bundles / scipy 2.0.0.dev0+git20260907.41eeb59 / scipy / stats / _stats_py / kendalltau

function

`scipy.stats._stats_py:kendalltau`

source: build-install/usr/lib/python3.14/site-packages/scipy/stats/_stats_py.py:5597

Signature

def kendalltau(x, y, *, nan_policy = propagate, method = auto, variant = b, alternative = two-sided, axis = None, keepdims = False)

Summary

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

Extended Summary

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.

Although a naive implementation has O(n^2) complexity, this implementation uses a Fenwick tree to do the computation in O(n log(n)) complexity.

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.

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'.

alternative : {'two-sided', 'less', 'greater'}, optional

Defines the alternative hypothesis. Default is 'two-sided'. The following options are available:

'two-sided': the rank correlation is nonzero
'less': the rank correlation is negative (less than zero)
'greater': the rank correlation is positive (greater than zero)

axis : int or None, default: None

If an int, the axis of the input along which to compute the statistic. The statistic of each axis-slice (e.g. row) of the input will appear in a corresponding element of the output. If None, the input will be raveled before computing the statistic.

keepdims : bool, optional

If this is set to True, the axes which are reduced are left in the result as dimensions with length one. With this option, the result will broadcast correctly against the input array.

Returns

res : SignificanceResult

An object containing attributes:

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

Raises

: ValueError: If nan_policy is 'omit' and variant is not 'b' or if method is 'exact' and there are ties between x and y.

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 tied pairs only in x, and U the number of tied pairs 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.

Examples

from scipy import stats
x1 = [12, 2, 1, 12, 2]
x2 = [1, 4, 7, 1, 0]
res = stats.kendalltau(x1, x2)

✓

res.statistic
res.pvalue

✗

np.float64(0.28274545993277467)

For a more detailed example, see :ref:`hypothesis_kendalltau`.

Aliases

scipy.stats.kendalltau