The LOGSELECT Procedure
Log-Likelihood Functions
The LOGSELECT procedure forms the log-likelihood functions of the various models as
where is the probability that the ith observation is an event, is the log-likelihood contribution of the ith observation, is the weight, and is its frequency. For the determination of and , see the WEIGHT and FREQ statements. The individual log-likelihood contributions for the various distributions are described in the following sections.
Binary Distribution
The LOGSELECT procedure computes the log-likelihood function for the ith binary observation as
where the variable takes the value 1 for an event and 0 for a nonevent. The inverse link function maps from the scale of the linear predictor to the scale of the mean. For example, for the logit link (the default),
You can control which binary outcome in your data is modeled as the event by using the response-options in the MODEL statement, and you can choose the link function by using the LINK= option in the MODEL statement.
If a WEIGHT statement is specified and denotes the weight for the current observation, the log-likelihood function is computed as
Binomial Distribution
The LOGSELECT procedure computes the log-likelihood function for the ith binomial observation as
where and are the values of the events and trials of the ith observation, respectively. The value measures the probability of events (successes) in the underlying Bernoulli distribution whose aggregate follows the binomial distribution.
Multinomial Distribution
The multinomial distribution that the LOGSELECT procedure models is a generalization of the binary distribution; it is the distribution of a single draw from a discrete distribution that has J possible values. Thus, the log-likelihood function for the ith observation is
In this expression, J denotes the number of response categories (the number of possible outcomes) and is the probability that the ith observation takes on the response value associated with category j. The category probabilities must satisfy
and the constraint is satisfied by modeling categories. In models that have ordered response categories, the probabilities are expressed in cumulative form, so the last category is redundant. In generalized logit models (multinomial models that have unordered categories), one category is chosen as the reference category, and the linear predictor in the reference category is set to 0. For more information, see the REF= response-option in the MODEL statement.