POISSON_MEANS_TEST

Overview

The POISSON_MEANS_TEST function performs the Poisson means test (E-test) to compare the means of two Poisson distributions. This test is useful for determining whether the observed event rates in two samples are statistically different, accounting for sample sizes and a hypothesized difference in means. The test statistic and p-value are computed using the method described in scipy.stats.poisson_means_test :

E = \frac{(k_1/n_1) - (k_2/n_2) - \text{diff}}{\sqrt{(k_1/n_1^2) + (k_2/n_2^2)}}

where $k_1$ and $k_2$ are the number of events, $n_1$ and $n_2$ are the sample sizes, and $\text{diff}$ is the hypothesized difference in means. The function exposes all commonly used parameters and supports the three alternative hypotheses: two-sided, less, and greater. The wrapper simplifies the output to a single-row 2D array for Excel compatibility.

This example function is provided as-is without any representation of accuracy.

Usage

To use the function in Excel:


=POISSON_MEANS_TEST(k_one, n_one, k_two, n_two, [diff], [alternative])

k_one (int, required): Number of events observed from distribution 1.
n_one (float, required): Size of sample from distribution 1.
k_two (int, required): Number of events observed from distribution 2.
n_two (float, required): Size of sample from distribution 2.
diff (float, optional, default=0.0): Hypothesized difference in means.
alternative (str, optional, default=two-sided): Defines the alternative hypothesis (two-sided, less, greater).

The function returns a single-row 2D array [[statistic, pvalue]] if successful, or an error message (string) if the input is invalid. The statistic quantifies the difference between means, and the p-value indicates the probability of observing such a difference under the null hypothesis.

Examples

Example 1: Basic Two-Sided Test

Inputs:

k_one	n_one	k_two	n_two	diff	alternative
10	5.0	15	7.0

Excel formula:


=POISSON_MEANS_TEST(10, 5.0, 15, 7.0)

Expected output:

statistic	pvalue
-0.170	0.877

Example 2: Greater Alternative

Inputs:

k_one	n_one	k_two	n_two	diff	alternative
20	10.0	10	10.0		greater

Excel formula:


=POISSON_MEANS_TEST(20, 10.0, 10, 10.0, , "greater")

Expected output:

statistic	pvalue
1.826	0.034

Example 3: Less Alternative with Diff

Inputs:

k_one	n_one	k_two	n_two	diff	alternative
5	2.0	8	3.0	0.5	less

Excel formula:


=POISSON_MEANS_TEST(5, 2.0, 8, 3.0, 0.5, "less")

Expected output:

statistic	pvalue
-0.456	0.348

Example 4: All Arguments Specified

Inputs:

k_one	n_one	k_two	n_two	diff	alternative
12	6.0	18	9.0	0.0	two-sided

Excel formula:


=POISSON_MEANS_TEST(12, 6.0, 18, 9.0, 0.0, "two-sided")

Expected output:

statistic	pvalue
0.000	1.000

Python Code


from scipy.stats import poisson_means_test as scipy_poisson_means_test
from typing import Union
 
def poisson_means_test(k_one: int, n_one: float, k_two: int, n_two: float, diff: float = 0.0, alternative: str = 'two-sided') -> Union[list[list[float]], str]:
    """
    Performs the Poisson means test (E-test) to compare the means of two Poisson distributions.
 
    Args:
        k_one: Number of events observed from distribution 1 (integer).
        n_one: Size of sample from distribution 1 (float).
        k_two: Number of events observed from distribution 2 (integer).
        n_two: Size of sample from distribution 2 (float).
        diff: Hypothesized difference in means (float, default 0.0).
        alternative: Defines the alternative hypothesis ('two-sided', 'less', 'greater'). Default is 'two-sided'.
 
    Returns:
        2D list [[statistic, pvalue]] if successful, or an error message (str) if input is invalid.
 
    This example function is provided as-is without any representation of accuracy.
    """
    # Validate input types and values
    try:
        k1 = int(k_one)
        n1 = float(n_one)
        k2 = int(k_two)
        n2 = float(n_two)
        diff_val = float(diff)
    except Exception:
        return "Invalid input: k_one, n_one, k_two, n_two, and diff must be numeric."
    if n1 <= 0 or n2 <= 0:
        return "Invalid input: n_one and n_two must be positive."
    if k1 < 0 or k2 < 0:
        return "Invalid input: k_one and k_two must be non-negative."
    if alternative not in ['two-sided', 'less', 'greater']:
        return "Invalid input: alternative must be 'two-sided', 'less', or 'greater'."
    try:
        result = scipy_poisson_means_test(
            k1=k1,
            n1=n1,
            k2=k2,
            n2=n2,
            diff=diff_val,
            alternative=alternative
        )
        stat = float(result.statistic)
        pval = float(result.pvalue)
        # Disallow nan, inf, -inf
        if any([
            stat is None, pval is None,
            stat != stat or stat in [float('inf'), float('-inf')],
            pval != pval or pval in [float('inf'), float('-inf')]
        ]):
            return "Invalid output: statistic or pvalue is not a finite number."
        return [[stat, pval]]
    except Exception as e:
        return f"scipy.poisson_means_test error: {e}"

Example Workbook

Link to Workbook