# Quasi-Likelihood Model for Counts The function fits the log linear model (Procedure II ) proposed by Breslow (1984) accounting for overdispersion in counts $y$. ```r 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 $\phi$ 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 $\chi{^2}[r+1] - \chi{^2}[r] <= tol$ is met by the $chi-squared$ statistics . ## Details For a given count $y$, the model is: $$ y~|~\lambda \sim Poisson(~\lambda)y | \lambda ~ Poisson(\lambda) $$ with $\lambda$ a random variable of mean $E[\lambda] = \mu$ and variance $Var[\lambda] = \phi * \mu^2$. The marginal mean and variance are: $$ E[y] = \mu $$ $$ Var[y] = \mu + \phi * \mu^2 $$ The function uses the function `glm` and the parameterization: $\mu = exp(X b) = exp(\eta)$, where $X$ is a design-matrix, $b$ is a vector of fixed effects and $\eta = X b$ is the linear predictor. The estimate of $b$ maximizes the quasi log-likelihood of the marginal model. The parameter $\phi$ is estimated with the moment method or can be set to a constant (a regular **glim** is fitted when $\phi$ is set to 0). The literature recommends to estimate $\phi$ with the saturated model. Several explanatory variables are allowed in $b$. None is allowed in $\phi$. 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 $\mu = exp(log(T) + \eta)$, respectively. ## Returns An object of formal class glimQL : see `glimQL-class` for details. ## 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. ## Author(s) Matthieu Lesnoff matthieu.lesnoff@cirad.fr , Renaud Lancelot renaud.lancelot@cirad.fr ## See Also `glm`, `negative.binomial` in the recommended package `MASS`, `geese` in the contributed package `geepack`, `glm.poisson.disp` in the contributed package `dispmod`. ## Examples ```r # 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) ```

quasipois function