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 {y_i}, Aitchison and Shen (1980) and Bartlett (1937) modeled logit (Y_i)???N(?_i,??²). Then Y_i itself is said to have a logistic-normal distribution.

a. Expressing a N(?,??²) variate as ? + ?Z, where Z is standard normal, show that Y_i = exp(?_i + ?Z)/[1 + exp(?_i + ?Z)].

b. Show that for small ?,

c. Letting ?_i = e^?i/(1 + e^?i), when ? is close to 0 show that E(Y_i) ? ?_i, var(Y_i) ? [?_i(1??_i)]² ?².

d. For independent continuous proportions {y_i}, let ?_i = E(Y_i). For a GLM, it is sensible to use an inverse cdf link for ?_i, but it is unclear how to choose a distribution for Y_i. The approximate moments for the logistic-normal motivate a quasi-likelihood approach (Wedder-burn 1974) with variance function ?(?_i) = ?[?_i(1 ? ?_i)]² for unknown ?. Explain why this provides similar results as fitting a normal regression model 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 don?t exist.)

e. Wedderburn (1974) gave an example with response the proportion of a leaf showing a type of blotch. Envision an approximation of binomial from based on cutting each leaf into a large number of small regions of the same size and observing for each region whether it is mostly covered with blotch. Explain why this suggests that ?(?_i) = ??_i(1 ? ?_i). What violation of the binomial assumptions might make this questionable? [The parametric family of beta distributions has variance function of this form. Barndorff-Nielsen and Jorgensen (1991) proposed a distribution having ?(?_i) = ?[?_i(1 ? ?_i)]³].

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