# KSample¶

class hyppo.ksample.KSample(indep_test, compute_distkern='euclidean', bias=False, **kwargs)

Nonparametric K-Sample Testing test statistic and p-value.

A k-sample test tests equality in distribution among groups. Groups can be of different sizes, but generally have the same dimensionality. There are not many non-parametric k-sample tests, but this version cleverly leverages the power of some of the implemented independence tests to test this equality of distribution.

Parameters
• indep_test ("CCA", "Dcorr", "HHG", "RV", "Hsic", "MGC", "KMERF", or list) -- A string corresponding to the desired independence test from hyppo.independence. This is not case sensitive. If using "MaxMargin" then this must be a list containing "MaxMargin" in the first index and another indep_test in the second index.

• compute_distkern (str, callable, or None, default: "euclidean" or "gaussian") -- A function that computes the distance among the samples within each data matrix. Valid strings for compute_distance are, as defined in sklearn.metrics.pairwise_distances,

• From scikit-learn: ["euclidean", "cityblock", "cosine", "l1", "l2", "manhattan"] See the documentation for scipy.spatial.distance for details on these metrics.

• From scipy.spatial.distance: ["braycurtis", "canberra", "chebyshev", "correlation", "dice", "hamming", "jaccard", "kulsinski", "mahalanobis", "minkowski", "rogerstanimoto", "russellrao", "seuclidean", "sokalmichener", "sokalsneath", "sqeuclidean", "yule"] See the documentation for scipy.spatial.distance for details on these metrics.

Alternatively, this function computes the kernel similarity among the samples within each data matrix. Valid strings for compute_kernel are, as defined in sklearn.metrics.pairwise.pairwise_kernels,

["additive_chi2", "chi2", "linear", "poly", "polynomial", "rbf", "laplacian", "sigmoid", "cosine"]

Note "rbf" and "gaussian" are the same metric.

• bias (bool, default: False) -- Whether or not to use the biased or unbiased test statistics (for indep_test="Dcorr" and indep_test="Hsic").

• **kwargs -- Arbitrary keyword arguments for compute_distkern.

Notes

The formulation for this implementation is as follows 1:

The k-sample testing problem can be thought of as a generalization of the two sample testing problem. Define $$\{ u_i \stackrel{iid}{\sim} F_U,\ i = 1, ..., n \}$$ and $$\{ v_j \stackrel{iid}{\sim} F_V,\ j = 1, ..., m \}$$ as two groups of samples deriving from different distributions with the same dimensionality. Then, problem that we are testing is thus,

$\begin{split}H_0: F_U &= F_V \\ H_A: F_U &\neq F_V\end{split}$

The closely related independence testing problem can be generalized similarly: Given a set of paired data $$\{\left(x_i, y_i \right) \stackrel{iid}{\sim} F_{XY}, \ i = 1, ..., N\}$$, the problem that we are testing is,

$\begin{split}H_0: F_{XY} &= F_X F_Y \\ H_A: F_{XY} &\neq F_X F_Y\end{split}$

By manipulating the inputs of the k-sample test, we can create concatenated versions of the inputs and another label matrix which are necessarily paired. Then, any nonparametric test can be performed on this data.

Letting $$n = \sum_{i=1}^k n_i$$, define new data matrices $$\mathbf{x}$$ and $$\mathbf{y}$$ such that,

\begin{split}\begin{align*} \mathbf{x} &= \begin{bmatrix} \mathbf{u}_1 \\ \vdots \\ \mathbf{u}_k \end{bmatrix} \in \mathbb{R}^{n \times p} \\ \mathbf{y} &= \begin{bmatrix} \mathbf{1}_{n_1 \times 1} & \mathbf{0}_{n_1 \times 1} & \ldots & \mathbf{0}_{n_1 \times 1} \\ \mathbf{0}_{n_2 \times 1} & \mathbf{1}_{n_2 \times 1} & \ldots & \mathbf{0}_{n_2 \times 1} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{n_k \times 1} & \mathbf{0}_{n_k \times 1} & \ldots & \mathbf{1}_{n_k \times 1} \\ \end{bmatrix} \in \mathbb{R}^{n \times k} \end{align*}\end{split}

\begin{split}\begin{align*} \mathbf{x} &= \begin{bmatrix} \mathbf{u}_1 \\ \mathbf{u}_2 \end{bmatrix} \in \mathbb{R}^{n \times p} \\ \mathbf{y} &= \begin{bmatrix} \mathbf{0}_{n_1 \times 1} \\ \mathbf{1}_{n_2 \times 1} \end{bmatrix} \in \mathbb{R}^n \end{align*}\end{split}

Given $$\mathbf{u}$$ and $$\mathbf{v}$$ as defined above, to perform a $$w$$-way test where $$w < k$$,

$\begin{split}\mathbf{y} = \begin{bmatrix} \mathbf{1}_{n_1 \times 1} & \mathbf{0}_{n_1 \times 1} & \ldots & \mathbf{1}_{n_1 \times 1} \\ \mathbf{1}_{n_2 \times 1} & \mathbf{1}_{n_2 \times 1} & \ldots & \mathbf{0}_{n_2 \times 1} \\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}_{n_k \times 1} & \mathbf{1}_{n_k \times 1} & \ldots & \mathbf{1}_{n_k \times 1} \\ \end{bmatrix} \in \mathbb{R}^{n \times k}.\end{split}$

where each row of $$\mathbf{y}$$ contains $$w$$ $$\mathbf{1}_{n_i}$$ elements. This leads to label matrix distances proportional to how many labels (ways) samples differ by, a hierarchy of distances between samples thought to be true if the null hypothesis is rejected.

Performing a multilevel test involves constructing $$x$$ and $$y$$ using either of the methods above and then performing a block permutation 2. Essentially, the permutation is striated, where permutation is limited to be within a block of samples or between blocks of samples, but not both. This is done because the data is not freely exchangeable, so it is necessary to block the permutation to preserve the joint distribution 2.

The p-value returned is calculated using a permutation test uses hyppo.tools.perm_test. The fast version of the test uses hyppo.tools.chi2_approx.

References

1

Sambit Panda, Cencheng Shen, Ronan Perry, Jelle Zorn, Antoine Lutz, Carey E. Priebe, and Joshua T. Vogelstein. Nonpar MANOVA via Independence Testing. arXiv:1910.08883 [cs, stat], April 2021. arXiv:1910.08883.

2(1,2)

Anderson M. Winkler, Matthew A. Webster, Diego Vidaurre, Thomas E. Nichols, and Stephen M. Smith. Multi-level block permutation. NeuroImage, 123:253–268, December 2015. doi:10.1016/j.neuroimage.2015.05.092.

Methods Summary

 KSample.statistic(*args) Calulates the k-sample test statistic. KSample.test(*args[, reps, workers, auto, ...]) Calculates the k-sample test statistic and p-value.

KSample.statistic(*args)

Calulates the k-sample test statistic.

Parameters

*args (ndarray of float) -- Variable length input data matrices. All inputs must have the same number of dimensions. That is, the shapes must be (n, p) and (m, p), ... where n, m, ... are the number of samples and p is the number of dimensions.

Returns

stat (float) -- The computed k-sample statistic.

KSample.test(*args, reps=1000, workers=1, auto=True, random_state=None)

Calculates the k-sample test statistic and p-value.

Parameters
• *args (ndarray of float) -- Variable length input data matrices. All inputs must have the same number of dimensions. That is, the shapes must be (n, p) and (m, p), ... where n, m, ... are the number of samples and p is the number of dimensions.

• reps (int, default: 1000) -- The number of replications used to estimate the null distribution when using the permutation test used to calculate the p-value.

• workers (int, default: 1) -- The number of cores to parallelize the p-value computation over. Supply -1 to use all cores available to the Process.

• auto (bool, default: True) -- Only applies to "Dcorr" and "Hsic". Automatically uses fast approximation when n and size of array is greater than 20. If True, and sample size is greater than 20, then hyppo.tools.chi2_approx will be run. Parameters reps and workers are irrelevant in this case. Otherwise, hyppo.tools.perm_test will be run.

Returns

Examples

>>> import numpy as np
>>> from hyppo.ksample import KSample
>>> x = np.arange(7)
>>> y = x
>>> z = np.arange(10)
>>> stat, pvalue = KSample("Dcorr").test(x, y)
>>> '%.3f, %.1f' % (stat, pvalue)
'-0.136, 1.0'