boundsKendall {bosfr} | R Documentation |
Bounds of Kendall's tau in the Presence of Missing Data
Description
Computes bounds of Kendall's tau in the presence of missing data. Suitable only for univariate distinct data where no ties is allowed.
Usage
boundsKendall(X, Y)
Arguments
X , Y |
Numeric vectors of data values with potential missing data. No ties in the data is allowed. Inf and -Inf values will be omitted. |
Details
boundsKendall()
computes bounds of Kendall's tau
for partially observed univariate, distinct data. The bounds are computed
by first calculating the bounds of Spearman's footrule (Zeng et al., 2025), and then applying
the combinatorial inequality between Kendall's tau and Spearman's footrule
(Kendall, 1948). See Zeng et al., 2025 for more details.
Let X = (x_1, \ldots, x_n)
and Y = (y_1, \ldots, y_n)
be
two vectors of univariate, distinct data.
Kendall's tau is defined as the number of discordant pairs between X
and Y
:
\tau(X,Y) = \sum\limits_{i < j} \{I(x_i < x_j)I(y_i > y_j) + I(x_i > x_j)I(y_i < y_j)\}.
Scaled Kendall's tau \tau_{Scale}(X,Y) \in [0,1]
is defined as (Kendall, 1948):
\tau_{Scale}(X,Y) = 1 - 4\tau(X,Y)/(n(n-1)).
Value
bounds |
bounds of Kendall's tau. |
bounds.scaled |
bounds of scaled Kendall's tau. |
References
Zeng Y., Adams N.M., Bodenham D.A. Exact Bounds of Spearman's footrule in the Presence of Missing Data with Applications to Independence Testing. arXiv preprint arXiv:2501.11696. 2025 Jan 20.
Kendall, M.G. (1948) Rank Correlation Methods. Charles Griffin, London.
Diaconis, P. and Graham, R.L., 1977. Spearman's footrule as a measure of disarray. Journal of the Royal Statistical Society Series B: Statistical Methodology, 39(2), pp.262-268.
Examples
### compute bounds of Kendall's tau between incomplete ranked lists
X <- c(1, 2, NA, 4, 3)
Y <- c(3, NA, 4, 2, 1)
boundsKendall(X, Y)
### compute bounds of Kendall's tau between incomplete vectors of distinct data
X <- c(1.3, 2.6, NA, 4.2, 3.5)
Y <- c(5.5, NA, 6.5, 2.6, 1.1)
boundsKendall(X, Y)