p(10) is suggested.

nnodes, copula

Details at N1binomial.

var.arg

See uninormal.

Author

T. W. Yee

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 \(\lambda_2\).

See Also

rN1pois, N1binomial, binormalcop, uninormal, poissonff, dpois.

Examples

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)