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)
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.
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)