# quasipois: Quasi-Likelihood Model for Counts In aod: Analysis of Overdispersed Data

## Description

The function fits the log linear model (“Procedure II”) proposed by Breslow (1984) accounting for overdispersion in counts y.

## Usage

 `1` ```quasipois(formula, data, phi = NULL, tol = 0.001) ```

## Arguments

 `formula` A formula for the fixed effects. The left-hand side of the formula must be the counts `y` i.e., positive integers (`y >= 0`). The right-hand side can involve an offset term. `data` A data frame containing the response (`y`) and explanatory variable(s). `phi` When `phi` is NULL (the default), the overdispersion parameter φ is estimated from the data. Otherwise, its value is considered as fixed. `tol` A positive scalar (default to 0.001). The algorithm stops at iteration r + 1 when the condition χ{^2}[r+1] - χ{^2}[r] <= tol is met by the chi-squared statistics .

## Details

For a given count y, the model is:

y | λ ~ Poisson(λ)

with λ a random variable of mean E[λ] = μ and variance Var[λ] = φ * μ^2.
The marginal mean and variance are:

E[y] = μ

Var[y] = μ + φ * μ^2

The function uses the function `glm` and the parameterization: μ = exp(X b) = exp(η), where X is a design-matrix, b is a vector of fixed effects and η = X b is the linear predictor.
The estimate of b maximizes the quasi log-likelihood of the marginal model. The parameter φ is estimated with the moment method or can be set to a constant (a regular glim is fitted when φ is set to 0). The literature recommends to estimate φ with the saturated model. Several explanatory variables are allowed in b. None is allowed in φ.
An offset can be specified in the argument `formula` to model rates y/T (see examples). The offset and the marginal mean are log(T) and μ = exp(log(T) + η), respectively.

## Value

An object of formal class “glimQL”: see `glimQL-class` for details.

## Author(s)

Matthieu Lesnoff [email protected], Renaud Lancelot [email protected]

## References

Breslow, N.E., 1984. Extra-Poisson variation in log-linear models. Appl. Statist. 33, 38-44.
Moore, D.F., Tsiatis, A., 1991. Robust estimation of the variance in moment methods for extra-binomial and extra-poisson variation. Biometrics 47, 383-401.

`glm`, `negative.binomial` in the recommended package MASS, `geese` in the contributed package geepack, `glm.poisson.disp` in the contributed package dispmod.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ``` # without offset data(salmonella) quasipois(y ~ log(dose + 10) + dose, data = salmonella) quasipois(y ~ log(dose + 10) + dose, data = salmonella, phi = 0.07180449) summary(glm(y ~ log(dose + 10) + dose, family = poisson, data = salmonella)) quasipois(y ~ log(dose + 10) + dose, data = salmonella, phi = 0) # with offset data(cohorts) i <- cohorts\$age ; levels(i) <- 1:7 j <- cohorts\$period ; levels(j) <- 1:7 i <- as.numeric(i); j <- as.numeric(j) cohorts\$cohort <- j + max(i) - i cohorts\$cohort <- as.factor(1850 + 5 * cohorts\$cohort) fm1 <- quasipois(y ~ age + period + cohort + offset(log(n)), data = cohorts) fm1 quasipois(y ~ age + cohort + offset(log(n)), data = cohorts, phi = fm1@phi) ```