# SomersDelta

0th

Percentile

##### Somers' Delta

Calculate Somers' Delta statistic, a measure of association for ordinal factors in a two-way table. The function has interfaces for a table (matrix) and for single vectors.

Keywords
nonparametric, multivar
##### Usage
SomersDelta(x, y = NULL, direction = c("row", "column"), conf.level = NA, ...)
##### Arguments
x
a numeric vector or a table. A matrix will be treated as table.
y
NULL (default) or a vector with compatible dimensions to x. If y is provided, table(x, y, ...) is calculated.
direction
direction of the calculation. Can be "row" (default) or "column", where "row" calculates Somers' D (R | C) ("column dependent").
conf.level
confidence level of the interval. If set to NA (which is the default) no confidence interval will be calculated.
...
further arguments are passed to the function table, allowing i.e. to set useNA. This refers only to the vector interface.
##### Details

Somers' D(C | R) and Somers' D(R | C) are asymmetric modifications of $\tau_b$. C | R indicates that the row variable x is regarded as the independent variable and the column variable y is regarded as dependent. Similarly, R | C indicates that the column variable y is regarded as the independent variable and the row variable x is regarded as dependent. Somers' D differs from tau-b in that it uses a correction only for pairs that are tied on the independent variable. Somers' D is appropriate only when both variables lie on an ordinal scale. Somers' D is computed as $$D(C | R) = \frac{P-Q}{n^2 - \sum(n_i.^2)}$$where P equals twice the number of concordances and Q twice the number of discordances and $n_i.$ rowSums(tab). Its range lies [-1, 1].

##### Value

• a single numeric value if no confidence intervals are requested and otherwise a numeric vector with 3 elements for the estimate, the lower and the upper confidence interval

##### References

Agresti, A. (2002) Categorical Data Analysis. John Wiley & Sons, pp. 57--59. Goodman, L. A., & Kruskal, W. H. (1954) Measures of association for cross classifications. Journal of the American Statistical Association, 49, 732-764. Somers, R. H. (1962) A New Asymmetric Measure of Association for Ordinal Variables, American Sociological Review, 27, 799--811. Goodman, L. A., & Kruskal, W. H. (1963) Measures of association for cross classifications III: Approximate sampling theory. Journal of the American Statistical Association, 58, 310--364.

There's an implementation of Somers's D in Frank Harrell's Hmisc somers2, which is quite fast for large sample sizes. However it is restricted to computing Somers' Dxy rank correlation between a variable x and a binary (0-1) variable y. ConDisPairs yields concordant and discordant pairs Other association measures: KendallTauA (tau-a), KendallTauB (tau-b), cor (method="kendall") for tau-b, StuartTauC (tau-c), GoodmanKruskalGamma Lambda, GoodmanKruskalTau, UncertCoef, MutInf

• SomersDelta
##### Examples
# example in:
# http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf
# pp. S. 1821

tab <- as.table(rbind(c(26,26,23,18,9),c(6,7,9,14,23)))

# Somers' D C|R
SomersDelta(tab, direction="column", conf.level=0.95)
# Somers' D R|C
SomersDelta(tab, direction="row", conf.level=0.95)
Documentation reproduced from package DescTools, version 0.99.16, License: GPL (>= 2)

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