Modeling and Multivariate Methods - SAS

654 Statistical Details Appendix A
The Response Models
Fx ( )
1
= ----------------
1 + e – x =
e x
-------------
1 + e x
For r response levels, JMP fits the probabilities that the response is one of r different response levels given by
the data values. The probability estimates must all be positive. For a given configuration of X’s, the
probability estimates must sum to 1 over the response levels. The function that JMP uses to predict
probabilities is a composition of a linear model and a multi-response logistic function. This is sometimes
called a log-linear model because the logs of ratios of probabilities are linear models. JMP relates each
response probability to the rth probability and fit a separate set of design parameters to these r – 1 models.
P( y = j)
log-------------------
P( y = r)
= Xβ () j forj = 1 , …,
r – 1
Fitting Principle For Nominal Response
Base Model
The fitting principle is called maximum likelihood. It estimates the parameters such that the joint probability
for all the responses given by the data is the greatest obtainable by the model. Rather than reporting the
joint probability (likelihood) directly, it is more manageable to report the total of the negative logs of the
likelihood.
The uncertainty (–log-likelihood) is the sum of the negative logs of the probabilities attributed by the model
to the responses that actually occurred in the sample data. For a sample of size n, it is often denoted as H and
written
n
H = –log( P( y = y i
))
i = 1
If you attribute a probability of 1 to each event that did occur, then the sum of the negative logs is zero for a
perfect fit.
The nominal model can take a lot of time and memory to fit, especially if there are many response levels.
JMP tracks the progress of its calculations with an iteration history, which shows the –log-likelihood values
becoming smaller as they converge to the estimates.
The simplest model for a nominal response is a set of constant response probabilities fitted as the occurrence
rates for each response level across the whole data table. In other words, the probability that y is response
level j is estimated by dividing the total sample count n into the total of each response level nj, and is written
n j
p j
= ---
n
All other models are compared to this base model. The base model serves the same role for a nominal
response as the sample mean does for continuous models.