Anderson-Darling k-Sample Test

Stefan Kloppenborg, Jeffrey Borlik

20-Jan-2019

This vignette explores the Anderson–Darling k-Sample test. CMH-17-1G [1] provides a formulation for this test that appears different than the formulation given by Scholz and Stephens in their 1987 paper [2].

Both references use different nomenclature, which is summarized as follows:

Given the possibility of ties in the data, the discrete version of the test must be used Scholz and Stephens (1987) give the test statistic as:

$$A_{a k N}^2 = \frac{N - 1}{N}\sum_{i=1}^k \frac{1}{n_i}\sum_{j=1}^{L}\frac{l_j}{N}\frac{\left(N M_{a i j} - n_i B_{a j}\right)^2}{B_{a j}\left(N - B_{a j}\right) - N l_j / 4}$$

CMH-17-1G gives the test statistic as:

$$ADK = \frac{n - 1}{n^2\left(k - 1\right)}\sum_{i=1}^k\frac{1}{n_i}\sum_{j=1}^L h_j \frac{\left(n F_{i j} - n_i H_j\right)^2}{H_j \left(n - H_j\right) - n h_j / 4}$$

By inspection, the CMH-17-1G version of this test statistic contains an extra factor of $\frac{1}{\left(k - 1\right)}$.

Scholz and Stephens indicate that one rejects $H_0$ at a significance level of $\alpha$ when:

$$\frac{A_{a k N}^2 - \left(k - 1\right)}{\sigma_N} \ge t_{k - 1}\left(\alpha\right)$$

This can be rearranged to give a critical value:

$$A_{c r i t}^2 = \left(k - 1\right) + \sigma_N t_{k - 1}\left(\alpha\right)$$

CHM-17-1G gives the critical value for $ADK$ for $\alpha=0.025$ as:

$$ADC = 1 + \sigma_n \left(1.96 + \frac{1.149}{\sqrt{k - 1}} - \frac{0.391}{k - 1}\right)$$

The definition of $\sigma_n$ from the two sources differs by a factor of $\left(k - 1\right)$.

The value in parentheses in the CMH-17-1G critical value corresponds to the interpolation formula for $t_m\left(\alpha\right)$ given in Scholz and Stephen’s paper. It should be noted that this is not the student’s t-distribution, but rather a distribution referred to as the $T_m$ distribution.

The cmstatr package use the package kSamples to perform the k-sample Anderson–Darling tests. This package uses the original formulation from Scholz and Stephens, so the test statistic will differ from that given software based on the CMH-17-1G formulation by a factor of $\left(k-1\right)$.

For comparison, SciPy’s implementation also uses the original Scholz and Stephens formulation. The statistic that it returns, however, is the normalized statistic, $\left[A_{a k N}^2 - \left(k - 1\right)\right] / \sigma_N$, rather than kSamples’s $A_{a k N}^2$ value. To be consistent, SciPy also returns the critical values $t_{k-1}(\alpha)$ directly. (Currently, SciPy also floors/caps the returned p-value at 0.1% / 25%.) The values of $k$ and $\sigma_N$ are available in cmstatr’s ad_ksample return value, if an exact comparison to Python SciPy is necessary.

The conclusions about the null hypothesis drawn, however, will be the same, whether R or CMH-17-1G or SciPy.