1. Much of the earlier work on the phenomenon of overdispersion in data from toxicological experiments is reviewed by Haseman and Kupper (1979). Anderson (1988) gives a summary of models for overdispersion, and illustrates their use in the analysis of two data sets. The books of Aitkin et al. (1989), Cox and Snell (1989) and McCullagh and Neider (1989) also contain sections on overdispersion.
2. A key paper in the literature on overdispersion is Williams (1982a), which includes the material on which Sections 6.2, 6.4 and 6.7.2 of this chapter are based. The use of the beta-binomial model for overdispersed proportions was proposed by Williams (1975) and Crowder (1978). Pierce and Sands (1975) proposed linear logistic models with a random effect for the analysis of binary data. Follmann and Lambert (1989) describe a non-parametric method for fitting models with random effects, in which the distribution of the random term remains unspecified.
3. An alternative way of obtaining the estimates β̂, γ̂ which minimise the marginal likelihood function in (6.10) is to use what is known as the E-M algorithm, due to Dempster, Laird and Rubin (1977). This algorithm can be implemented in a GLIM macro, following the method outlined by Anderson and Aitkin (1985). Hinde (1982) discusses the use of this algorithm in the analysis of overdispersed Poisson data, and gives GLIM code which can be easily modified for use with binary data. However, the algorithm does tend to be rather slow, which is why it has not been discussed in this chapter.
4. The model in Section 6.5 for the variance of the number of successes for the i th observation can be generalized to give Var (y
i) = σ2
n
i
p
i(1 − p
i), where the n
i are no longer assumed constant. Moreover, by allowing σ2 to lie in the range (0, ∞), underdispersion can be modelled by values of σ2 less than unity. This model can be fitted using the notion of quasi-likelihood, which is discussed in detail in Chapter 9 of McCullagh and Neider (1989). In fact, the quasi-likelihood approach leads to the same procedure for adjusting for overdispersion as that described in Section 6.4.