Modeling and Multivariate Methods - SAS

Chapter 7 Performing Logistic Regression on Nominal and Ordinal Responses 199
Introduction to Logistic Models
Introduction to Logistic Models
Logistic regression fits nominal Y responses to a linear model of X terms. To be more precise, it fits
probabilities for the response levels using a logistic function. For two response levels, the function is:
PY ( = r 1
) = ( 1 + e – Xb
) – 1
where r 1 is the first response
or equivalently:
PY ( = r 1
)
log-----------------------
PY ( = r 2
)
=
Xb
where r 1 and r 2 are the two responses
For r nominal responses, where r > 2 , it fits r – 1 sets of linear model parameters of the following form:
PY ( = j)
log--------------------
PY
( = r)
=
Xb j
The fitting principal of maximum likelihood means that the βs are chosen to maximize the joint probability
attributed by the model to the responses that did occur. This fitting principal is equivalent to minimizing
the negative log-likelihood (–LogLikelihood) as attributed by the model:
n
Loss = – logLikelihood = –log( Prob( ith row has the y j
th response)
)
i = 1
For example, consider an experiment that was performed on metal ingots prepared with different heating
and soaking times. The ingots were then tested for readiness to roll. See Cox (1970). The Ingots.jmp data
table in the Sample Data folder has the experimental results. The categorical variable called ready has values
1 and 0 for readiness and not readiness to roll, respectively.
The Fit Model platform fits the probability of the not readiness (0) response to a logistic cumulative
distribution function applied to the linear model with regressors heat and soak:
1
Probability (not ready to roll) = ---------------------------------------------------------------
–( β 0 + β 1 heat + β 2 soak)
1 + e
The parameters are estimated by minimizing the sum of the negative logs of the probabilities attributed to
the observations by the model (maximum likelihood).
To analyze this model, select Analyze > Fit Model. The ready variable is Y, the response, and heat and soak
are the model effects. The count column is the Freq variable. When you click Run, iterative calculations
take place. When the fitting process converges, the nominal or ordinal regression report appears. The
following sections discuss the report layout and statistical tables, and show examples.