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Module « scipy.stats »

Fonction barnard_exact - module scipy.stats

Signature de la fonction barnard_exact

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

Description

help(scipy.stats.barnard_exact)

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

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 : float
        The Wald statistic with pooled or unpooled variance, depending
        on the user choice of `pooled`.

    pvalue : float
        P-value, the probability of obtaining a distribution at least as
        extreme as the one that was actually observed, assuming that the
        null hypothesis is true.

See Also
--------
chi2_contingency : Chi-square test of independence of variables in a
    contingency table.
fisher_exact : Fisher exact test on a 2x2 contingency table.
boschloo_exact : Boschloo's exact test on a 2x2 contingency table,
    which is an uniformly more powerful alternative to Fisher's exact test.

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 :math:`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 :math:`p_1, p_2` the theoretical binomial
probabilities for  :math:`x_{11}` and :math:`x_{12}`. When using
Barnard exact test, we can assert three different null hypotheses :

- :math:`H_0 : p_1 \geq p_2` versus :math:`H_1 : p_1 < p_2`,
  with `alternative` = "less"

- :math:`H_0 : p_1 \leq p_2` versus :math:`H_1 : p_1 > p_2`,
  with `alternative` = "greater"

- :math:`H_0 : p_1 = p_2` versus :math:`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 [3]_ with pooled or unpooled variance.
Under the default assumption that both variances are equal
(``pooled = True``), the statistic is computed as:

.. math::

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

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