`scipy.stats._hypotests:poisson_means_test`

source: /scipy/stats/_hypotests.py :178

Signature

   def     poisson_means_test (    k1    ,    n1    ,    k2    ,    n2    ,  * ,     diff    =  0   ,    alternative    =  two-sided     )

Summary

Performs the Poisson means test, AKA the "E-test".

Extended Summary

This is a test of the null hypothesis that the difference between means of two Poisson distributions is diff. The samples are provided as the number of events k1 and k2 observed within measurement intervals (e.g. of time, space, number of observations) of sizes n1 and n2.

Parameters

k1 : int

Number of events observed from distribution 1.

n1: float

Size of sample from distribution 1.

k2 : int

Number of events observed from distribution 2.

n2 : float

Size of sample from distribution 2.

diff : float, default=0

The hypothesized difference in means between the distributions underlying the samples.

alternative : {'two-sided', 'less', 'greater'}, optional

Defines the alternative hypothesis. The following options are available (default is 'two-sided'):

'two-sided': the difference between distribution means is not equal to diff
'less': the difference between distribution means is less than diff
'greater': the difference between distribution means is greater than diff

Returns

statistic : float: The test statistic (see ^[1] equation 3.3).
pvalue : float: The probability of achieving such an extreme value of the test statistic under the null hypothesis.

Notes

Let:

X_1 \sim \mbox{Poisson}(\mathtt{n1}\lambda_1)

be a random variable independent of

X_2 \sim \mbox{Poisson}(\mathtt{n2}\lambda_2)

and let k1 and k2 be the observed values of $X_{1}$ and $X_{2}$ , respectively. Then poisson_means_test uses the number of observed events k1 and k2 from samples of size n1 and n2, respectively, to test the null hypothesis that

H_{0} : λ_{1} - λ_{2} = diff

A benefit of the E-test is that it has good power for small sample sizes, which can reduce sampling costs ^[1]. It has been evaluated and determined to be more powerful than the comparable C-test, sometimes referred to as the Poisson exact test.

Array API Standard Support

poisson_means_test has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

====================  ====================  ====================
Library               CPU                   GPU
====================  ====================  ====================
NumPy                 ✅                     n/a                 
CuPy                  n/a                   ⛔                   
PyTorch               ⛔                     ⛔                   
JAX                   ⛔                     ⛔                   
Dask                  ⛔                     n/a                 
====================  ====================  ====================

See dev-arrayapi for more information.

Examples

Suppose that a gardener wishes to test the number of dodder (weed) seeds in a sack of clover seeds that they buy from a seed company. It has previously been established that the number of dodder seeds in clover follows the Poisson distribution. A 100 gram sample is drawn from the sack before being shipped to the gardener. The sample is analyzed, and it is found to contain no dodder seeds; that is, `k1` is 0. However, upon arrival, the gardener draws another 100 gram sample from the sack. This time, three dodder seeds are found in the sample; that is, `k2` is 3. The gardener would like to know if the difference is significant and not due to chance. The null hypothesis is that the difference between the two samples is merely due to chance, or that :math:`\lambda_1 - \lambda_2 = \mathtt{diff}` where :math:`\mathtt{diff} = 0`. The alternative hypothesis is that the difference is not due to chance, or :math:`\lambda_1 - \lambda_2 \ne 0`. The gardener selects a significance level of 5% to reject the null hypothesis in favor of the alternative [2]_.

import scipy.stats as stats
res = stats.poisson_means_test(0, 100, 3, 100)

✓

res.statistic, res.pvalue

✗

The p-value is .088, indicating a near 9% chance of observing a value of the test statistic under the null hypothesis. This exceeds 5%, so the gardener does not reject the null hypothesis as the difference cannot be regarded as significant at this level.