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barnard_exact(table, alternative='two-sided', pooled=True, n=32)

Notes

Barnard's test is an exact test used in the analysis of contingency tables. It examines the association of two categorical variables, and is a more powerful alternative than Fisher's exact test for 2x2 contingency tables.

Let's define X_0 a 2x2 matrix representing the observed sample, where each column stores the binomial experiment, as in the example below. Let's also define p_1, p_2 the theoretical binomial probabilities for x_{11} and x_{12}. When using Barnard exact test, we can assert three different null hypotheses :

H_0 : p_1 \geq p_2 versus H_1 : p_1 < p_2, with alternative = "less"
H_0 : p_1 \leq p_2 versus H_1 : p_1 > p_2, with alternative = "greater"
H_0 : p_1 = p_2 versus H_1 : p_1 \neq p_2, with alternative = "two-sided" (default one)

In order to compute Barnard's exact test, we are using the Wald statistic with pooled or unpooled variance. Under the default assumption that both variances are equal (pooled = True), the statistic is computed as:

T(X) = \frac{
    \hat{p}_1 - \hat{p}_2
}{
    \sqrt{
        \hat{p}(1 - \hat{p})
        (\frac{1}{c_1} +
        \frac{1}{c_2})
    }
}

with \hat{p}_1, \hat{p}_2 and \hat{p} the estimator of p_1, p_2 and p, the latter being the combined probability, given the assumption that p_1 = p_2.

If this assumption is invalid (pooled = False), the statistic is:

T(X) = \frac{
    \hat{p}_1 - \hat{p}_2
}{
    \sqrt{
        \frac{\hat{p}_1 (1 - \hat{p}_1)}{c_1} +
        \frac{\hat{p}_2 (1 - \hat{p}_2)}{c_2}
    }
}

The p-value is then computed as:

\sum
    \binom{c_1}{x_{11}}
    \binom{c_2}{x_{12}}
    \pi^{x_{11} + x_{12}}
    (1 - \pi)^{t - x_{11} - x_{12}}

where the sum is over all 2x2 contingency tables X such that: * T(X) \leq T(X_0) when alternative = "less", * T(X) \geq T(X_0) when alternative = "greater", or * T(X) \geq |T(X_0)| when alternative = "two-sided". Above, c_1, c_2 are the sum of the columns 1 and 2, and t the total (sum of the 4 sample's element).

The returned p-value is the maximum p-value taken over the nuisance parameter \pi, where 0 \leq \pi \leq 1.

This function's complexity is O(n c_1 c_2), where n is the number of sample points.

Parameters

table : array_like of ints: A 2x2 contingency table. Elements should be non-negative integers.
alternative : {'two-sided', 'less', 'greater'}, optional: Defines the null and alternative hypotheses. Default is 'two-sided'. Please see explanations in the Notes section below.
pooled : bool, optional: Whether to compute score statistic with pooled variance (as in Student's t-test, for example) or unpooled variance (as in Welch's t-test). Default is True.
n : int, optional: Number of sampling points used in the construction of the sampling method. Note that this argument will automatically be converted to the next higher power of 2 since scipy.stats.qmc.Sobol is used to select sample points. Default is 32. Must be positive. In most cases, 32 points is enough to reach good precision. More points comes at performance cost.

Returns

ber : BarnardExactResult

A result object with the following attributes.

statistic: statistic
pvalue: pvalue

Perform a Barnard exact test on a 2x2 contingency table.

Examples

An example use of Barnard's test is presented in [2]_.

Consider the following example of a vaccine efficacy study (Chan, 1998). In a randomized clinical trial of 30 subjects, 15 were inoculated with a recombinant DNA influenza vaccine and the 15 were inoculated with a placebo. Twelve of the 15 subjects in the placebo group (80%) eventually became infected with influenza whereas for the vaccine group, only 7 of the 15 subjects (47%) became infected. The data are tabulated as a 2 x 2 table::

Vaccine Placebo Yes 7 12 No 8 3

When working with statistical hypothesis testing, we usually use a threshold probability or significance level upon which we decide to reject the null hypothesis :math:`H_0`. Suppose we choose the common significance level of 5%.

Our alternative hypothesis is that the vaccine will lower the chance of becoming infected with the virus; that is, the probability :math:`p_1` of catching the virus with the vaccine will be *less than* the probability :math:`p_2` of catching the virus without the vaccine. Therefore, we call `barnard_exact` with the ``alternative="less"`` option:

import scipy.stats as stats
res = stats.barnard_exact([[7, 12], [8, 3]], alternative="less")
res.statistic

res.pvalue

Under the null hypothesis that the vaccine will not lower the chance of becoming infected, the probability of obtaining test results at least as extreme as the observed data is approximately 3.4%. Since this p-value is less than our chosen significance level, we have evidence to reject :math:`H_0` in favor of the alternative.

Suppose we had used Fisher's exact test instead:

_, pvalue = stats.fisher_exact([[7, 12], [8, 3]], alternative="less")
pvalue

With the same threshold significance of 5%, we would not have been able to reject the null hypothesis in favor of the alternative. As stated in [2]_, Barnard's test is uniformly more powerful than Fisher's exact test because Barnard's test does not condition on any margin. Fisher's test should only be used when both sets of marginals are fixed.

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.stats._hypotests:boschloo_exact scipy.stats.contingency:chi2_contingency scipy.stats._stats_py:chisquare scipy.stats._stats_py:fisher_exact

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