gp.reg {gp}

Generalized Poisson regression

Description

Generalized Poisson regression.

Usage

gp.reg(y, x, tol = 1e-7)
gp.reg2(y, x, tol = 1e-7)

Arguments

y

The response variable, a vector with non negative integer values.

x

A data.frame or a matrix with the independent variables.

tol

The tolerance value to terminate the optimization.

Details

The loglikelihood of the generalised Poisson distribution when covariates are present is the following (Consul & Famoye, 1992):

\ell(\beta, \phi)=\sum_{i=1}^n\log(\mu_i) + \sum_{i=1}^n(y_i-1)\log{[\mu_i+(\phi-1)y_i]}- \log{\phi}\sum_{i=1}^ny_i-\frac{1}{\phi}\sum_{i=1}^n[\mu_i+(\phi-1)y_i]-\sum_{i=1}^n\log{(y_i)},

where \mu_i=e^{\sum_{j=0}^kX_{ij}\beta_j}, n denotes the sample size, k is the number of \beta coefficients, and \phi > 0.

Breslow (1984) suggested the (moment) estimation of a dispersion parameter by equating the chi-square statistic to its degrees of freedom. For the generalised Poisson regression model, this leads to \sum_{i=1}^n\frac{(y_i-\mu_i)^2}{\mu_i\phi^2}=n-k and we solve this for \phi.

According to Consul and Famoye (1992) we begin by fitting a Poisson regression model and obtain initial values for \beta_s and \phi. If \hat{\phi} \approx 1, it implies that the Poisson regression model is appropriate and no further estimation needs to be done. However, if \hat{\phi} \neq 1, this is used to obtain new values of the estimated \beta_s by maximizing the log-likelihood. This process is iterated until we obtain a stable solution.

The function as seen below returns the log-likelihood of the initial Poisson regression as well. This is useful if one wants to test, via the log-likelihood ratio test as 1 degree of freedom, if the generalized Poisson regression is to be preferred over the Poisson regression.

gp.reg() estimates the beta coefficients using Newton-Raphson, whereas gp.reg2() uses the optim function. For some reason these two do not always agree. One might yield higher log-likelihood than the other and this is why I offer both ways.

Value

A list including:

pois.loglik

The initial Poisson regression log-likelihood.

gp.loglik

The generalized Poisson regression log-likelihood.

be

The estimated beta coefficients.

phi

The estimated \phi parameter.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Consul P.C. & Famoye F. (1992). Generalized poisson regression model. Communications in Statistics - Theory and Methods, 21(1): 89–109.

Breslow N. E. (1984). Extra-Poisson variation in log-linear models. Journal of the Royal Statistical Society: Series C (Applied Statistics), 33(1): 38–44.

See Also

gp.mle

Examples

n <- 500
x <- matrix (rnorm(n * 2), nrow = n, ncol = 2)
be <- c(1, 1)
mi <- x[, 1] * be[1] + x[, 2] * be[2] + 1
mi <- exp(mi)
y <- numeric(n)
for (i in 1:n)  y[i] <- rgp(2, mi[i], 0.5, method = "Inversion")[1]
gp.reg(y, x)
gp.reg2(y, x)