Design-based Causal Inference for Incomplete Block Designs

A complete class of balanced incomplete block designs [formula omitted] (7, 35, 15, 3, 5) with repeated blocks

Schwager, Adico sign fuel additive promotion deal for Chile

In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.

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.

By an incomplete block design we will mean an arrangement of v different varieties of objects in b distinct blocks or sets such that a block does not contain all the varieties and a variety appears at most once in a block. An incomplete block design which satisfies two extra conditions, viz. (i) every pair of varieties occurs together in ' S 0 of the blocks and (ii) each block contains the same number of objects, say k, is called a balanced incomplete block (b.i.b. for conciseness) design. A b.i.b. design with b=v is sometimes called a v-k-X configuration. It is known [1] and can be seen below that in a b.i.b. design every variety occurs the same number, say r, of times. Clearly then bk = vr, X(v-1) =r(k-1). For a b.i.b. design Fisher's inequality b_v must also hold [1 ]. An incomplete block design satisfying condition (i) is not necessarily a b.i.b. design. Ryser [3 ] has proved (in an essentially equivalent form) that if in a symmetrical incomplete block design (that is to say, one in which b = v) every variety appears a constant number of times, say r, also condition (i) holds, then X(v 1) = r(k 1) and (ii) holds and it is a symmetrical b.i.b. design. We give here conditions under which any incomplete block design becomes a b.i.b. design. An extension of Ryser's result in another direction has been given in [2].

In comparative experiments involving a fairly large number of varieties, when for lack of homogeneous experimental units, complete block designs are not available. Balanced Incomplete Block (BIB) designs prove very convenient in two respects. Firstly, the analysis is very simple and secondly, comparison between any two varieties has the same precision. On the other hand, a BIB design requires a large number of experimental units, because balancing is not possible unless the number of blocks is atleast as large as the number of varieties. To obviate this dif ficulty, different types of incomplete block designs have been introduced, of which the most notable is the Partially Balanced Incomplete Block (PBIB) design introduced by Bose and Nair (1939). Another class of incomplete block designs, called Linked Block (LB) designs was introduced by Youden (1951) which he obtained by dualising several BIB designs, that is by taking the varieties and blocks of the BIB design respectively as blocks and varieties in the LB design. The LB designs so constructed by Youden happen to be all PBIB designs, but this is not necessarily true. A great advantage of LB designs is that the analysis can be easily worked out. In this paper, we show how the intra-blockTanalysis of a LB design can be neatly carried out. The efficiency factor of a LB design is obtained. The methods are illustrated with a numerical example. An exhaustive list of all LB designs with ten or less plots per block and involving ten or less replications is given. These designs fall into three groups: (1) symmetrical BIB designs, (2) PBIB designs and (3) Irregular designs For plans of LB designs belonging to group (2) reference is made to the serial number of the two-associate class PBIB designs enumerated by Bose, Clatworthy and Shrikhande (1954). Plans for other designs belonging to groups (2) and (3) are given in detail. Certain theorems are derived which are useful in determining from the parameters of a given two-associate class PBIB design whether the design is of the LB type or not.

An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The present investigation involves the methods of construction of complete diallel cross plans using balanced incomplete block (BIB) designs. Furthermore, the analysis of complete diallel crosses plans are carried out to estimate the general combining ability of the ith line (i=1, r 2, r …, r v) where the intra- block analysis of the adjusted sum of squares for GCA and the unadjusted block sum of squares are also obtained, thereafter the relationship between the estimates of BIB design and the estimates of the GCA effect of CDC plan has been established. Moreover, it has also been shown that the complete diallel crosses design obtained through two BIB designs satisfying v1=b1= 4 5 1+3=v2=b2, r r1=2 5 1+1=r2=k1=k2 and 5 1= 5 2 are universally optimum. These results are further supported by a suitable example of each. However, the need of this study is to show that the analysis of the CDC plan is reducible to the analysis of generating the BIB design.

: The structure and the size of the supports of balanced incomplete block (BIB) designs are explored. The concepts of fundamental BIB designs is introduced and its usefulness in can be reduced via a technique call trade on a design. A new graphical method of studying the supports of BIB designs with blocks of size three is introduced. Several useful results are obtained via this graphical method. In particular, it is shown that no BIB design with seven varieties in blocks of size three can be built based on sixteen distinct blocks. Contributions made here have immediate applications in controlled experimental designs and survey samplings.

In agricultural, animal, fisheries and industrial experimentation under block design setup, systematic trend may affect the response under consideration. Although remote, these effects may still have high influence on response and hence should be incorporated in the model for proper model specification. Completely trend resistant block designs, also known as trend free block designs, are an important class of designs where the analysis can be done in usual manner as in case of normal block designs since trend effects and treatment effects are orthogonal. Literature is available on different aspects of trend resistant incomplete block designs mainly focusing on balanced incomplete block designs. This article deals with the construction of completely trend resistant Partially Balanced Incomplete Block (TR-PBIB) designs. Two methods of constructing such designs have been discussed. The information matrices for the class of designs developed have been derived and SAS code for the generation of the developed designs has also been provided.

Whenever respondents must rank-order a large number of items and/or the reliability of their rankings may be questionable, balanced incomplete block designs (BIBDs) represent a more effective means for doing so than either complete rankings or paired comparisons for business and marketing researchers. By providing a type of balancing and replication across items and respondents, BIBDs significantly reduce the number of subjective evaluations each individual must make. But, at the same time, BIBDs allow a limited number of respondents as a group to rank many items. This balancing and replication in BIBDs also reduces standard deviation, which increases the precision of a study. BIBDs, therefore, can improve response rates as well as increase the accuracy and reliability of the data collected. After discussing the general nature of BIBDs and statistical techniques for analyzing preference data collected by BIBDs, three business-related applications are presented to illustrate the benefits of BIBDs. Next, caveats concerning the use of BIBDs are presented. In the last section, advantages of BIBDs are discussed

On group divisible treatment designs for comparing test treatments with a standard treatment in blocks of size 3

Construction of nested balanced incomplete block (BIB) designs, nested balanced ternary designs and rectangular designs from given nested BIB designs and resolvable BIB designs are described. New constructions ofq-ary codes from nested BIB designs and balanced bipartite weighing designs are also given.

Construction of Balanced Incomplete Block Designs (BIBD) is a combination problem that involves the arrangement of \(\mathit{v}\) treatments into b blocks each of size \(\mathit{k}\) such that each treatment is replicated exactly \(\mathit{r}\) times in the design and a pair of treatments occur together in \(\lambda\) blocks. Several methods of constructing BIBDs exist. However, these methods still cannot be used to design all BIBDs. Therefore, several BIBDs are still unknown because a definite construction method for all BIBDs is still unknown. The study aimed to develop a new construction method that could aid in constructing more BIBDs. The study derived a new class of BIBD from un-reduced BIBD with parameters \(\mathit{v}\) and \(\mathit{k}\) such that \(\mathit{k} \ge\) 3 through selection of all blocks within the un-reduced BIBD that contains a particular treatment \(\mathit{i}\) then in the selected blocks treatment delete treatment \(\mathit{i}\) and retain all the other treatments. The resulting BIBD was Derived Reduced BIBD with parameters \(v^*=v-1, b^*=\left(\begin{array}{c}v-1 \\ k-1\end{array}\right), k^*=k-1, r^*=\left(\begin{array}{c}v-2 \\ k-2\end{array}\right), \lambda=\left(\begin{array}{c}v-3 \\ k-3\end{array}\right)\). In conclusion, the construction method was simple and could be used to construct several BIBDs, which could assist in solving the problem of BIBD, whose existence is still unknown.

# Linear Estimation and Tests for Linear Hypotheses # General Analysis of Block Designs # Randomized Block Designs # Balanced Incomplete Block Designs -- Analysis and Combinatorics # Balanced Incomplete Block Designs -- Applications # t-Designs # Linked Block Designs: Partially Balanced Incomplete Block Designs # Lattice Designs: Miscellaneous Designs