Description Usage Arguments Details Value Warning See Also Examples

View source: R/family.bivariate.R View source: R/family.bivariate.R

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

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

`x, y` |
Numeric vectors. Must be of equal length.
Ideally their values are continuous and not too discrete.
Let |

`exact` |
Logical. If |

`max.n` |
Numeric. If |

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.

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

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.

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
``` |

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