Non-Parametric Statistics

The goodness-of-fit test checks whether observed category frequencies match a hypothesised distribution. For $k$ categories with observed counts $O_i$ and expected counts $E_i = n\cdot p_i$ (where $p_i$ are the hypothesised probabilities), the test statistic is $\chi^2 = \sum_{i=1}^k \frac{(O_i-E_i)^2}{E_i}$.

Under $H_0$, $\chi^2 \sim \chi^2_{k-1}$ (degrees of freedom $= k-1$; subtract one more for each parameter estimated from the data, e.g., if $\mu$ and $\sigma$ are estimated for a normal fit).

The test is always right-tailed: a large $\chi^2$ means the observed counts deviate substantially from expected. A small $\chi^2$ (not significant) means the data is consistent with $H_0$.

Validity condition: all expected counts $E_i \geq 5$. If some are too small, merge adjacent categories or use the exact multinomial test.

Applications: testing whether a die is fair, whether genetic ratios follow Mendelian predictions, or whether a sample comes from a specified distribution.

A two-way contingency table displays the joint frequencies of two categorical variables. The test of independence asks $H_0$: the two variables are independent (knowing one gives no information about the other).

Under independence, the expected count in cell $(i,j)$ is $E_{ij} = (\text{row}_i \text{ total}) \times (\text{col}_j \text{ total})/n$. The test statistic is $\chi^2 = \sum_{i,j} (O_{ij}-E_{ij})^2/E_{ij}$ with $(r-1)(c-1)$ degrees of freedom.

The chi-square test detects association but not direction or causation. For a $2\times 2$ table, the odds ratio or relative risk quantifies the strength of association beyond just significance.

Fisher's exact test: for small samples where $\chi^2$ is unreliable (expected counts $< 5$), compute the exact probability of the observed table and all more extreme ones. It is exact regardless of sample size.

The $\chi^2$ test for independence is equivalent to the $z$-test for comparing two proportions in a $2\times 2$ table: $z^2 = \chi^2$.

The Mann-Whitney U test (Wilcoxon rank-sum test) is the non-parametric alternative to the two-sample $t$-test. It tests whether one group tends to produce larger values than the other — formally, whether $P(X_1 > X_2) = 0.5$.

Procedure: combine both groups, rank all observations, sum the ranks for each group. The test statistic $U = R_1 - n_1(n_1+1)/2$ where $R_1$ is the rank sum for group $1$.

The Wilcoxon signed-rank test is the paired version. Compute differences $d_i = x_i - y_i$, rank their absolute values, and compare the sum of positive ranks to the sum of negative ranks. It is the non-parametric alternative to the paired $t$-test.

Rank-based tests are resistant to outliers because they use only the ordering of observations, not their actual values. A single extreme outlier changes only one rank.

Efficiency: when the normal distribution holds, the Mann-Whitney U test has about $95.5\%$ efficiency relative to the $t$-test (i.e., it needs about $5\%$ more observations to achieve the same power). For non-normal data, it can be more powerful than the $t$-test.

The Kruskal-Wallis test is the non-parametric equivalent of one-way ANOVA. It tests whether $k$ independent groups have the same distribution (equivalently, the same median). It ranks all $N$ observations and compares rank sums across groups.

Test statistic: $H = \frac{12}{N(N+1)}\sum_{j=1}^k \frac{R_j^2}{n_j} - 3(N+1)$, where $R_j$ is the rank sum for group $j$. Under $H_0$, $H \approx \chi^2_{k-1}$.

If Kruskal-Wallis is significant, use pairwise Mann-Whitney tests with Bonferroni correction for post-hoc comparisons.

Spearman's rank correlation $r_s$: compute ranks of $x$ and $y$ separately, then apply the Pearson correlation formula to the ranks. It measures monotone (not just linear) relationships and is robust to outliers and non-normality.

Use non-parametric tests when: the data is ordinal (e.g., satisfaction ratings on a $1$–$5$ scale); the sample size is small and normality cannot be assumed; there are severe outliers that resist transformation; or the outcome is inherently rank-based.

With large samples and continuous data, parametric tests are usually robust by the CLT. Non-parametric tests are most valuable for small $n$ with non-normal data.

Power trade-off: when parametric assumptions hold, non-parametric tests have somewhat lower power (need larger $n$ to detect the same effect). When assumptions are violated, non-parametric tests are often more powerful.

Parametric vs. non-parametric summary: $t$-test $\leftrightarrow$ Mann-Whitney U; paired $t$-test $\leftrightarrow$ Wilcoxon signed-rank; one-way ANOVA $\leftrightarrow$ Kruskal-Wallis; Pearson $r$ $\leftrightarrow$ Spearman $r_s$.