UPHESC Assistant Professor Agricultural Statistics PYQs

Concept: A Youden square is an incomplete Latin square: a symmetric BIBD (b = v, r = k) whose blocks are laid out as rows, with each treatment appearing exactly once in each column (position).

1. Taking rows as blocks, every pair of treatments occurs together in λ rows ⇒ the rows form a (symmetric) BIBD.
2. The other classification is complete (each treatment once per column), so two-way elimination of heterogeneity is possible, as in an LSD.
3. Example: 7 treatments in a 7 × 3 arrangement (v = b = 7, r = k = 3, λ = 1).

Short trick: Youden square = symmetric BIBD arranged like an incomplete Latin square.

Concept: BIBD relations: $bk=vr$ and $\lambda (v-1)=r(k-1)$; Fisher's inequality $b\ge v$.

1. Symmetric BIBD: b = v (statement 1 true), and then bk = vr gives r = k.
2. So statement 2 is wrong: in a symmetric BIBD r = k, and the relation is $\lambda (v-1)=k(k-1)$, not $v(\lambda -1)=k(k-1)$.

Short trick: Symmetric ⇒ b = v and r = k.

Concept: Split-plot design: factors needing large plots (sowing methods, irrigation, tillage) go to main plots; factors that can use small plots (varieties, fertiliser doses) go to sub-plots.

1. Main-plot factor is estimated less precisely; sub-plot factor and interaction more precisely.
2. Youden square/lattice handle many treatments in incomplete blocks; cross-over is for animal/clinical trials with periods.

Short trick: Big-plot factor → main plot; small-plot factor → sub-plot.

Concept: Euler (1782) conjectured that no pair of orthogonal Latin squares exists for orders 6, 10, 14, … (4n + 2).

1. Order 6 is truly impossible (Tarry, 1900; the 36 officers problem).
2. In 1959–60 R.C. Bose, S.S. Shrikhande and E.T. Parker constructed orthogonal Latin squares of order 22 and 10 and showed they exist for every order except 2 and 6.
3. They became known as the 'Euler's spoilers'.

Short trick: Euler's spoilers = Bose, Shrikhande (and Parker).

Concept: When two effects are confounded, their generalised interaction (product with squares cancelled) is also confounded.

1. ABC × BCD = A B² C² D = AD.
2. 16 combinations in blocks of 4 ⇒ 4 blocks ⇒ 3 d.f. confounded: ABC, BCD and AD.

Short trick: Multiply the letters and delete any letter that appears twice.

Concept: Stratification forms internally homogeneous, mutually heterogeneous strata; a sample is drawn from every stratum.

1. Precision gain comes from small within-stratum variance.
2. Cluster sampling is the opposite ideal: clusters should be internally heterogeneous (each a mini-population).

Short trick: Strata: homogeneous within, different between. Clusters: heterogeneous within.

Concept: Sampling error arises only because a part (sample) is observed instead of the whole; non-sampling errors arise in observation, response, recording, processing, etc.

1. Cluster sampling is a probability method ✓; quota sampling is non-probability ✓.
2. Processing error is a non-sampling error ✓.
3. Observational (measurement) error is a non-sampling error, so matching it with sampling error is wrong ✗.

Short trick: Sampling error vanishes in a complete census; non-sampling errors do not.

Concept: India's Agriculture Census (operational holdings) began with reference year 1970-71 under the FAO World Census of Agriculture programme and is held every five years.

1. Five-yearly interval ✓ (1970-71, 1976-77, 1980-81, … 2015-16, 2021-22).
2. 1980-81 was the third census, not the first ✗.
3. Crop cutting experiments (Mahalanobis, Sukhatme) give yield estimates ✓; in systematic sampling only the first unit is random ✓.

Short trick: First Agriculture Census = 1970-71.

Concept: Ratio estimator beats the sample mean when $\rho >\frac{1}{2}\frac{C_x}{C_y}$.

1. $V({\overline{y}}_R)\approx \frac{1-f}{n}{\overline{Y}}^2(C_y^2+C_x^2-2\rho C_x{C}_y)$; $V(\overline{y})=\frac{1-f}{n}{\overline{Y}}^2{C}_y^2$.
2. $V({\overline{y}}_R)<V(\overline{y})⟺C_x^2<2\rho C_x{C}_y⟺\rho >\frac{C_x}{2C_y}$.
3. With $C_x=C_y$: $\rho >0.5$ (strict; at 0.5 the two are equally efficient).

Short trick: Ratio estimator pays off when ρ > ½ (Cx/Cy).

Concept: In linear systematic sampling a random start r is chosen from 1 … K, and then every K-th unit is taken.

1. Only K distinct samples are possible, each fully determined by the random start.
2. So each possible sample has probability 1/K.
3. n/N is the inclusion probability of a single unit, not of the whole sample.

Short trick: K possible samples ⇒ probability 1/K each.

Concept: Murthy (1957) obtained his unordered estimator by applying the Rao–Blackwell idea to an ordered estimator (Des Raj's), i.e. averaging over all orderings of the same set of units.

1. The unordered sample is a sufficient statistic; conditioning on it cannot increase variance.
2. Hence $V(M)\le V(O)$, with equality only in trivial cases.

Short trick: Unordering = Rao–Blackwellisation ⇒ variance never larger.

Concept: With σ known: $\overline{x}\pm 1.96\frac{\sigma }{\sqrt{n}}$.

1. σ = √25 = 5, n = 100 ⇒ SE = 5/10 = 0.5.
2. 1.96 × 0.5 = 0.98 ⇒ 15 ± 0.98 = (14.02, 15.98).

Short trick: The interval must be symmetric about 15 — only (D) is.

Concept: Genetic advance $GA=k{\sigma }_P{h}^2$ (k = 2.06 at 5% selection intensity).

1. High h² with high GA (as % of mean) indicates additive gene action ⇒ simple selection is effective — statement 1 true (Panse, 1957).
2. Johnson, Robinson and Comstock (1955): heritability is more useful when used together with genetic advance to predict response — statement 2 true.

Short trick: High h² + high GA ⇒ additive genes ⇒ select.

Concept: A full diallel includes all p² crosses — parents, F₁s and their reciprocals (Griffing's method 1).

1. Only a full diallel lets reciprocal (maternal/cytoplasmic) effects be estimated and removed.
2. Half diallel (p(p − 1)/2 or p(p + 1)/2 crosses) assumes no reciprocal differences; partial diallel samples only some crosses.

Short trick: Reciprocals matter ⇒ make reciprocals ⇒ full diallel.

Concept: In an idealised population of size N: $F_t=\frac{1}{2N}+(1-\frac{1}{2N})F_{t-1}$, so $1-F_t=(1-\frac{1}{2N}{)}^t(1-F_0)$.

1. $\mathrm{\Delta }F=\frac{1}{2N}>0$ every generation, so F rises steadily towards 1.
2. Smaller N ⇒ faster rise (random drift + relatedness).

Short trick: Finite N ⇒ F increases by 1/(2N) per generation.

Concept: Comstock and Robinson's NC designs differ in how many parents are sampled for a given number of crosses.

1. NC I (nested): each male is mated to a different set of females ⇒ m males + mf females — the largest number of parents, widest sample of the population.
2. NC III: n random F₂ plants backcrossed to the two parental lines ⇒ n + 2 parents for 2n crosses — intermediate.
3. NC II (factorial): every male × every female ⇒ only m + f parents for mf crosses — the narrowest coverage.
4. Order of coverage: NC I > NC III > NC II.

Short trick: Nested design uses fresh females for every male ⇒ most parents ⇒ best coverage.

Concept: Equal parent index assumes the daughter is the average of sire and dam: $\overline{D}=\frac{S+\overline{M}}{2}$ ⇒ $S=2\overline{D}-\overline{M}$.

1. So the correct equal-parent index is $I=2\overline{D}-\overline{M}$; the printed $2\overline{M}-\overline{D}$ has D and M interchanged — statement 1 false.
2. Krishnan's modified index corrects the daughters' average for the dams' deviation from the herd average using the intra-sire regression: $I=\overline{D}-b(\overline{M}-A)$ — statement 2 true.

Short trick: Sire = 2 × daughters − dams (not the other way).

Concept: Genetic correlation is a ratio involving the genetic covariance and two genetic variances, each estimated with error.

1. $r_G=\frac{{\mathrm{cov}}_G(x,y)}{\sqrt{{\sigma }_{G_x}^2{\sigma }_{G_y}^2}}$ combines three sampling errors, while $h^2$ involves one variance ratio.
2. Hence, from the same data, genetic correlations are estimated far less precisely: SE(h²) < SE(r_G).
3. This is why large samples are needed for reliable genetic correlations (Falconer).

Short trick: More components in the ratio ⇒ larger standard error ⇒ r_G is the least precise.

Concept: The three observations are the labels 1, 2 and 1, so $L(\theta )=p(1)p(2)p(1)={\theta }^2\cdot 2\theta (1-\theta )\cdot {\theta }^2=2{\theta }^5(1-\theta )$.

1. $\log L=\log 2+5\log \theta +\log (1-\theta )$.
2. $\frac{d}{d\theta }\log L=\frac{5}{\theta }-\frac{1}{1-\theta }=0\Rightarrow 5-5\theta =\theta \Rightarrow \hat{\theta }=\frac{5}{6}$.
3. Check by gene counting: allele '1' appears 2 + 1 + 2 = 5 times out of 6 genes ⇒ 5/6.

Short trick: HW model: MLE of θ = (2·n₁ + n₂)/(2n) — simple allele counting.

Concept: $d=\frac{\sum (e_t-e_{t-1}{)}^2}{\sum e_t^2}\approx 2(1-\hat{\rho })$, so 0 ≤ d ≤ 4.

1. DW statistic tests first-order autocorrelation of regression residuals: (a)-(2).
2. d = 2 ⇔ ρ̂ = 0, no autocorrelation: (b)-(4).
3. d below 2 (towards 0) ⇔ ρ̂ > 0, positive autocorrelation: (c)-(1).
4. d above 2 (towards 4) ⇔ ρ̂ < 0, negative autocorrelation: (d)-(3).

Short trick: d ≈ 2(1 − ρ): 0 = perfect positive, 2 = none, 4 = perfect negative.