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Somers' D

Source: R/assoc.R
somers_d.Rd

somers_d() computes Somers' D for a two-way contingency table of ordinal variables.

Usage

somers_d(
  x,
  direction = c("row", "column", "symmetric"),
  detail = FALSE,
  conf_level = 0.95,
  digits = 3L
)

Arguments

x

A contingency table (of class table).

direction

Direction of prediction: "row" (default, column predicts row), "column" (row predicts column), or "symmetric" (harmonic mean of both directions).

detail

Logical. If FALSE (default), return the estimate as a numeric scalar. If TRUE, return a named numeric vector including confidence interval and p-value.

conf_level

A single number strictly between 0 and 1 giving the confidence level (default 0.95). Only used when detail = TRUE. Set to NULL to omit the confidence interval. Any other value – including percentages such as 95 – raises a classed error (spicy_invalid_input).

digits

Number of decimal places used when printing the result (default 3). Only affects the detail = TRUE output.

Value

Same structure as cramer_v(): a scalar when detail = FALSE, a named vector when detail = TRUE. The p-value tests H0: D = 0 (Wald z-test).

Details

Somers' D is an asymmetric ordinal measure defined as \(d = (C - D) / (C + D + T)\), where \(T\) is the number of pairs tied on the independent variable. The symmetric version (direction = "symmetric") is the harmonic mean of the two asymmetric values, matching the SPSS / PSPP convention; this is not identical to Kendall's Tau-b (which is the geometric mean of the same two quantities), although the two often agree to two decimals. It is computed via the equivalent closed form \(2(C - D)\) divided by the sum of the two asymmetric denominators, so a table with exactly as many concordant as discordant pairs (e.g. an independence pattern) yields the well-defined value 0 – the harmonic-mean form is 0/0 there – as printed by SPSS / PSPP. The symmetric estimate is NA only when one of the asymmetric directions is itself undefined (with the same spicy_undefined_stat warning). No analytic SE / CI is reported for the symmetric form: its se is always NA, matching PSPP, which prints no ASE for it (DescTools offers no symmetric form at all).

The default direction = "row" differs deliberately from lambda_gk() and uncertainty_coef() (which default to "symmetric"): Somers' D is intrinsically asymmetric, its symmetric form is a derived convention without inference, and the reference implementation (DescTools::SomersDelta()) offers only the two asymmetric directions, defaulting to "row".

Standard error formulas for the asymmetric directions follow the DescTools implementations (Signorell et al., 2024); see cramer_v() for full references.

References

Somers, R. H. (1962). A new asymmetric measure of association for ordinal variables. American Sociological Review, 27(6), 799-811. doi:10.2307/2090408

Brown, M. B., & Benedetti, J. K. (1977). Sampling behavior of tests for correlation in two-way contingency tables. Journal of the American Statistical Association, 72(358), 309-315. doi:10.1080/01621459.1977.10480995

See also

kendall_tau_b(), gamma_gk(), assoc_measures()

Other association measures: assoc_measures(), contingency_coef(), cramer_v(), gamma_gk(), goodman_kruskal_tau(), kendall_tau_b(), kendall_tau_c(), lambda_gk(), phi(), uncertainty_coef(), yule_q()

Examples

tab <- table(sochealth$education, sochealth$self_rated_health)
somers_d(tab, direction = "row")
#> [1] 0.2015369
somers_d(tab, direction = "column", detail = TRUE)
#> Estimate     SE  CI lower  CI upper      p
#>    0.208  0.026     0.157     0.258  <.001