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Do the Mann-Kendall and Kendall Tau-b use very similar test statistics? It seems that everytime I perform both tests, they always provide the same p-value and same conclusion.

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The Mann-Kendall test calculates a Kendall correlation between the time series and time (up to a constant scaling factor).

If $S$ is the usual Mann-Kendall statistic, then $\tau=\frac{S}{D}$, where $D^2= {n\choose 2}({n\choose 2}-\sum_i {t_i \choose 2})$, where $t_i$ is the number of values tied at the $i$-th set of tied values.

Since (presumably) the times are all distinct (i.e. cannot be tied), the formula for $\tau_b$ will also take this form.

i.e. $S$ is merely a scaled $\tau_b$, where the scaling constant that depends only on the sample size and the pattern of ties (how many ties there are for each set of tied values, not the values taken).

If the p-values are calculated in corresponding fashion for both, they should be identical.

[However, since programs may use different methods to obtain p-values for one or the other (one might use an exact calculation where the other uses a normal approximation, or one might apply a continuity correction the other doesn't), it's not absolutely guaranteed the value returned by some package would always exactly correspond.]

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