Common Errors in Statistics (and How to Avoid Them), 4th Edition

COMMON SOURCES OF ERROR

The caveats of previous chapters also apply to the specification of the link and variance functions of GLMs. The pair of functions that define a specific model should be determined on the basis of cause-and-effect relationships and not by inspecting the data.

For example, when deciding among a Poisson, negative binomial, or binomial model for counts, the wrong approach to model specification is to make function choices based on the ratio of the mean to the variance of the sample. As Bruce Tabor notes in a personal communication,

In a contagious process, such as an infectious disease outbreak, the probability of a subsequent event will increase after the occurrence of a preceding event. A person carrying an infection is likely to infect additional persons. This results in positive correlation between events and overdispersion. A negative binomial model has this property and may provide a suitable model (or may not, as the case may be).

In a count process with negative contagion (underdispersion), the occurrence of an event makes subsequent events less likely—events are negatively correlated. One example might be house burglaries in a neighborhood. After an initial burglary, residents and police are alerted to subsequent burglaries and thieves respond appropriately, targeting other neighbourhoods for a while.

The other common sources of error in applying generalized linear models are the use of an inappropriate or erroneous link function, the wrong choice ...