|The Four Types of Estimable Functions|
Given a response or dependent variable , predictors or independent variables , and a linear expectation model relating the two, a primary analytical goal is to estimate or test for the significance of certain linear combinations of the elements of . For least squares regression and analysis of variance, this is accomplished by computing linear combinations of the observed s. An unbiased linear estimate of a specific linear function of the individual s, say , is a linear combination of the s that has an expected value of . Hence, the following definition:
A linear combination of the parameters is estimable if and only if a linear combination of the s exists that has expected value .
Any linear combination of the s, for instance , will have expectation . Thus, the expected value of any linear combination of the s is equal to that same linear combination of the rows of multiplied by . Therefore,
is estimable if and only if there is a linear combination of the rows of that is equal to —that is, if and only if there is a such that .
Thus, the rows of form a generating set from which any estimable can be constructed. Since the row space of is the same as the row space of , the rows of also form a generating set from which all estimable s can be constructed. Similarly, the rows of also form a generating set for .
Therefore, if can be written as a linear combination of the rows of , , or , then is estimable.
In the context of least squares regression and analysis of variance, an estimable linear function can be estimated by , where . From the general theory of linear models, the unbiased estimator is, in fact, the best linear unbiased estimator of , in the sense of having minimum variance as well as maximum likelihood when the residuals are normal. To test the hypothesis that , compute the sum of squares
and form an test with the appropriate error term. Note that in contexts more general than least squares regression (for example, generalized and/or mixed linear models), linear hypotheses are often tested by analogous sums of squares of the estimated linear parameters .