## Statistical Assumptions for Using PROC GLM

The basic statistical assumption underlying the least-squares approach
to general linear modeling is that the observed values of each
dependent variable can be written as the sum of two parts: a fixed
component , which is a linear function of the independent
coefficients, and a random noise, or error, component :

The independent coefficients *x* are constructed from the model
effects as described in the "Parameterization of PROC GLM Models" section. Further, the errors for different observations are
assumed to be uncorrelated with identical variances. Thus, this
model can be written

where *Y* is the vector of dependent variable values, *X* is the matrix
of independent coefficients, *I* is the identity matrix, and is the common variance for the errors. For multiple dependent variables,
the model is similar except that the errors for different dependent
variables within the same observation are not assumed to be
uncorrelated. This yields a multivariate linear model of the form

where *Y* and *B* are now matrices, with one column for each dependent
variable, vec(*Y*) strings *Y* out by rows, and indicates the Kronecker matrix product.
Under the assumptions thus far discussed, the least-squares approach
provides estimates of the linear parameters that are unbiased and
have minimum variance among linear estimators. Under the further
assumption that the errors have a normal (or Gaussian) distribution,
the least-squares estimates are the maximum likelihood estimates and
their distribution is known. All of the significance levels
("*p* values") and confidence limits calculated by the GLM
procedure require this assumption of normality in order to be exactly
valid, although they are good approximations in many other cases.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.