### Balanced Incomplete Block Design with Control Treatment

#### Abstract

The theory of design and analysis of experiments has been primarily developed by statisticians engaged mostly in agricultural research. The theory has now found applications in other fields of research, because it is based on general principles concerning the statistical behavior of observations which arise either freely in nature or in artificial laboratory conditions. Incomplete block designs were developed to suit experiments where the number of experimental units per block is less than the number of treatments. The concept of incomplete block designs augmented by a control was first introduced by Das (1954). He considered the case where (q ³ 1) new treatments are introduced into the design and all included in each block. He concluded that q should be chosen as low as possible to keep the block size to a reasonable level. Pearce (1960) considered the case where a *Balanced Incomplete Block* (B. I. B.) design is supplemented by a control, where the test treatments are replicated r times except the control which is replicated r_{0 }times, while all pairs of treatments occurs l times in blocks, the supplemented treatment occurs l_{0} times with any other treatment. Pesek (1974) considered the case of a B. I. B. design having an extra control in each block. He utilized Rao’s general formulas to obtain the variances of the elementary treatment contrast between any pair of test treatment and any treatment and the control. He also obtained the efficiency factor of this design and showed that this design is more efficient than a B. I. B. design for comparing treatments with a control, but is less efficient for pair wise comparisons between the test treatments. Many times, for important new drugs and for serious diseases, investigations are carried out simultaneously at various locations under different climatic conditions, for a series of treatments. In such cases, the result of all such investigations need to be combined to produce an overall estimate of the effect of treatment contrast to the control.

In this study the case where a control is added to each block in a B. I. B. design is considered. The variances of the elementary treatment contrast between any pair of test treatments and any treatment and the control were obtained. The general theory of inter & intra -block estimates of treatment effect when a B. I. B. design is augmented by a control was investigated, the general theory of a balanced incomplete block design is given where a control is added to each block. The estimation of weight for combining inter and intra-block estimates is also dealt with, and the weight for combining inter-block estimates of treatment contrast was also considered. The advantage of this study is to get a minimum variance of the treatment contrast between the inter and intra block estimates when a control treatment is added to each block in B. I. B. design.

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