Statistics: An Introduction Using R, 2nd Edition

12Other Response Variables

Up to now, the response variable has been a continuous real number such as a weight, length or concentration. We made several important assumptions about the behaviour of the response variable, and it is worth reiterating those assumptions here, ranked in order of importance:

• random sampling
• constant variance
• normal errors
• independent errors
• additive effects

So far, when we found that one or more of the assumptions was wrong, our typical resort was transformation of the response variable, coupled perhaps with transformation of one or more of the explanatory variables.

In our line of work, we often meet with response variables where the key assumptions are rarely if ever met. In these cases, it is sensible to look for alternatives to transformation that might improve our ability to model these systems effectively. In this book we cover four new kinds of response variable that are very common in practice:

• count data
• proportion data
• binary response data
• age-at-death data

all of which routinely fail the assumptions about constancy of variance and normality of errors. It turns out that these different kinds of response variables have more in common than you might first imagine, and that they can all be dealt with in the framework of generalized linear models (GLMs). These models allow variance to be non-constant and errors to be non-normally distributed. It is worth noting, however, that they still assume random sampling and independence of errors. ...