# SomersDelta

##### 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.

##### See Also

There's an implementation of Somers's D in Frank Harrell's `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)
```

*Documentation reproduced from package DescTools, version 0.99.16, License: GPL (>= 2)*