(\frac{1}{c_1} +
            \frac{1}{c_2})
        }
    }

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

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

.. math::

    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:

.. math::

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

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

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

References
----------
.. [1] Barnard, G. A. "Significance Tests for 2x2 Tables". *Biometrika*.
       34.1/2 (1947): 123-138. :doi:`dpgkg3`

.. [2] Mehta, Cyrus R., and Pralay Senchaudhuri. "Conditional versus
       unconditional exact tests for comparing two binomials."
       *Cytel Software Corporation* 675 (2003): 1-5.

.. [3] "Wald Test". *Wikipedia*. https://en.wikipedia.org/wiki/Wald_test

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
-1.894
>>> res.pvalue
0.03407

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
0.0640

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.

r