Modeling and Multivariate Methods - SAS

Appendix A Statistical Details 653
The Response Models
The Response Models
JMP fits linear models to three different kinds of responses: continuous, nominal, and ordinal. The models
and methods available in JMP are practical, are widely used, and suit the need for a general approach in a
statistical software tool. As with all statistical software, you are responsible for learning the assumptions of
the models you choose to use, and the consequences if the assumptions are not met. For more information
see “The Usual Assumptions” on page 672 in this chapter.
Continuous Responses
When the response column (column assigned the Y role) is continuous, JMP fits the value of the response
directly. The basic model is that for each observation,
Y = (some function of the X’s and parameters) + error
Statistical tests are based on the assumption that the error term in the model is normally distributed.
Fitting Principle for Continuous Response
Base Model
The Fitting principle is called least squares. The least squares method estimates the parameters in the model
to minimize the sum of squared errors. The errors in the fitted model, called residuals, are the difference
between the actual value of each observation and the value predicted by the fitted model.
The least squares method is equivalent to the maximum likelihood method of estimation if the errors have a
normal distribution. This means that the analysis estimates the model that gives the most likely residuals.
The log-likelihood is a scale multiple of the sum of squared errors for the normal distribution.
The simplest model for continuous measurement fits just one value to predict all the response values. This
value is the estimate of the mean. The mean is just the arithmetic average of the response values. All other
models are compared to this base model.
Nominal Responses
Nominal responses are analyzed with a straightforward extension of the logit model. For a binary (two-level)
response, a logit response model is
which can be written
where
log--------------------
P( y = 1)
P( y = 2)
=
Xβ
P( y = 1) = F( Xβ)
Fx ( )
is the cumulative distribution function of the standard logistic distribution