GLM Hypothesis Testing - Estimability of Hypotheses
Before considering tests of specific hypotheses of this sort, it is important to address the issue of estimability. A test of a specific hypothesis using the general linear model must be framed in terms of the regression coefficients for the solution of the normal equations. If the X'X matrix is less than full rank, the regression coefficients depend on the particular g2 inverse used for solving the normal equations, and the regression coefficients will not be unique. When the regression coefficients are not unique, linear functions (f) of the regression coefficients having the form
f = Lb
where L is a vector of coefficients, will also in general not be unique. However, Lb for an L which satisfies
L = L(X'X)`X'X
is invariant for all possible g2 inverses, and is therefore called an estimable function.
The theory of estimability of linear functions is an advanced topic in the theory of algebraic invariants (Searle, 1987, provides a comprehensive introduction), but its implications are clear enough. One instance of non-estimability of a hypothesis has been encountered in tests of the effective hypothesis which have zero degrees of freedom. On the other hand, Type III sums of squares for categorical predictors variable effects in ANOVA designs with no missing cells (and the least squares means in such designs) provide an example of estimable functions which do not depend on the model parameterization (i.e., the particular g2 inverse used to solve the normal equations). The general implication of the theory of estimability of linear functions is that hypotheses which cannot be expressed as linear combinations of the rows of X (i.e., the combinations of observed levels of the categorical predictors variables) are not estimable, and therefore cannot be tested. Stated another way, we simply cannot test specific hypotheses that are not represented in the data. The notion of estimability is valuable because the test for estimability makes explicit which specific hypotheses can be tested and which cannot.
Whole Model Tests
Error Terms for Tests
Testing Hypotheses for Repeated Measures and Dependent Variables