scipy.stats.kendalltau

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

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, values close to -1 indicate strong disagreement. This is the 1945 “tau-b” version of Kendall’s tau [2], which can account for ties and which reduces to the 1938 “tau-a” version [1] in absence of ties.

Parameters

x, yarray_like

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

initial_lexsortbool, optional

Unused (deprecated).

nan_policy{‘propagate’, ‘raise’, ‘omit’}, optional

Defines how to handle when input contains nan. ‘propagate’ returns nan, ‘raise’ throws an error, ‘omit’ performs the calculations ignoring nan values. Default is ‘propagate’. Note that if the input contains nan ‘omit’ delegates to mstats_basic.kendalltau(), which has a different implementation.

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

Defines which method is used to calculate the p-value [5]. ‘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. ‘auto’ is the default and selects the appropriate method based on a trade-off between speed and accuracy.

Returns

correlationfloat

The tau statistic.

pvaluefloat

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

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.

References

1

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

2(1,2)

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