quasibin {aod} | R Documentation |
The function fits the generalized linear model “II” proposed by Williams (1982) accounting for overdispersion in clustered binomial data (n, y).
quasibin(formula, data, link = c("logit", "cloglog"), phi = NULL, tol = 0.001)
formula |
A formula for the fixed effects. The left-hand side of the formula must be of the form
cbind(y, n - y) where the modelled probability is y/n . |
link |
The link function for the mean p: “logit” or “cloglog”. |
data |
A data frame containing the response (n and 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 cluster (n, y), the model is:
y | \lambda ~ Binomial(n, \lambda)
with \lambda a random variable of mean E[\lambda] = p
and variance Var[\lambda] = \phi * p * (1 - p).
The marginal mean and variance are:
E[y] = p
Var[y] = p * (1 - p) * [1 + (n - 1) * \phi]
The overdispersion parameter \phi corresponds to the intra-cluster correlation coefficient,
which is here restricted to be positive.
The function uses the function glm
and the parameterization: p = h(X b) = h(\eta), where h is the
inverse of a given link function, 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 zero). The literature recommends to estimate \phi
from the saturated model. Several explanatory variables are allowed in b. None is allowed in \phi.
An object of formal class “glimQL”: see glimQL-class
for details.
Matthieu Lesnoff matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr
Moore, D.F., 1987, Modelling the extraneous variance in the presence of extra-binomial variation.
Appl. Statist. 36, 8-14.
Williams, D.A., 1982, Extra-binomial variation in logistic linear models. Appl. Statist. 31, 144-148.
glm
, geese
in the contributed package geepack,
glm.binomial.disp
in the contributed package dispmod.
data(orob2) fm1 <- glm(cbind(y, n - y) ~ seed * root, family = binomial, data = orob2) fm2 <- quasibin(cbind(y, n - y) ~ seed * root, data = orob2, phi = 0) fm3 <- quasibin(cbind(y, n - y) ~ seed * root, data = orob2) rbind(fm1 = coef(fm1), fm2 = coef(fm2), fm3 = coef(fm3)) # show the model fm3 # dispersion parameter and goodness-of-fit statistic c(phi = fm3@phi, X2 = sum(residuals(fm3, type = "pearson")^2)) # model predictions predfm1 <- predict(fm1, type = "response", se = TRUE) predfm3 <- predict(fm3, type = "response", se = TRUE) New <- expand.grid(seed = levels(orob2$seed), root = levels(orob2$root)) predict(fm3, New, se = TRUE, type = "response") data.frame(orob2, p1 = predfm1$fit, se.p1 = predfm1$se.fit, p3 = predfm3$fit, se.p3 = predfm3$se.fit) fm4 <- quasibin(cbind(y, n - y) ~ seed + root, data = orob2, phi = fm3@phi) # Pearson's chi-squared goodness-of-fit statistic # compare with fm3's X2 sum(residuals(fm4, type = "pearson")^2)