statisticalrethinkin..

9.1. GENERALIZED LINEAR MODELS 271an additive model. Some other link is needed. And many fundamental parameters are notcontinuous like µ. Instead, they must be positive (like σ) or bounded between zero and one.ese facts can make establishing a useful link a little more complicated.As this book introduces each outcome distribution, it will also introduce common linkingstrategies for associating distribution parameters and predictor variables. e importantprinciple to remember is that there is no single right way to accomplish this goal. Your hypothesestell you how you want the predictor variables to relate the aspects of the outcome,like its mean. e link is a mathematical strategy for modeling that hypothesis. Like allmodels, all you need to hope for is that it be useful.But while there is no one right way to build a link, there are well practiced and usefulways to accomplish it. Depending upon the distribution chosen—exponential, binomial oranother—there are usually common links that provide a time-tested way to embed an additivemodel. Still, it’s worth being critical of link choices. Just like with other aspects of amodel, such as the prior and the likelihood function, there is always some subjective choiceinvolved in the choice of a link. While scientists tend to be much more suspicious of priorsthan links, both can alter inference in powerful ways.9.1.3. Living with the family. Moving from Gaussian models to models based upon otherdistributions does have some costs. We need GLM’s, because there are many important typesof data that are not even approximately Gaussian and cannot be usefully coerced into Gaussianform through transformation. But these non-Gaussian distributions all share the inconvenientfeature of having patterns of variation that are related to the mean. For example, ifthe average count of some event is close to zero, then the variation around that mean mustextend further above the mean than below it. is is just because counts cannot go belowzero. e distribution will be skewed. As the mean increases, the pattern of variation willchange, usually becoming more symmetrical.Such relationships between the mean and variation in a collection of measures is natural.Distances and counts really cannot be negative, and their means and variances really docovary. It’s how nature actually works. But in the practice of statistics, it presents some newproblems for us.9.1.3.1. Needle in a non-linear haystack. Search is harder.9.1.3.2. Ceilings, floors and interpreting parameters. Everything interacts.9.1.3.3. e weakest link. Link functions matter and even smuggle in assumptions.