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

(a) RBD versus BIBD; proof of the BIBD relations

Difference: in a randomised block design every block contains all v treatments (k = v, r = b), so it is a complete-block design. In a BIBD each block holds only k < v treatments, so blocks are incomplete, and it is used when block size cannot be as large as the number of treatments. In a BIBD each treatment occurs in r blocks and each pair of treatments occurs together in exactly λ blocks, which keeps all treatment comparisons equally precise. Notation: v treatments, b blocks, r replications per treatment, k units per block, λ common pair-count. (i) bk = vr: the total number of units is counted by blocks (b blocks of k units each) and by treatments (v treatments of r units each), so bk = vr. (ii) λ(v − 1) = r(k − 1): fix one treatment. It occurs in r blocks, each containing k − 1 other treatments, so it meets other treatments r(k − 1) times. It meets each of the other v − 1 treatments λ times, so λ(v − 1) = r(k − 1). (iii) b ≥ v (Fisher's inequality): Step 1: Let N be the v × b incidence matrix. Then NN′ has diagonal entries r and off-diagonal entries λ, so NN′ = (r − λ)I + λJ. Step 2: From (ii), r > λ because k < v. Its determinant is (r − λ)^(v−1)(r + λ(v − 1)) = (r − λ)^(v−1) rk, which is non-zero. Step 3: Hence rank(NN′) = v. Since rank(NN′) ≤ rank(N) ≤ b, we get b ≥ v.

(b) Confounding and partial confounding in a 2³ experiment

Concept: confounding deliberately mixes (aliases) a treatment effect, usually a higher-order interaction, with block effects, so that blocks of a smaller size can be used while the other effects remain estimable. When the same effect is confounded in all replicates it is completely confounded and lost; when different effects are confounded in different replicates (partial confounding), each effect is estimated from the replicates where it is not confounded. Plan: the eight treatment combinations are (1), a, b, ab, c, ac, bc, abc. Each replicate has two blocks of four units. The two blocks separate combinations by the sign of the confounded contrast. Replicate 1, confounding ABC: Block 1 holds a, b, c, abc (odd number of letters); Block 2 holds (1), ab, ac, bc. Replicate 2, confounding AC: Block 1 holds (1), b, ac, abc; Block 2 holds a, c, ab, bc. Replicate 3, confounding BC: Block 1 holds (1), a, bc, abc; Block 2 holds b, c, ab, ac. Thus ABC, AC and BC are each lost in one replicate only. Their partial information, along with full information on A, B, C, AB, is obtained from the other replicates.

(c) Fixed, random and mixed models; basic principles in RBD

Fixed-effects model: the factor levels are chosen deliberately, effects are constants and inference applies only to those levels. Example: comparing three named paddy varieties, or urea doses of 0, 50 and 100 kg N/ha. Random-effects model: the levels are a random sample from a larger population, effects are random variables, and inference concerns variance components for the whole population. Example: randomly chosen villages, fields or years. Mixed model: some factors are fixed and others random. Example: fixed fertiliser levels tested on randomly selected farms. Basic principles in RBD: Replication: each treatment appears once in every block, so it is repeated b times, which gives an estimate of error. Randomisation: within each block treatments are allotted to plots at random, which avoids systematic bias and makes the error term valid. Local control: blocks group homogeneous plots (similar fertility, slope or moisture), so block-to-block variation is removed from the error.