# scipy.stats.kendalltau

scipy.stats.kendalltau(x, y, initial_lexsort=None, nan_policy='propagate', method='auto', variant='b')

  Calculate Kendall’s tau, a correlation measure for ordinal data.

  Kendall’s tau is a measure of the correspondence between two rankings. Values close to 1 indicate strong agreement, and values close to -1 indicate strong disagreement. This implements two variants of Kendall’s tau: tau-b (the default) and tau-c (also known as Stuart’s tau-c). These differ only in how they are normalized to lie within the range -1 to 1; the hypothesis tests (their p-values) are identical. Kendall’s original tau-a is not implemented separately because both tau-b and tau-c reduce to tau-a in the absence of ties.

  Parameters
    x, y array_like

      Arrays of rankings, of the same shape. If arrays are not 1-D, they will be flattened to 1-D.

    initial_lexsort bool, optional

      Unused (deprecated).

    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

    method{‘auto’, ‘asymptotic’, ‘exact’}, optional

      Defines which method is used to calculate the p-value [5]. The following options are available (default is ‘auto’):

      > - ‘auto’: selects the appropriate method based on a trade-off between speed and accuracy

      > - ‘asymptotic’: uses a normal approximation valid for large samples

      > - ‘exact’: computes the exact p-value, but can only be used if no ties are present. As the sample size increases, the ‘exact’ computation time may grow and the result may lose some precision.

    variant: {‘b’, ‘c’}, optional

      Defines which variant of Kendall’s tau is returned. Default is ‘b’.

  Returns
    correlation float

      The tau statistic.

    pvalue float

      The two-sided p-value for a hypothesis test whose null hypothesis is an absence of association, tau = 0.

  See also

  `spearmanr`

    Calculates a Spearman rank-order correlation coefficient.

  `theilslopes`

    Computes the Theil-Sen estimator for a set of points (x, y).

  `weightedtau`

    Computes a weighted version of Kendall’s tau.

  Notes

  The definition of Kendall’s tau that is used is [2]:

      tau_b = (P - Q) / sqrt((P + Q + T) * (P + Q + U))

      tau_c = 2 (P - Q) / (n**2 * (m - 1) / m)

  where P is the number of concordant pairs, Q the number of discordant pairs, T the number of ties only in x, and U the number of ties only in y. If a tie occurs for the same pair in both x and y, it is not added to either T or U. n is the total number of samples, and m is the number of unique values in either x or y, whichever is smaller.

  References

  1

    Maurice G. Kendall, “A New Measure of Rank Correlation”, Biometrika Vol. 30, No. 1/2, pp. 81-93, 1938.

  2

    Maurice G. Kendall, “The treatment of ties in ranking problems”, Biometrika Vol. 33, No. 3, pp. 239-251. 1945.

  3

    Gottfried E. Noether, “Elements of Nonparametric Statistics”, John Wiley & Sons, 1967.

  4

    Peter M. Fenwick, “A new data structure for cumulative frequency tables”, Software: Practice and Experience, Vol. 24, No. 3, pp. 327-336, 1994.

  5

    Maurice G. Kendall, “Rank Correlation Methods” (4th Edition), Charles Griffin & Co., 1970.

  Examples

      >>> from scipy import stats
      >>> x1 = [12, 2, 1, 12, 2]
      >>> x2 = [1, 4, 7, 1, 0]
      >>> tau, p_value = stats.kendalltau(x1, x2)
      >>> tau
      -0.47140452079103173
      >>> p_value
      0.2827454599327748

scipy.stats.pointbiserialr scipy.stats.weightedtau