quasipois {aod} | R Documentation |
The function fits the log linear model (“Procedure II”) proposed by Breslow (1984) accounting for overdispersion in counts y.
quasipois(formula, data, phi = NULL, tol = 0.001)
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 . |
For a given count y, the model is:
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.
An object of formal class “glimQL”: see glimQL-class
for details.
Matthieu Lesnoff matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr
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.
# 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)