CSIR-UGC NET Nonparametric & Rank-Based Methods: practice questions

CSIR-UGC NET - Mathematical Sciences (statistics option) · India · National Testing Agency for CSIR

Original practice questions written to the published syllabus · not past papers

10worked here, free

37written to this exam in this subject

11syllabus units

What this subject covers

Methods 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)

Empirical distribution function and its properties
Goodness of fit: chi-square and Kolmogorov-Smirnov tests
Run test for randomness
Sign test and Wilcoxon signed-rank test
Mann-Whitney U test and two-sample Kolmogorov-Smirnov test
Kruskal-Wallis test
Friedman's test for randomised blocks
Cochran's Q test
Spearman's and Kendall's rank correlations
Asymptotic relative efficiency
Analysing ordinal scores: disease ratings and sensory panels

Where the 37 questions fall

How our questions for CSIR-UGC NET spread across the units of Nonparametric & Rank-Based Methods. Difficulty: 0 easy, 12 medium, 25 hard.

Empirical distribution function and its properties	5
Sign test and Wilcoxon signed-rank test	4
Mann-Whitney U test and two-sample Kolmogorov-Smirnov test	4
Kruskal-Wallis test	4
Spearman's and Kendall's rank correlations	4
Asymptotic relative efficiency	4
Goodness of fit: chi-square and Kolmogorov-Smirnov tests	3
Cochran's Q test	3
Run test for randomness	2
Friedman's test for randomised blocks	2
Analysing ordinal scores: disease ratings and sensory panels	2

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Run test for randomness · hard

Under randomness, how does the number of runs R behave for large n₁ and n₂?

Approximately χ² with N − 1 df
Approximately Poisson with mean N/2
Approximately normal with mean 2n₁n₂/N + 1
Exactly binomial with N trials and p = 1/2

Answer and working

C. Approximately normal with mean 2n₁n₂/N + 1

R is asymptotically normal (Wald and Wolfowitz), with the stated mean and variance 2n₁n₂(2n₁n₂ − N)/[N²(N − 1)], which justifies the Z approximation.

Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods

2 · Sign test and Wilcoxon signed-rank test · medium

Which estimator of the centre of symmetry is associated with the Wilcoxon signed-rank test?

The sample mean of the differences
The mode of the differences
The median of the Walsh averages
The midrange of the differences

Answer and working

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

3 · Mann-Whitney U test and two-sample Kolmogorov-Smirnov test · hard

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)?

4.33 g
6.00 g
4.00 g
−2.00 g

Answer and working

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

4 · Kruskal-Wallis test · hard

For k = 2 groups, how is the Kruskal-Wallis H related to the Mann-Whitney test?

They are unrelated statistics
H is half the Mann-Whitney U
H equals the Mann-Whitney U itself, unstandardised
H equals the square of the Mann-Whitney Z

Answer and working

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

5 · Friedman's test for randomised blocks · hard

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?

Mean b/2, variance b/4
Mean k(k + 1)/2, variance b(k² − 1)/12
Mean b(k + 1)/2, variance b(k + 1)/12
Mean b(k + 1)/2, variance b(k² − 1)/12

Answer and working

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

6 · Cochran's Q test · medium

What is the null hypothesis of Cochran's Q test?

All responses are independent Bernoulli(0.5) trials in every block
Treatment and block effects are independent
The blocks have equal success probabilities
The success probability is the same for all k treatments

Answer and working

D. The success probability is the same for all k treatments

Q tests equality of the treatment marginal success proportions, allowing blocks to differ. It does not assume p = 0.5 or test blocks.

Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods

7 · Spearman's and Kendall's rank correlations · medium

Under independence, what is the variance of Kendall's tau for n = 10?

0.1000
0.0617
0.5556
0.1111

Answer and working

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

8 · Asymptotic relative efficiency · hard

Hodges and Lehmann proved a lower bound for the ARE of the Wilcoxon (or Mann-Whitney) test relative to the t-test, over all continuous symmetric distributions. What is it?

0.864
0.500
0.955
0.637

Answer and working

A. 0.864

108/125 = 0.864; it can never be lower, while it can be arbitrarily large for heavy-tailed distributions. 0.955 is the value at the normal, not the minimum.

Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods

9 · Analysing ordinal scores: disease ratings and sensory panels · hard

How is the Mann-Whitney test related to the proportional-odds model for comparing two groups on an ordinal scale?

It is closely related to the score test for the group effect in that model
It is unrelated, because the model is parametric and the test is not
It is identical to the Wald test for the cut-point parameters θⱼ
It is the likelihood-ratio test of the proportional-odds assumption itself

Answer and working

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

10 · Sign test and Wilcoxon signed-rank test · medium

In the Wilcoxon signed-rank test, how is the null variance of T⁺ adjusted when some |dᵢ| are tied and given mid-ranks?

It is reduced by Σ(t³ − t)/48 over tie groups of size t
It is unchanged, since mid-ranks keep the same sum
It is divided by the number of tie groups
It is increased by Σ(t³ − t)/48 over tie groups

Answer and working

A. It is reduced by Σ(t³ − t)/48 over tie groups of size t

Mid-ranks keep E(T⁺) but lower the variance of the ranks, so Var(T⁺) = n(n + 1)(2n + 1)/24 − Σ(t³ − t)/48.

Further reading: Gibbons and Chakraborti, Nonparametric Statistical Inference; Conover, Practical Nonparametric Statistics; Hollander, Wolfe and Chicken, Nonparametric Statistical Methods

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