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.

### See Also

`glm`, `geese` in the contributed package geepack, `glm.binomial.disp` in the contributed package dispmod.

### 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]