Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 655
The Response Models
The R 2
statistic measures the portion of the uncertainty accounted for by the model, which is
H( full model)
1 – ----------------------------------
H( base model)
However, it is rare in practice to get an R 2 near 1 for categorical models.
Ordinal Responses
With an ordinal response (Y), as with nominal responses, JMP fits probabilities that the response is one of r
different response levels given by the data.
Ordinal data have an order like continuous data. The order is used in the analysis but the spacing or
distance between the ordered levels is not used. If you have a numeric response but want your model to
ignore the spacing of the values, you can assign the ordinal level to that response column. If you have a
classification variable and the levels are in some natural order such as low, medium, and high, you can use
the ordinal modeling type.
Ordinal responses are modeled by fitting a series of parallel logistic curves to the cumulative probabilities.
Each curve has the same design parameters but a different intercept and is written
P( y≤
j) = F( α j
+ Xβ)
forj = 1 , …,
r – 1
where r response levels are present and
Fx ( )
is the standard logistic cumulative distribution function
Fx ( )
1
= ----------------
1 + e – x =
e x
-------------
1 + e x
Another way to write this is in terms of an unobserved continuous variable, z, that causes the ordinal
response to change as it crosses various thresholds
y
=
r α r – 1
≤ z
j α j – 1
≤ z < α j
1 z ≤ α 1
where z is an unobservable function of the linear model and error
z = Xβ+
ε
and ε has the logistic distribution.
These models are attractive in that they recognize the ordinal character of the response, they need far fewer
parameters than nominal models, and the computations are fast even though they involve iterative
maximum likelihood calculation.
A different but mathematically equivalent way to envision an ordinal model is to think of a nominal model
where, instead of modeling the odds, you model the cumulative probability. Instead of fitting functions for
all but the last level, you fit only one function and slide it to fit each cumulative response probability.