ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR
Original practice questions written to the published syllabus · not past papersMethods that hold without a normal model: the empirical distribution, goodness-of-fit, sign and rank tests for one, two and many samples, rank correlation, and tests used for scored or ranked agricultural data such as disease ratings and sensory panels.
In the exams: AICE-JRF Ag Stat Unit 2 (heavy: run, sign, rank, Wilcoxon, Kruskal-Wallis, Mann-Whitney, Cochran, Friedman named); GATE ST 'Non-parametric Statistics' (heavy); UPSC ISS Statistics II (present)
ICAR AICE-JRF/SRF Agricultural Statistics (Unit 2)
Run, sign, Wilcoxon, Mann-Whitney, Kruskal-Wallis, Friedman and Cochran's Q tests by name: when each applies, the statistic, null mean and variance, and degrees of freedom of the chi-square approximation.
How our questions for ICAR AICE-JRF/SRF spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 11 easy, 24 medium, 25 hard.
| Sign test and Wilcoxon signed-rank test | 7 | |
| Kruskal-Wallis test | 7 | |
| Empirical distribution function and its properties | 6 | |
| Friedman's test for randomised blocks | 6 | |
| Spearman's and Kendall's rank correlations | 6 | |
| Run test for randomness | 5 | |
| Mann-Whitney U test and two-sample Kolmogorov-Smirnov test | 5 | |
| Cochran's Q test | 5 | |
| Asymptotic relative efficiency | 5 | |
| Goodness of fit: chi-square and Kolmogorov-Smirnov tests | 4 | |
| Analysing ordinal scores: disease ratings and sensory panels | 4 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What are E[Fₙ(x)] and Var[Fₙ(x)] for the empirical distribution function at a fixed x?
B. F(x) and F(x)[1 − F(x)]/n
Since nFₙ(x) ~ binomial(n, F(x)), Fₙ(x) is unbiased for F(x) with variance F(x)[1 − F(x)]/n. 1/(4n) is only the maximum of this variance, reached at the median.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
What does the Wilcoxon signed-rank test assume about the distribution of the paired differences that the sign test does not?
A. Symmetry of the differences
Both need independence and continuity; the signed-rank test also assumes symmetry, which is what lets each rank carry either sign with probability 1/2. Neither assumes normality.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Which estimator of the centre of symmetry is associated with the Wilcoxon signed-rank test?
C. The median of the Walsh averages
The Hodges-Lehmann one-sample estimator is the median of all (dᵢ + dⱼ)/2, i ≤ j. It inverts the signed-rank test and is robust to outliers.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Treated plants yield 12, 16 and 18 g; untreated plants yield 9, 10 and 14 g. What is the Hodges-Lehmann estimate of the shift (treated − untreated)?
C. 4.00 g
The nine differences xᵢ − yⱼ, sorted, are −2, 2, 2, 3, 4, 6, 7, 8, 9; their median is 4 g. The difference of sample medians is 16 − 10 = 6 g; the difference of means is 15.33 − 11 = 4.33 g; −2 g is the smallest pairwise difference.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
For k = 2 groups, how is the Kruskal-Wallis H related to the Mann-Whitney test?
D. H equals the square of the Mann-Whitney Z
With two groups H is the squared standardised rank-sum statistic, Z², and χ²₁ is the distribution of Z² for normal Z.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Treatments are increasing doses of a nutrient, and the alternative of interest is a monotone increase in response. Which rank test is more powerful than Kruskal-Wallis here?
C. The Jonckheere-Terpstra test
Jonckheere-Terpstra targets ordered alternatives across independent groups (Page's test does the same for blocked designs). Kruskal-Wallis spends power on all possible patterns.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Which design is Friedman's test intended for?
C. An RCBD with one observation per treatment per block
Friedman ranks treatments within blocks, the rank analogue of the RCBD ANOVA. Binary blocked data use Cochran's Q.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under H₀ in Friedman's test with b blocks and k treatments, what are the mean and variance of a single treatment's rank sum Rⱼ?
D. Mean b(k + 1)/2, variance b(k² − 1)/12
In each block a treatment's rank is uniform on 1, ..., k, with mean (k + 1)/2 and variance (k² − 1)/12; summing over b independent blocks multiplies both by b.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
Under independence, what is the variance of Kendall's tau for n = 10?
B. 0.0617
Var(τ) = 2(2n + 5)/[9n(n − 1)] = 50/810 = 0.0617. 1/(n − 1) = 0.1111 is the null variance of Spearman's ρ_s.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
How is the Mann-Whitney test related to the proportional-odds model for comparing two groups on an ordinal scale?
A. It is closely related to the score test for the group effect in that model
For two groups, the score test of β = 0 in the cumulative logit model is essentially the Wilcoxon rank-sum test with mid-ranks, which is why the two usually agree; the model adds odds-ratio estimates and covariates.
Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods
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