Estimability Effects model: Linear combinations of parameters are estimable only if they can be expressed as linear combinations of the cell means {mi}

Estimability Linear combinations of cell means are estimable. The parameters in our effects model are not estimable. On the other hand, the quantities ai-aj are estimable, since

Estimability LS solutions for non-estimable parameters are not unique: Minimizing SS leads to the following normal equations:

Estimability These normal equations simplify to the following equations, which have no unique solution:

Estimability Linear constraints can lead to unique estimates, but the effects model parameters are still not estimable As an example, consider a=3 and

Estimability Our normal equations could now be written:

Estimability Which leads to the familiar estimates: Different constraints (e.g., a3=0) lead to other expressions for the parameters in the effects model.

Estimability These constraints can also be incorporated into our expression for a linear model:

Estimability SAS can provide a general form for estimable functions that can be used to test whether certain parameters are estimable Let us look at the Soil example and confirm some of our earlier estimability results Yandell discusses estimability for two-factor models—we’ll save that for our discussion of unbalanced designs (which is when things get really interesting).