(a) Differentiate between randomised block design and balanced incomplete block design. In usual notations, for a balanced…

Part (a). In the usual notation, v treatments, b blocks, r replications, k plots per block and λ pairwise concurrence, RBD and BIBD differ in block completeness. In RBD every block contains all v treatments, so k=v, r=b, and each pair of treatments occurs in every block; local control is obtained by grouping homogeneous units. In BIBD each block contains only k<v treatments, so blocks are incomplete; each treatment appears r times and every pair appears in λ blocks, with λ constant. RBD is therefore a complete-block design, while BIBD is an incomplete-block design used when block size is restricted. Let N be the v×b incidence matrix, N_ij=1 if treatment i is in block j. Counting treatment occurrences gives bk=vr. Counting treatment pairs within blocks gives b k(k−1)/2 = λ v(v−1)/2; using bk=vr gives r(k−1)=λ(v−1). For b≥v, since v>k, r>λ. Then NN^T=(r−λ)I+λJ is nonsingular, where I and J are identity and all-ones matrices, so rank(NN^T)=v. Since rank(NN^T)≤rank(N)≤b, Fisher’s inequality gives b≥v. The first two identities are double-counting identities; the third shows that incomplete blocks require at least as many blocks as treatments.

Part (b). Confounding occurs when a treatment effect is deliberately aliased with a block effect, so that effect cannot be estimated separately; in factorial experiments a higher-order interaction is usually sacrificed to obtain smaller, more homogeneous blocks. Complete confounding would confound the same interaction in all replications, making it wholly unestimable; partial confounding confounds different interactions in different replications, so each is lost only in part. For a 2^3 factorial, write combinations in Yates notation as (1), a, b, ab, c, ac, bc, abc. In each replication, put combinations having the same sign of the confounded interaction in one block; the block contrast is that interaction. Replication I, confounding ABC: Block 1: a, b, c, abc; Block 2: (1), ab, ac, bc. Replication II, confounding AC: Block 1: (1), b, ac, abc; Block 2: a, c, ab, bc. Replication III, confounding BC: Block 1: (1), a, bc, abc; Block 2: b, c, ab, ac. Thus ABC is lost in Rep I, AC in Rep II and BC in Rep III, while main effects and the other interactions remain estimable from the remaining replications.

Part (c). Fixed, random and mixed models differ in the nature of factor levels, the error structure, and the target of inference. In a fixed-effect model the levels are specifically chosen and conclusions apply only to those levels; the treatment effects are constants, e.g. comparing three named paddy varieties or fixed doses of urea at 0, 50 and 100 kg N/ha. In a random-effect model the levels are a random sample from a larger population, and the interest is in variance components and generalisation, e.g. randomly selected villages, fields, years or batches. A mixed model contains both, e.g. fixed fertilizer levels applied over randomly selected farms, or fixed varieties tested over random locations. In Indian agricultural trials, fixed varieties and random districts illustrate such a mixed model. RBD fulfils the three basic principles of experimental design. Replication is achieved because each treatment is repeated in every block, so r=b, giving independent error estimates. Randomisation is applied within each block by assigning treatments to plots at random, preventing systematic bias and justifying the error term. Local control is obtained by forming blocks of homogeneous units, such as plots with similar soil fertility, slope or moisture, so block-to-block variation is removed from experimental error. Hence RBD is a complete-block design, BIBD an incomplete-block design, confounding is controlled aliasing, and the effect model is chosen according to whether factor levels are fixed, random or mixed.