Note

Click here to download the full example code

# K-Sample Testing¶

A common problem experienced in research is the k-sample testing problem. Conceptually, it can be described as follows: consider k groups of data where each group had a different treatment. We can ask, are these groups the similar to one another or statistically different? More specifically, supposing that each group has a distribution, are these distributions equivalent to one another, or is one of them different?

If you are interested in questions of this mold, this module of the package is for you!
All our tests can be found in `hyppo.ksample`

, and will be elaborated in
detail below. But before that, let's look at the mathematical formulations:

Consider random variables \(U_1, U_2, \ldots, U_k\) with distributions \(F_{U_1}, F_{U_2}, \ldots F_{U_k}\). When performing k-sample testing, we are seeing whether or not these distributions are equivalent. That is, we are testing

Like all the other tests within hyppo, each method has a `statistic`

and
`test`

method. The `test`

method is the one that returns the test statistic
and p-values, among other outputs, and is the one that is used most often in the
examples, tutorials, etc.
The p-value returned is calculated using a permutation test using
`hyppo.tools.perm_test`

unless otherwise specified.

Specifics about how the test statistics are calculated for each in
`hyppo.ksample`

can be found the docstring of the respective test.
let's look at unique properties of some of these tests:

## Multivariate Analysis of Variance (MANOVA) and Hotelling¶

**MANOVA** is the current standard for k-sample testing in the literature.
More details can be found in `hyppo.ksample.MANOVA`

.
**Hotelling** is 2-sample MANOVA.
More details can be found in `hyppo.ksample.Hotelling`

.

Note

- Pros
Very fast

Similar to tests found in scientific literature

- Cons
Not accurate when compared to other tests in most situations

Assumes data is derived from a multivariate Gaussian

Assumes data is has same covariance matrix

Neither of these test are distance based, and so do not have a `compute_distance`

parameter and are not nonparametric, so they don't have `reps`

nor `workers`

.
Otherwise, these test runs like any other test.

## K-Sample Testing via Independence Testing¶

**Nonparametric MANOVA via Independence Testing** is a k-sample test that addresses
the aforementioned k-sample testing problem as follow: reduce the k-sample testing
problem to the independence testing problem (see Independence Testing).
To solve this, we create a new matrix of concatenated inputs and a matrix that labels
which of the concatenated data comes from which input [2].
Because independence tests have high finite sample testing power in some cases, this
method has a number of advantages.
More details can be found in `hyppo.ksample.KSample`

.
The following applies to both:

Note

If you want use 2-sample MGC, we have added that functionality to SciPy!
Please see `scipy.stats.multiscale_graphcorr`

.

Note

- Pros
Highly accurate

No additional computation complexity added

Not many assumptions of the data (only must be i.i.d.)

Has fast implementations (for

`indep_test="Dcorr"`

and`indep_test="Hsic"`

)

- Cons
Can be a little slower than some of the other tests in the package

The `indep_test`

parameter accepts a string corresponding to the name of the class
in the `hyppo.independence`

.
Other parameters are those in the corresponding independence test.
Since this this process is nearly the same for all independence tests, we are going
to use `hyppo.independence.MGC`

as the example independence test.

```
from hyppo.ksample import KSample
from hyppo.tools import rot_ksamp
# 100 samples, 1D cubic independence simulation, 3 groups sim, 60 degree rotation, no
# noise
sims = rot_ksamp("linear", n=100, p=1, k=3, degree=[60, -60], noise=True)
```

The data are points simulating a 1D linear relationship between random variables
\(X\) and \(Y\). It the concatenates these two matrices, and then rotates
the simulation by 60 degrees, generating the second and, in this case, the third
sample. It returns realizations as `numpy.ndarray`

.

```
import matplotlib.pyplot as plt
import seaborn as sns
# make plots look pretty
sns.set(color_codes=True, style="white", context="talk", font_scale=1)
# look at the simulation
plt.figure(figsize=(5, 5))
for sim in sims:
plt.scatter(sim[:, 0], sim[:, 1])
plt.xticks([])
plt.yticks([])
sns.despine(left=True, bottom=True, right=True)
plt.show()
# run k-sample test on the provided simulations. Note that *sims just unpacks the list
# we got containing our simulated data
stat, pvalue = KSample(indep_test="Dcorr").test(*sims)
print(stat, pvalue)
```

Out:

```
0.020257572395244467 0.007806778339791604
```

This was a general use case for the test, but there are a number of intricacies that
depend on the type of independence test chosen. Those same parameters can be modified
in this class. For a full list of the parameters, see the desired test in
`hyppo.independence`

and for examples on how to use it, see Independence Testing.

## Distance (and Kernel) Equivalencies¶

It turns out that a number of test statistics are multiples of one another and so, their p-values are equivalent to the above K-Sample Testing via Independence Testing. [1] goes through the distance and kernel equivalencies and [2] goes through the independence and two-sample (and by extension k-sample) equivalences in far more detail.

**Energy** is a powerful distance-based two sample test,
**Distance components (DISCO)** is the k-sample analogue to Energy,
and **Maximal mean discrepency (MMD)** is a powerful kernel-based two sample test,
These are equivalent to `hyppo.ksample.KSample`

using `indep_test="Dcorr"`

for Energy and DISCO and `indep_test="Hsic"`

for MMD.
More information can be found at `hyppo.ksample.Energy`

,
`hyppo.ksample.DISCO`

, and
`hyppo.ksample.MMD`

.
However, the test statistics have been modified to make it more in tune with other
implementations.

Note

- Pros
Highly accurate

Has similar test statistics to the literature

Has fast implementations

- Cons
Lower power than more computationally complex algorithms

For MMD, kernels are used instead of distances with the `compute_kernel`

parameter.
Any addition, if the bias variant of the test statistic is required, then the `bias`

parameter can be set to `True`

. In general, we do not recommend doing this.
Otherwise, these tests runs like any other test.

## Smooth Characteristic Function Test¶

The **Smooth Characteristic Function Test** (Smooth CF), is a form of non-parametric two-sample
tests. The Smooth CF test utilizes smoothed empirical characteristic functions to represent
two data distributions. Characteristic functions completely define the probability distribution
of a random variable. In hypothesis testing, it is useful to estimate characteristic functions
for given data. However, empirical characteristic functions can be very complex and therefore
expensive to compute. The smooth characteristic function can serve as a heuristic in place of
the empirical function which is much faster w.r.t. computation times. More information can be
found at `hyppo.ksample.SmoothCFTest`

.

Note

- Pros
Very fast computation time

Faster than current, state-of-the-art quadratic-time kernel-based tests

- Cons
Heuristic method, checking more frequencies will give more power.

This test is also initialized with the `num_randfreq`

parameter. This parameter can be
thought of as the degrees of freedom associated with the test and also dictates the number
of test points used in the test (see `hyppo.ksample.SmoothCFTest`

). If data
is kept constant, increasing the magnitude of this parameter will generally result in
larger magnitude test statistics while magnitude of the p-value will fluctuate:

```
import numpy as np
from hyppo.ksample import SmoothCFTest
np.random.seed(1234)
x = np.random.randn(500, 10)
y = np.random.randn(500, 10)
stat1, pvalue1 = SmoothCFTest(num_randfreq=5).test(x, y, random_state=1234)
stat2, pvalue2 = SmoothCFTest(num_randfreq=10).test(x, y, random_state=1234)
print("5 degrees of freedom (stat, pval):\n", stat1, pvalue1)
print("10 degrees of freedom (stat, pval):\n", stat2, pvalue2)
```

Out:

```
5 degrees of freedom (stat, pval):
4.698068715588362 0.9104136782024711
10 degrees of freedom (stat, pval):
14.494540898474115 0.8045631415568775
```

## Mean Embedding Test¶

The **Mean Embedding Test** is another non-parametric two-sample statistical test. This test
is based on analytic mean embeddings of data distributions in a reproducing kernel hilbert
space (RKHS). Hilbert spaces allow the representation of functions as points; thus, if
mean embeddings can be determined for two data distributions then the distance between
these two distributions in the hilbert space can be determined. In other words, the RKHS
allows the mapping of probability measures into a finite dimensional Euclidean space. More
details can be found at `hyppo.ksample.MeanEmbeddingTest`

.

Note

- Pros
Very fast computation time

Faster than current, state-of-the-art quadratic-time kernel-based tests

- Cons
Heuristic method, checking more frequencies will give more power.

This test is also initialized with the `num_randfreq`

parameter. This parameter can be
thought ofas the degrees of freedom associated with the test and also dictates the number
of test points used in the test (see `hyppo.ksample.MeanEmbeddingTest`

). If data
is kept constant, increasing the magnitude of this parameter will generally result in
larger magnitude test statistics while magnitude of the p-value will fluctuate:

```
from hyppo.ksample import MeanEmbeddingTest
np.random.seed(1234)
x = np.random.randn(500, 10)
y = np.random.randn(500, 10)
stat1, pval1 = MeanEmbeddingTest(num_randfreq=5).test(x, y, random_state=1234)
stat2, pval2 = MeanEmbeddingTest(num_randfreq=10).test(x, y, random_state=1234)
print("5 degrees of freedom (stat, pval):\n", stat1, pval1)
print("10 degrees of freedom (stat, pval):\n", stat2, pval2)
```

Out:

```
5 degrees of freedom (stat, pval):
5.32713968958162 0.377273001086016
10 degrees of freedom (stat, pval):
10.150092396142151 0.42742539627581594
```

## Univariate-Based Test¶

The **Heller Heller Gorfine (HHG) 2-Sample Test** is a non-parametric two-sample
statistical test. This test is based on testing the independence of the distances of sample vectors
from a center point by a univariate K-sample test. If the distribution of samples differs
across categories, then so does the distribution of distances of the vectors from almost every
point z. The univariate test used is the Kolmogorov-Smirnov 2-sample Test, which looks
at the largest absolute deviation between the cumulative distribution functions of
the samples.
More info can found at `hyppo.ksample.KSampleHHG`

.

Note

- Pros
Very fast computation time

- Cons
Lower power than more computationally complex algorithms

Inherits the assumptions of the KS univariate test

```
from hyppo.ksample import KSampleHHG
np.random.seed(1234)
x, y = rot_ksamp("linear", n=100, p=1, k=2, noise=False)
stat, pvalue = KSampleHHG().test(x, y)
print(stat, pvalue)
```

Out:

```
0.5757575757575758 3.3189957106557175e-13
```

**Total running time of the script:** ( 0 minutes 0.766 seconds)