A.18. Checking the Estimability of a Linear Function p'β

Often, one needs to check if a given linear function of β say, p'β, in the linear model set up is estimable. There is a theorem (Searle, 1972) which says that it is estimable if and only if X'X(X'X)−p = p where (X'X)− is any g-inverse of (X'X). Also, it so happens that checking the above requirement for a given g-inverse is as good as checking it for all g-inverses and hence with just one choice of a g-inverse, we can determine, using above, the estimability of a linear function.

Consider the one way classification model, described above. Suppose we want to know (a) if μ+t₂ is estimable and ( b) if t₁+t₂ is estimable. The corresponding choices for p' are p'₁ = (1 0 1 0) and p'₂ = (0 1 1 0). With X as obtained earlier, for p₁ and p₂, the left sides of the estimability condition can be calculated using the SAS statements

left_p1 = (x_with)`*x_with*(ginv( (x_with)`*x_with ) )*p1;
left_p2 = (x_with)`*x_with*(ginv( (x_with)`*x_with ) )*p2;

resulting in

and

Since LEFT_P1 is identically equal to p₁, estimability of μ+t₂ is established. However, since LEFT_P2 is not equal to p₂, t₁+t₂ is not estimable.