Modeling for response variables that are proportions
Maarten L. Buis
When dealing with response variables that are proportions, people often use regress. This approach can be problematic since the model can lead to predicted proportions less than zero or more than one and errors that are likely to be heteroskedastic and nonnormally distributed. This talk will discuss three more appropriate methods for proportions as response variables: betafit, dirifit, and glm. betafit is a maximum likelihood estimator using a beta likelihood, dirifit is a maximum likelihood estimator using a Dirichlet likelihood, and glm can be used to create a quasi–maximum likelihood estimator using a binomial likelihood. On an applied level, a difference between dirifit and the others is that the others can handle only one response variable, whereas dirifit can handle multiple response variables. For instance, betafit and glm can model the proportion of city budget spent on the category security (police and fire department), whereas dirifit can simultaneously model the proportions spent on categories security, social policy, infrastructure, and other. Another difference between betafit and glm is that glm can handle a proportion of exactly zero and one, whereas betafit can handle only proportions between zero and one. Special attention will be given on how to fit these models in Stata and on how to interpret the results. This presentation will end with a warning not to use any of these techniques for ecological inference, i.e., using aggregated data to infer about individual units. To use a classic example: In the United States in the 1930s, states with a high proportion of immigrants also had a high literacy rate (in the English language), whereas immigrants were on average less literate than nonimmigrants. Regressing state level literacy rate on state level proportion of immigrants would thus give a completely wrong picture about the relationship between individual immigrant status and literacy.
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