Kendall Tau Rank Correlation Coefficient - Significance Tests

Significance Tests

When two quantities are statistically independent, the distribution of is not easily characterizable in terms of known distributions. However, for the following statistic, is approximately distributed as a standard normal when the variables are statistically independent:

Thus, to test whether two variables are statistically dependent, one computes, and finds the cumulative probability for a standard normal distribution at . For a 2-tailed test, multiply that number by two to obtain the p-value. If the p-value is below a given significance level, one rejects the null hypothesis (at that significance level) that the quantities are statistically independent.

Numerous adjustments should be added to when accounting for ties. The following statistic, has the same distribution as the distribution, and is again approximately equal to a standard normal distribution when the quantities are statistically independent:

where

\begin{array}{ccl} v & = & (v_0 - v_t - v_u)/18 + v_1 + v_2 \\ v_0 & = & n (n-1) (2n+5) \\ v_t & = & \sum_i t_i (t_i-1) (2 t_i+5)\\ v_u & = & \sum_j u_j (u_j-1)(2 u_j+5) \\ v_1 & = & \sum_i t_i (t_i-1) \sum_j u_j (u_j-1) / (2n(n-1)) \\ v_2 & = & \sum_i t_i (t_i-1) (t_i-2) \sum_j u_j (u_j-1) (u_j-2) / (9 n (n-1) (n-2)) \end{array}

Read more about this topic:  Kendall Tau Rank Correlation Coefficient

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