Balanced Incomplete Blocks

In the ordinary Randoimsed Block the block must be large enough to take one complete set of treatments. It may happen, however, that we wish to compare more treatments than can be got into one block. 

Suppose we have raw material in batches that are only large enough to allow 4 experiments to be carried out on one batch, and further suppose that there is an appreciable variance between batches. If we wish to carry out a comparison of 2, 3, or 4 treatments this between batch variance would not worry us, as each set of 2, 3 or 4 treatments could be carried out on the same batch. If, however, we wished to compare 5 or more treatments, then one or more of these-would have to be carried out on a different batch from that on which the first 4 treatmoits were tested, and the comparison would be invalidated by the confusing element of the between batch variance. The standard method of avoiding this difficulty is to select one of the treatments as a standard, and include it in each batch. All treatments are then compared directly with the standard, and of the other comparison some are direct through the treatments occurring in the same block and others are indirect, from treatment M to the standard in its block to the standard in the other block concerned to the treatment N. 

Although this procedure does work it is by no means very satisfactory, as there are three different errors for the three different types of comparison. Further, it is not the most efficient possible. 

Consider the alternative design in Table 1.6 where we have the five treatments replicated four times in five batches of four. 

These experimental designs are known as balanced incomplete blocks. They are balanced because each treatment occurs to exactly the same extent they are incomplete because no block contains the full number of treatments. They suffer from the restriction that balanced arrangements are not possible for all experimental set-ups. Broadly speaking, if we fix the number of treatments that we wish to compare, and if the number of experiments per batch (or block ) is also fixed, then the number of replication of each treatment is thereby determined. This is the principal disadvantage of these designs the number of replications they require may be greater than we think are necessary to attain sufficient accuracy. 

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When the block size is smaller than the number of treatments to be evaluated, incomplete block designs may be used [Yates (1936)]. Balanced incomplete block designs give approximately the same precision to all pairs of treatments. 

The following is a symmetrical (i.e., the number of blocks equals the number of treatments) balanced incomplete block design ... 

What is the relationship - Youden square designs Latin square designs balanced incomplete block designs randomized complete block designs ... 

The ruggedization of the analytical procedure was performed by applying statistical screening techniques to minimize the effort required and, therefore, reduce the time and the cost substantially. The statistical approaches used in this study were those first introduced by Plackett-Burman (3.) and Youden-Steiner (4). Both techniques reduce the required effort since they use balanced incomplete block design experiments which can clearly indicate the non-affecting parameters from those that may have an effect. In this study the important variables of the analytical method were identified by using the Plackett-Burman technique. 

One such approach is the Plackett-Burman (2) design in which a large number of variables can be screened efficiently by a small number of experiments. This design is based on balanced incomplete blocks which allow the statistical identification of the nonaffecting variables. The remaining relatively small number of variables that may be important can be examined in further detail. In the... 

The optimization of the atomic absorption method of determining metals in particulates found in the air of workplace is described. The Plackett-Burman Youden-Steiner balanced incomplete block designs as well as single-factor experiments were utilized with ten metals Be, Cd, Co, Cr, Cu, Mn, Mo, Ni, Pb, and Pd. Of the parameters tested, perchloric acid digestion, flame-stoichiometry, and the composition of the calibration standards were the most significant. Perchloric acid affected the recoveries of chromium. This was attributed to the formation of volatile chromylchloride. Flame-related phenomena and interelemental effects were brought under control using lanthanum flame buffer. 

Nguyen (1996) showed that E(s2 )-optimal supersaturated designs could be obtained from balanced incomplete block designs for n - 1 treatments in blocks of size n/2 - 1, provided that there are no repeated blocks. Each block generates a... 

The principal additions to this Edition are a substantial enlargement of Chapter I and two new chapters, Chapter XIII on balanced incomplete blocks amd Chapter XIV on confounding. Further additions are the components of variance for unequal column size in Chapter VII (d), the exact formula for the residual variance about a regression line in Chapter IX (h), the Doolittle method of computation in multiple regression in Chapter X (d), and the partitioning of sums of squares in Chapter XII (c). 

The outstanding disadvantages of the Balanced Incomplete Blocks is that to obtain balance the number of replications necessary becomes rather high as the number of treatments increases. Generally, p treatments require (p -f- 1) replications, e.g. 36 treatments require 7 replications, and this we may consider to be more than is justified. 

A general description of the experimental design known as balanced incomplete blocks have been given in Chapter I Section (f), and will not be repeated here. This chapter will give the methods of computation for such experiments, the method and notation being based on Yates( ). 

The alternative classical solution of this problem would be to carry out the treatments Tj to T7 with B constant at Ba and Ba to Bg with T constant at Tx which would require 13 units or runs. In the absence of replication the experiment is most unsatisfactory, for there is no estimate of error and-no means of testing for significance. If we replicate all treatments, making a total of 26 runs, we then get the comparison of our means with a variance error variance 14% larger. 

Consumers received 30 samples, each coded with a three-digit number, in a single session. For each subject, the 30 samples were selected from the 34 samples according to a balanced incomplete block design (BIBD). The order of presentation of the samples was randomized for each subject. 

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