Comparing Two Populations: Independent Samples

Nonparametric methods for testing the independence of samples include Spearman's rank correlation coefficient, the Kendall tau rank correlation coefficient, the Kruskal–Wallis one-way analysis of variance, and the Walk–Wolfowitz runs test.

Spearman's Rank Correlation Coefficient

Spearman's rank correlation coefficient, often denoted by the Greek letter $\rho$ (rho), is a nonparametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can be described using a monotonic function. If there are no repeated data values, a perfect Spearman correlation of $1$ or $-1$ occurs when each of the variables is a perfect monotone function of the other.

For a sample of size $n$, the $n$ raw scores $X_i$, $Y_i$ are converted to ranks $x_i$, $y_i$, and $\rho$ is computed from these:

$\displaystyle{\rho = \frac{\sum_i (x_i - \bar{x}) (y_i - \bar{y})}{\sqrt{\sum_i(x_i - \bar{x})^2 \sum_i(y_i - \bar{y})^2}}}$

The sign of the Spearman correlation indicates the direction of association between $X$ (the independent variable) and $Y$ (the dependent variable). If $Y$ tends to increase when $X$ increases, the Spearman correlation coefficient is positive. If $Y$ tends to decrease when $X$ increases, the Spearman correlation coefficient is negative. A Spearman correlation of zero indicates that there is no tendency for $Y$ to either increase or decrease when $X$ increases.

The Kendall Tau Rank Correlation Coefficient

Kendall's tau ($\tau$) coefficient is a statistic used to measure the association between two measured quantities. A tau test is a non-parametric hypothesis test for statistical dependence based on the tau coefficient.

Let $(x_1, y_1), (x_2, y_2), \cdots, (x_n, y_n)$ be a set of observations of the joint random variables $X$ and $Y$ respectively, such that all the values of ($x_i$) and ($y_i$) are unique. Any pair of observations are said to be concordant if the ranks for both elements agree. The Kendall $\tau$ coefficient is defined as:

$\displaystyle{\tau = \frac{(\text{number of concordant pairs}) - (\text{number of discordant pairs})}{\frac{1}{2} n (n-1)}}$

The denominator is the total number pair combinations, so the coefficient must be in the range $-1 \leq \tau \leq 1$. If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value $1$. If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value $-1$. If $X$ and $Y$ are independent, then we would expect the coefficient to be approximately zero.

The Kruskal–Wallis One-Way Analysis of Variance

The Kruskal–Wallis one-way ANOVA by ranks is a nonparametric method for testing whether samples originate from the same distribution. It is used for comparing more than two samples that are independent, or not related. When the Kruskal–Wallis test leads to significant results, then at least one of the samples is different from the other samples. The test does not identify where the differences occur or how many differences actually occur.

Since it is a non-parametric method, the Kruskal–Wallis test does not assume a normal distribution, unlike the analogous one-way analysis of variance. However, the test does assume an identically shaped and scaled distribution for each group, except for any difference in medians.

The Walk–Wolfowitz Runs Test

The Walk–Wolfowitz runs test is a non-parametric statistical test that checks a randomness hypothesis for a two-valued data sequence. More precisely, it can be used to test the hypothesis that the elements of the sequence are mutually independent.

A "run" of a sequence is a maximal non-empty segment of the sequence consisting of adjacent equal elements. For example, the 22-element-long sequence

$++++−+++−++++++−$

consists of 6 runs, 3 of which consist of $+$ and the others of $-$. The run test is based on the null hypothesis that the two elements $+$ and $-$ are independently drawn from the same distribution.

The mean and variance parameters of the run do not assume that the positive and negative elements have equal probabilities of occurring, but only assume that the elements are independent and identically distributed. If the number of runs is significantly higher or lower than expected, the hypothesis of statistical independence of the elements may be rejected.