Use of Inter-Block Information to Obtain Uniformly Better Estimators

Abstract

Yates (1939, 1940) suggested use of information about treatment differences contained in differences of block totals. The procedure given by Yates for three dimensional lattice designs (1939) and for balanced incomplete block (BIB) designs was adopted by Nair (1944) for partially balanced incomplete block (PBIB) designs and was later generalized by Rao (1947) for use with any incomplete block design. The procedure is called recovery of inter-block information and consists of the following stages. The method of least-squares is applied to both intra- and inter-block contrasts, assuming that the value of $\rho$, the ratio of the inter-block variance to the intra-block variance is known. This gives the so called "normal" equations for combined estimation. The equations involve $\rho$ which is estimated from the observations by equating the error sum of squares (intra-block) and the adjusted block sum of squares in the standard analysis of variance to their respective expected values. This estimate is substituted for $\rho$ in the normal equations and the combined estimates are obtained by solving these equations. A priori, the inter-block variance is expected to be larger than the intra-block variance and hence it is customary to use the above estimator of $\rho$, truncated at unity. The error sum of squares in the inter-block analysis has at times been used in place of the adjusted block sum of squares (Yates (1939) for three dimensional lattice designs, Graybill and Deal (1959) for BIB designs). If $\rho$ were known, the combined estimators would have all the good properties of least-squares estimates. Since only an estimate of $\rho$ is used, the properties of the combined estimators have to be critically examined. One would expect these to depend on the type of estimator of $\rho$ used. To use the combined estimator of a treatment contrast with confidence one would like to know if it is unbiased and if its variance is smaller than that of the corresponding intra-block estimator, uniformly in $\rho$. The question of unbiasedness has been examined by some authors. Graybill and Weeks (1959) showed that for a BIB design, the combined estimator of a treatment contrast based on the Yates' estimator of $\rho$ in its untruncated form is unbiased. Graybill and Seshadri (1960) proved the same with Yates' estimator of $\rho$ in its usual truncated form, again for BIB designs. Roy and Shah (1962) showed that for any incomplete block design, if the estimator of $\rho$ is the ratio of quadratic forms of a special type, the corresponding combined estimators of treatment contrasts are unbiased. The customary estimator of $\rho$ (as given by Yates (1939) and Rao (1947)) is of the above type and hence gives rise to unbiased combined estimators. The variance of the combined estimators has also been examined by some authors. Yates (1939) used the method of numerical integration to show that for a three dimensional lattice design with 27 treatments and with 6 replications or more, the combined estimator of a treatment contrast has variance smaller than that of the intra-block estimator, uniformly in $\rho$. For a BIB design for which the number of blocks exceeds the number of treatments by at least 10 (or by 9 if in addition, the number of degrees of freedom for intra-block error is not less than 18), Graybill and Deal (1959) used the exact expression for the variance to establish this property of the combined estimators. In both the cases, the estimator of $\rho$ is based on the inter-block error and thus differs from the usual one based on the adjusted block sum of squares. For BIB designs, Seshadri (1963) gave yet another estimator of $\rho$ which gives rise to more precise combined estimators provided that the number of treatments exceeds 8. Roy and Shah (1962) gave an expression for the variance of the combined estimator based on any estimator of $\rho$ belonging to the class described above. Shah (1964) used this expression to show that the combined estimator of any treatment contrast in any incomplete block design has variance smaller than that of the corresponding intra-block estimator if $\rho$ does not exceed 2. The question that now arises is whether a combined estimator for a treatment contrast can be constructed which is "uniformly better" than the intra-block estimator, in the sense of having a smaller variance for all values of $\rho$. It is shown in Section 4 that for a linked block (LB) design with 4 or 5 blocks, recovery of inter-block information by the Yates-Rao procedure may even result in loss of efficiency for large values of $\rho$. A method of constructing a certain estimator of $\rho$, applicable to any incomplete block design for which the association matrix has a nonzero latent root of multiplicity $p > 2$, is presented in Section 3. For any treatment contrast belonging to a sub-space associated with the multiple latent root, the combined estimator based on this estimator of $\rho$ is shown to be uniformly better than the intra-block estimator if and only if $(p - 4) \times (e_0 - 2) \geqq 8$, where $e_0$ is the number of degrees of freedom for error (inter-block). For almost all well-known designs, the association matrix has multiple latent roots and this method can therefore be applied to many of the standard designs, at least for some of the treatment contrasts. It may be noted that, in general, this estimator of $\rho$ is different from the customary one given by Yates (1939) and Rao (1947). For LB designs however, this estimator of $\rho$ coincides with the customary one. It is shown here that, for a LB design, the usual procedure of recovery of inter-block information gives uniformly better combined estimators for all treatment contrasts if the number of blocks exceeds 5. As was pointed out before, if the number of blocks is 4 or 5 and if $\rho$ is large, recovery of inter-block information by the usual procedure results in loss of efficiency. Using the above method, we obtain an estimator of $\rho$ which produces a combined estimator uniformly better than the intra-block estimator for any treatment contrast for the following designs: (i) a BIB design with more than five treatments (ii) a simple lattice design with sixteen treatments or more and (iii) a triple lattice design with nine treatments or more. Applications to some other two-associate partially balanced incomplete block designs and to inter- and intra-group balanced designs have also been worked out in Sections 4 and 5. A computational procedure for obtaining the estimate of $\rho$ has been given for each case.