ABSTRACT

Grouped logistic regression is based on the binomial probability distribution. Recall that standard logistic regression is based on the Bernoulli distribution, which is a subset of the binomial. As such, the standard logistic model is a subset of the grouped. The key concept involved is the binomial probability distribution function (PDF), which is defined as:

f y p n

n

y p py n y( ; , ) ( )=

 

  − −1

(5.1)

The product sign is assumed to be in front of the right-hand side terms. In exponential family form, the above expression becomes:

f y p n y p

p n p

n

y ( ; , ) exp ( )=

 

  + − +

 

 

 

ln ln ln 1

(5.2)

The symbol n represents the number of observations in a given covariate pattern. We have discussed covariate patterns before when dealing with residuals in the last chapter. For Bernoulli response logistic models, the model is estimated on the basis of observations. Only when analyzing the fit of the

model is the data put into covariate patterns and evaluated by observationbased residuals. Here the PDF itself is in covariate pattern structure.