On Combinability of Information from Uncorrelated Linear Models by Simple Weighting

Abstract

Within the set of all linear parametric functions, $\lambda'\beta$, estimable from either or both of two uncorrelated sets of data, $y_1 = X_1\beta + e_1$ and $y_2 = X_2\beta + e_2$ with known non-singular variances, a general characterization is presented of those $\lambda'\beta$'s for which the best linear unbiased estimator (b.l.u.e.) is obtainable from one source of information alone or by simple weighting of respective b.l.u.e.'s from each of the two sources. It is shown that if the intersection of the row spaces of $X_1$ and $X_2$ has rank $r$ then in the intersection space there are exactly $r$ independent $\lambda'$ vectors for which the b.l.u.e. of $\lambda'\beta$ is obtainable by simple weighting. Some related statements are made for $k > 2$ uncorrelated sources of information. In the case of incomplete block designs the reduced intrablock normal equations and the interblock normal equations may be regarded as originating from two uncorrelated sources of information on the treatment parameter vector $\tau$. It is shown that an estimable treatment contrast, $\gamma'\tau$, is best estimated from one source alone or by simple weighting of b.l.u.e.'s from the respective sources if and only if $\gamma$ is an eigenvector of $\Lambda = (\lambda_{ij})$, where $\lambda_{ij}$ is the number of times treatments $i$ and $j$ occur together in a block. For symmetric factorial or quasifactorial designs, it is shown that any effect or interaction degree of freedom contrast is an eigenvector of $\Lambda$, and hence is best estimated by simple weighting of its interblock and intrablock estimates.