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Fonction ttest_ind - module scipy.stats

Signature de la fonction ttest_ind

def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate', permutations=None, random_state=None, alternative='two-sided', trim=0) 

Description

ttest_ind.__doc__

    Calculate the T-test for the means of *two independent* samples of scores.

    This is a two-sided test for the null hypothesis that 2 independent samples
    have identical average (expected) values. This test assumes that the
    populations have identical variances by default.

    Parameters
    ----------
    a, b : array_like
        The arrays must have the same shape, except in the dimension
        corresponding to `axis` (the first, by default).
    axis : int or None, optional
        Axis along which to compute test. If None, compute over the whole
        arrays, `a`, and `b`.
    equal_var : bool, optional
        If True (default), perform a standard independent 2 sample test
        that assumes equal population variances [1]_.
        If False, perform Welch's t-test, which does not assume equal
        population variance [2]_.

        .. versionadded:: 0.11.0

    nan_policy : {'propagate', 'raise', 'omit'}, optional
        Defines how to handle when input contains nan.
        The following options are available (default is 'propagate'):

          * 'propagate': returns nan
          * 'raise': throws an error
          * 'omit': performs the calculations ignoring nan values

        The 'omit' option is not currently available for permutation tests or
        one-sided asympyotic tests.

    permutations : non-negative int, np.inf, or None (default), optional
        If 0 or None (default), use the t-distribution to calculate p-values.
        Otherwise, `permutations` is  the number of random permutations that
        will be used to estimate p-values using a permutation test. If
        `permutations` equals or exceeds the number of distinct partitions of
        the pooled data, an exact test is performed instead (i.e. each
        distinct partition is used exactly once). See Notes for details.

        .. versionadded:: 1.7.0

    random_state : {None, int, `numpy.random.Generator`,
            `numpy.random.RandomState`}, optional

        If `seed` is None (or `np.random`), the `numpy.random.RandomState`
        singleton is used.
        If `seed` is an int, a new ``RandomState`` instance is used,
        seeded with `seed`.
        If `seed` is already a ``Generator`` or ``RandomState`` instance then
        that instance is used.

        Pseudorandom number generator state used to generate permutations
        (used only when `permutations` is not None).

        .. versionadded:: 1.7.0

    alternative : {'two-sided', 'less', 'greater'}, optional
        Defines the alternative hypothesis.
        The following options are available (default is 'two-sided'):

          * 'two-sided'
          * 'less': one-sided
          * 'greater': one-sided

        .. versionadded:: 1.6.0

    trim : float, optional
        If nonzero, performs a trimmed (Yuen's) t-test.
        Defines the fraction of elements to be trimmed from each end of the
        input samples. If 0 (default), no elements will be trimmed from either
        side. The number of trimmed elements from each tail is the floor of the
        trim times the number of elements. Valid range is [0, .5).

        .. versionadded:: 1.7

    Returns
    -------
    statistic : float or array
        The calculated t-statistic.
    pvalue : float or array
        The two-tailed p-value.

    Notes
    -----
    Suppose we observe two independent samples, e.g. flower petal lengths, and
    we are considering whether the two samples were drawn from the same
    population (e.g. the same species of flower or two species with similar
    petal characteristics) or two different populations.

    The t-test quantifies the difference between the arithmetic means
    of the two samples. The p-value quantifies the probability of observing
    as or more extreme values assuming the null hypothesis, that the
    samples are drawn from populations with the same population means, is true.
    A p-value larger than a chosen threshold (e.g. 5% or 1%) indicates that
    our observation is not so unlikely to have occurred by chance. Therefore,
    we do not reject the null hypothesis of equal population means.
    If the p-value is smaller than our threshold, then we have evidence
    against the null hypothesis of equal population means.

    By default, the p-value is determined by comparing the t-statistic of the
    observed data against a theoretical t-distribution.
    When ``1 < permutations < binom(n, k)``, where

    * ``k`` is the number of observations in `a`,
    * ``n`` is the total number of observations in `a` and `b`, and
    * ``binom(n, k)`` is the binomial coefficient (``n`` choose ``k``),

    the data are pooled (concatenated), randomly assigned to either group `a`
    or `b`, and the t-statistic is calculated. This process is performed
    repeatedly (`permutation` times), generating a distribution of the
    t-statistic under the null hypothesis, and the t-statistic of the observed
    data is compared to this distribution to determine the p-value. When
    ``permutations >= binom(n, k)``, an exact test is performed: the data are
    partitioned between the groups in each distinct way exactly once.

    The permutation test can be computationally expensive and not necessarily
    more accurate than the analytical test, but it does not make strong
    assumptions about the shape of the underlying distribution.

    Use of trimming is commonly referred to as the trimmed t-test. At times
    called Yuen's t-test, this is an extension of Welch's t-test, with the
    difference being the use of winsorized means in calculation of the variance
    and the trimmed sample size in calculation of the statistic. Trimming is
    reccomended if the underlying distribution is long-tailed or contaminated
    with outliers [4]_.

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test

    .. [2] https://en.wikipedia.org/wiki/Welch%27s_t-test

    .. [3] http://en.wikipedia.org/wiki/Resampling_%28statistics%29

    .. [4] Yuen, Karen K. "The Two-Sample Trimmed t for Unequal Population
           Variances." Biometrika, vol. 61, no. 1, 1974, pp. 165-170. JSTOR,
           www.jstor.org/stable/2334299. Accessed 30 Mar. 2021.

    .. [5] Yuen, Karen K., and W. J. Dixon. "The Approximate Behaviour and
           Performance of the Two-Sample Trimmed t." Biometrika, vol. 60,
           no. 2, 1973, pp. 369-374. JSTOR, www.jstor.org/stable/2334550.
           Accessed 30 Mar. 2021.

    Examples
    --------
    >>> from scipy import stats
    >>> rng = np.random.default_rng()

    Test with sample with identical means:

    >>> rvs1 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)
    >>> rvs2 = stats.norm.rvs(loc=5, scale=10, size=500, random_state=rng)
    >>> stats.ttest_ind(rvs1, rvs2)
    Ttest_indResult(statistic=-0.4390847099199348, pvalue=0.6606952038870015)
    >>> stats.ttest_ind(rvs1, rvs2, equal_var=False)
    Ttest_indResult(statistic=-0.4390847099199348, pvalue=0.6606952553131064)

    `ttest_ind` underestimates p for unequal variances:

    >>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500, random_state=rng)
    >>> stats.ttest_ind(rvs1, rvs3)
    Ttest_indResult(statistic=-1.6370984482905417, pvalue=0.1019251574705033)
    >>> stats.ttest_ind(rvs1, rvs3, equal_var=False)
    Ttest_indResult(statistic=-1.637098448290542, pvalue=0.10202110497954867)

    When ``n1 != n2``, the equal variance t-statistic is no longer equal to the
    unequal variance t-statistic:

    >>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100, random_state=rng)
    >>> stats.ttest_ind(rvs1, rvs4)
    Ttest_indResult(statistic=-1.9481646859513422, pvalue=0.05186270935842703)
    >>> stats.ttest_ind(rvs1, rvs4, equal_var=False)
    Ttest_indResult(statistic=-1.3146566100751664, pvalue=0.1913495266513811)

    T-test with different means, variance, and n:

    >>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100, random_state=rng)
    >>> stats.ttest_ind(rvs1, rvs5)
    Ttest_indResult(statistic=-2.8415950600298774, pvalue=0.0046418707568707885)
    >>> stats.ttest_ind(rvs1, rvs5, equal_var=False)
    Ttest_indResult(statistic=-1.8686598649188084, pvalue=0.06434714193919686)

    When performing a permutation test, more permutations typically yields
    more accurate results. Use a ``np.random.Generator`` to ensure
    reproducibility:

    >>> stats.ttest_ind(rvs1, rvs5, permutations=10000,
    ...                 random_state=rng)
    Ttest_indResult(statistic=-2.8415950600298774, pvalue=0.0052)

    Take these two samples, one of which has an extreme tail.

    >>> a = (56, 128.6, 12, 123.8, 64.34, 78, 763.3)
    >>> b = (1.1, 2.9, 4.2)

    Use the `trim` keyword to perform a trimmed (Yuen) t-test. For example,
    using 20% trimming, ``trim=.2``, the test will reduce the impact of one
    (``np.floor(trim*len(a))``) element from each tail of sample `a`. It will
    have no effect on sample `b` because ``np.floor(trim*len(b))`` is 0.

    >>> stats.ttest_ind(a, b, trim=.2)
    Ttest_indResult(statistic=3.4463884028073513,
                    pvalue=0.01369338726499547)