# kendall.tau: Kendall's Tau Statistic In VGAM: Vector Generalized Linear and Additive Models

## Description

Computes Kendall's Tau, which is a rank-based correlation measure, between two vectors.

## Usage

 `1` ```kendall.tau(x, y, exact = FALSE, max.n = 3000) ```

## Arguments

 `x, y` Numeric vectors. Must be of equal length. Ideally their values are continuous and not too discrete. Let `length(x)` be N, say. `exact` Logical. If `TRUE` then the exact value is computed. `max.n` Numeric. If `exact = FALSE` and `length(x)` is more than `max.n` then a random sample of `max.n` pairs are chosen.

## Details

Kendall's tau is a measure of dependency in a bivariate distribution. Loosely, two random variables are concordant if large values of one random variable are associated with large values of the other random variable. Similarly, two random variables are disconcordant if large values of one random variable are associated with small values of the other random variable. More formally, if `(x[i] - x[j])*(y[i] - y[j]) > 0` then that comparison is concordant (i \neq j). And if `(x[i] - x[j])*(y[i] - y[j]) < 0` then that comparison is disconcordant (i \neq j). Out of `choose(N, 2`) comparisons, let c and d be the number of concordant and disconcordant pairs. Then Kendall's tau can be estimated by (c-d)/(c+d). If there are ties then half the ties are deemed concordant and half disconcordant so that (c-d)/(c+d+t) is used.

## Value

Kendall's tau, which lies between -1 and 1.

## Warning

If `length(x)` is large then the cost is O(N^2), which is expensive! Under these circumstances it is not advisable to set `exact = TRUE` or `max.n` to a very large number.

`binormalcop`, `cor`.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```N <- 5000; x <- 1:N; y <- runif(N) true.rho <- -0.8 ymat <- rbinorm(N, cov12 = true.rho) # Bivariate normal, aka N_2 x <- ymat[, 1] y <- ymat[, 2] ## Not run: plot(x, y, col = "blue") kendall.tau(x, y) # A random sample is taken here kendall.tau(x, y) # A random sample is taken here kendall.tau(x, y, exact = TRUE) # Costly if length(x) is large kendall.tau(x, y, max.n = N) # Same as exact = TRUE (rhohat <- sin(kendall.tau(x, y) * pi / 2)) # This formula holds for N_2 actually true.rho # rhohat should be near this value ```

### Example output

```Loading required package: stats4