The R Book

Proportion Data and Binomial Errors

Proportion data have three important properties that affect the way the data should be analysed:

• the data are strictly bounded;
• the variance is non-constant;
• errors are non-normal.

You cannot have a proportion greater than 1 or less than 0. This has obvious implications for the kinds of functions fitted and for the distributions of residuals around these fitted functions. For example, it makes no sense to have a linear model with a negative slope for proportion data because there would come a point, with high levels of the x variable, that negative proportions would be predicted. Likewise, it makes no sense to have a linear model with a positive slope for proportion data because there would come a point, with high levels of the x variable, that proportions greater than 1 would be predicted.

With proportion data, if the probability of success is 0, then there will be no successes in repeated trials, all the data will be zeros and hence the variance will be zero. Likewise, if the probability of success is 1, then there will be as many successes as there are trials, and again the variance will be 0. For proportion data, therefore, the variance increases with the mean up to a maximum (when the probability of success is one half) then declines again towards zero as the mean approaches 1. The variance–mean relationship is humped, rather than constant as assumed in the classical tests.

The final assumption is that the errors (the differences between ...