Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 677
Key Statistical Concepts
Pressure Cylinders
What if your response is categorical instead of continuous? For example, suppose that the response is the
country of origin for a sample of cars. For your sample, there are probabilities for the three response levels,
American, European, and Japanese. You can set these probabilities for country of origin to some estimate
and then evaluate the uncertainty in your data. This uncertainty is found by summing the negative logs of
the probabilities of the responses given by the data. It is written
H = h = yi ()
– log p yi ()
The idea of springs illustrates how a mean is fit to continuous data. When the response is categorical,
statistical methods estimate the response probabilities directly and choose the estimates that minimize the
total uncertainty of the data. The probability estimates must be nonnegative and sum to 1. You can picture
the response probabilities as the composition along a scale whose total length is 1. For each response
observation, load into its response area a gas pressure cylinder, for example, a tire pump. Let the partitions
between the response levels vary until an equilibrium of lowest potential energy is reached. The sizes of the
partitions that result then estimate the response probabilities.
Figure A.7 shows what the situation looks like for a single category such as the medium size cars (see the
mosaic column from Carpoll.jmp labeled medium in Figure A.8). Suppose there are thirteen responses
(cars). The first level (American) has six responses, the next has two, and the last has five responses. The
response probabilities become 6•13, 2•13, and 5•13, respectively, as the pressure against the response
partitions balances out to minimize the total energy.
Figure A.7 Effect of Pressure Cylinders in Partitions
As with springs for continuous data, you can divide your sample by some factor and fit separate sets of
partitions. Then test that the response rates are the same across the groups by measuring how much
additional energy you need to push the partitions to be equal. Imagine the pressure cylinders for car origin
probabilities grouped by the size of the car. The energy required to force the partitions in each group to
align horizontally tests whether the variables have the same probabilities. Figure A.8 shows these partitions.