R: Somers' Delta

SomersDelta {DescTools} | R Documentation

## Somers' Delta

### Description

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.

### 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 τ_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 - ∑(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

### Author(s)

Andri Signorell <andri@signorell.net>

### 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.

### See Also

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`

### 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)

[Package DescTools version 0.99.24 Index]