aod (version 1.3.3)

quasipois: Quasi-Likelihood Model for Counts

Description

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

Usage

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

Value

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

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 \(\chi^2\) statistics .

Author

Matthieu Lesnoff , Renaud Lancelot

Details

For a given count \(y\), the model is: $$y~|~\lambda \sim Poisson(~\lambda)$$ 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.

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.

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

  # 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)