GATE Statistics Non-parametric Statistics Flashcards

24 sample cards from the Non-parametric Statistics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. What is the null hypothesis of the Mann-Whitney U-test?

   That two independent samples come from the same distribution (equivalently $P(X<Y)=\tfrac{1}{2}$, often stated as equal medians/locations under a shift model): $$H_0:\;F=G.$$

2. Give the formula for the Mann-Whitney U statistic in terms of the rank sum.

   With sample sizes $n_1,n_2$ and $R_1$ the sum of ranks of sample 1 in the combined ranking, $$U_1=R_1-\frac{n_1(n_1+1)}{2},\qquad U_2=n_1n_2-U_1.$$ The test statistic is $U=\min(U_1,U_2)$.

3. What identity links the two Mann-Whitney statistics $U_1$ and $U_2$?

   $$U_1+U_2=n_1 n_2.$$

4. State the mean and variance of the Mann-Whitney $U$ statistic under $H_0$ (no ties).

   $$E[U]=\frac{n_1 n_2}{2},\qquad \operatorname{Var}[U]=\frac{n_1 n_2(n_1+n_2+1)}{12}.$$

5. What is the relationship between the Mann-Whitney U-test and the Wilcoxon rank-sum test?

   They are equivalent tests: the Wilcoxon rank-sum statistic (sum of ranks of one sample) and the Mann-Whitney $U$ differ only by the constant $\frac{n_1(n_1+1)}{2}$, so they yield identical p-values.

6. What is the parametric counterpart of the Mann-Whitney U-test, and when is U preferred?

   Its parametric counterpart is the two-sample (unpaired) $t$-test. The U-test is preferred when normality fails, for ordinal data, or with outliers, since it only uses ranks.

7. What is the parametric counterpart of the Wilcoxon signed-rank test?

   The paired-sample (one-sample) $t$-test. The signed-rank test replaces it when the differences are non-normal but symmetric.

8. What is Spearman's rank correlation coefficient measuring?

   It measures the strength and direction of a monotonic association between two variables, computed as the Pearson correlation applied to the ranks of the data.

9. Give the shortcut formula for Spearman's rank correlation coefficient $\rho_s$ when there are no tied ranks.

   $$\rho_s=1-\frac{6\sum_{i=1}^{n} d_i^{2}}{n(n^{2}-1)},$$ where $d_i$ is the difference between the ranks of the $i$-th pair.

10. What is the range of Spearman's $\rho_s$, and what do the extreme values mean?

    $-1\leq\rho_s\leq 1$. $\rho_s=+1$ indicates a perfect increasing monotonic relationship, $\rho_s=-1$ a perfect decreasing one, and $\rho_s=0$ no monotonic association.

11. How can the significance of Spearman's $\rho_s$ be tested for large $n$?

    Using the approximate $t$ statistic with $n-2$ degrees of freedom: $$t=\rho_s\sqrt{\frac{n-2}{1-\rho_s^{2}}}.$$ Alternatively $Z=\rho_s\sqrt{n-1}\sim N(0,1)$ for large $n$.

12. What is Kendall's tau ($\tau$) based on?

    It is based on counting concordant and discordant pairs of observations. A pair $(i,j)$ is concordant if the orderings of both variables agree, and discordant if they disagree.

13. Give the formula for Kendall's $\tau$ (tau-a, no ties).

    $$\tau=\frac{C-D}{\binom{n}{2}}=\frac{C-D}{\tfrac{1}{2}n(n-1)},$$ where $C$ is the number of concordant pairs and $D$ the number of discordant pairs.

14. What are the mean and variance of Kendall's $\tau$ under the null hypothesis of independence (no ties)?

    $$E[\tau]=0,\qquad \operatorname{Var}[\tau]=\frac{2(2n+5)}{9n(n-1)}.$$ The standardized statistic $Z=\tau/\sqrt{\operatorname{Var}[\tau]}$ is used for large-sample testing.

15. Compare Spearman's $\rho_s$ and Kendall's $\tau$ as measures of association.

    Both are rank-based measures of monotonic association in $[-1,1]$. Kendall's $\tau$ has a direct probabilistic interpretation (difference of concordance/discordance probabilities), is more robust to outliers, and converges faster to normality; Spearman's $\rho_s$ is easier to compute and typically larger in magnitude. Generally $|\tau|\leq|\rho_s|$.

16. What is Kendall's tau-b designed to handle, compared with tau-a?

    Tau-b adjusts for tied ranks by including tie corrections in the denominator: $$\tau_b=\frac{C-D}{\sqrt{(n_0-n_1)(n_0-n_2)}},$$ where $n_0=\tfrac{1}{2}n(n-1)$ and $n_1,n_2$ are tie terms for the two variables. Tau-a assumes no ties.

17. How are tied observations typically handled when assigning ranks in non-parametric tests?

    Tied observations are assigned the average (midrank) of the ranks they would otherwise occupy. This requires variance corrections in tests such as Wilcoxon, Mann-Whitney, and Spearman.

18. What does it mean for a test statistic to be 'distribution-free'?

    Its sampling distribution under $H_0$ does not depend on the specific (continuous) underlying distribution $F$ of the data. This is why rank-based statistics have universal tabulated critical values.

19. What is the asymptotic relative efficiency (ARE) of the sign test relative to the $t$-test under normality?

    $\frac{2}{\pi}\approx 0.637$. The sign test is comparatively inefficient under normality but can be more efficient for heavy-tailed distributions.

20. What is the asymptotic relative efficiency of the Wilcoxon signed-rank and Mann-Whitney tests relative to the $t$-test under normality?

    $\frac{3}{\pi}\approx 0.955$. They lose only about $5\%$ efficiency under normality but can be far more efficient for non-normal, heavy-tailed distributions (ARE never below $0.864$).

21. For the one-sample KS test, what is the limiting (Kolmogorov) distribution of $\sqrt{n}\,D_n$ under $H_0$?

    $$P\big(\sqrt{n}\,D_n\leq t\big)\to K(t)=1-2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}t^{2}},\quad t>0,$$ the Kolmogorov distribution.

22. In the sign test for paired data, how are observations exactly equal to the hypothesized median (ties/zeros) treated?

    They are discarded (excluded), and the effective sample size $n$ is reduced to the number of non-zero differences before applying the binomial distribution.

23. Which non-parametric test is the rank-based analogue of the one-way ANOVA, extending Mann-Whitney to $k>2$ groups?

    The Kruskal-Wallis $H$ test. It generalizes the Mann-Whitney U-test to compare more than two independent samples using combined-sample ranks.

24. Summarize which non-parametric test to use for: (a) one sample location, (b) two paired samples, (c) two independent samples.

    (a) Sign test or Wilcoxon signed-rank test (the latter if differences are symmetric). (b) Wilcoxon signed-rank test on the differences (or sign test). (c) Mann-Whitney U-test (equivalently Wilcoxon rank-sum test).

What this deck covers

Non-parametric Statistics flashcards FAQ