R: Quasi-Likelihood Model for Proportions

quasibin {aod}

R Documentation

Quasi-Likelihood Model for Proportions

Description

The function fits the generalized linear model “II” proposed by Williams (1982) accounting for overdispersion in clustered binomial data (n, y).

Usage

quasibin(formula, data, link = c("logit", "cloglog"), phi = NULL, tol =  0.001)

Arguments

`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 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 cluster (n, y), the model is:

y | λ ~ Binomial(n, λ)

with λ a random variable of mean E[λ] = p and variance Var[λ] = φ * p * (1 - p).
The marginal mean and variance are:

E[y] = p

Var[y] = p * (1 - p) * [1 + (n - 1) * φ]

The overdispersion parameter φ 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(η), where h is the inverse of a given link function, 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 zero). The literature recommends to estimate φ from the saturated model. Several explanatory variables are allowed in b. None is allowed in φ.

Value

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

Author(s)

Matthieu Lesnoff matthieu.lesnoff@cirad.fr, Renaud Lancelot renaud.lancelot@cirad.fr

References

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.

Examples

  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)

[Package aod version 1.3 Index]