Sometimes sample proportions are continuous rather than of the binomial form (number of successes)/(number of trials). Each observation is any real number between 0 and 1, such as the proportion of a tooth surface that is covered with plaque. For independent responses {yi}, Bartlett (1937) modeled logit(yi) ∼ N(xi????, ????2). Then yi itself has a logit-normal distribution.

a. Expressing a N(xi????, ????2) variate as xi???? + ????z, where z is a standard normal variate, show that yi = exp(xi???? + ????z)∕[1 + exp(xi???? + ????z)] and for small ????, yi = exi????

1 + exi???? + exi????

1 + exi????

1 1 + exi???? ????z + exi???? (1 − exi???? )

2(1 + exi???? )3 ????2z 2 + ⋯ .

b. Letting ????i = exi???? ∕(1 + exi???? ), when ???? is close to 0 show that E(yi) ≈ ????i, var(yi) ≈ [????i(1 − ????i)]2????2.

c. The approximate moments for the logit-normal motivate a QL approach with v(????i) = ????[????i

(1 − ????i)]2 for unknown ????. Explain why this approach provides similar results as fitting an ordinary linear model to the sample logits, assuming constant variance. (The QL approach has the advantage of not requiring adjustment of 0 or 1 observations, for which sample logits do not exist. Papke and Wooldridge (1996) proposed an alternative QL approach using a sandwich covariance adjustment.)

d. Wedderburn (1974) used QL to model the proportion of a leaf showing a type of blotch. Envision an approximation of binomial form based on cutting each leaf into a very large number of tiny regions of the same size and observing for each region whether it is covered with blotch. Explain why this suggests using v(????i) = ????????i(1 − ????i). What violation of the binomial assumptions might make this questionable? (Recall that the parametric family of beta distributions has variance function of this form.)