N1poisson: Linear Model and Poisson Mixed Data Type Family Function

In VGAM: Vector Generalized Linear and Additive Models

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

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.

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.

Value

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

Note

This VGAM family function is based on N1binomial and shares many properties with it. It pays to set trace = TRUE to monitor convergence, especially when abs(apar) is high.

Author(s)

T. W. Yee

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)

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