Generalized Linear Model (GLM) Hypothesis Testing - Lack-of-Fit Tests using Pure Error
Whole model tests and tests based on the 6 types of sums of squares use the mean square residual as the error term for tests of significance. For certain types of designs, however, the residual sum of squares can be further partitioned into meaningful parts which are relevant for testing hypotheses. One such type of design is a simple regression design in which there are subsets of cases all having the same values on the predictor variable. For example, performance on a task could be measured for subjects who work on the task under several different room temperature conditions. The test of significance for the Temperature effect in the linear regression of Performance on Temperature would not necessarily provide complete information on how Temperature relates to Performance; the regression coefficient for Temperature only reflects its linear effect on the outcome.
One way to glean additional information from this type of design is to partition the residual sums of squares into lack-of-fit and pure error components. In the example just described, this would involve determining the difference between the sum of squares that cannot be predicted by Temperature levels, given the linear effect of Temperature (residual sums of squares) and the pure error; this difference would be the sums of squares associated with the lack-of-fit (in this example, of the linear model). The test of lack-of-fit, using the mean square pure error as the error term, would indicate whether non-linear effects of Temperature are needed to adequately model Temperature's influence on the outcome. Further, the linear effect could be tested using the pure error term, thus providing a more sensitive test of the linear effect independent of any possible nonlinear effect.
Whole Model Tests
Error Terms for Tests
Testing Hypotheses for Repeated Measures and Dependent Variables