Ordinal Association Measures

Computes concordance-based association measures for two ordinal variables or a contingency table, optionally with confidence intervals.

Usage

ordAssocs(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  which = c("all", "gamma", "tauA", "tauB", "tauC", "somers", "cstat"),
  direction = c("row", "column")
)

gkGamma(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  direction = c("row", "column")
)

kendallTauA(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right")
)

kendallTauB(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right")
)

stuartTauC(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right")
)

somersDelta(
  x,
  y = NULL,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  direction = c("row", "column")
)

Arguments

x: numeric or ordinal vector, or a two-dimensional contingency table
y: optional second vector of the same length as x
conf.level: confidence level of the interval; NA returns only the estimate
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See details in ConfidenceIntervals(). The open side is closed at the boundary of the parameter's range - $\pm 1$ for gamma, the tau family and Somers' $D$, and $[0, 1]$ for the c statistic - rather than at an infinity none of them can reach.
which: measure returned by ordAssocs(): "all", "gamma", "tauA", "tauB", "tauC", "somers", or "cstat"
direction: direction of Somers' $D$; "row" or "column"

Value

The extractor functions return am unnamed numeric scalar if conf.level = NA, and otherwise a named numeric vector with elements:

est: point estimate
lci: lower confidence interval bound
uci: upper confidence interval bound

ordAssocs() returns the same structure in a named list containing the selected measures.

Details

Let $P$ and $Q$ denote the numbers of concordant and discordant pairs, $T_X$ and $T_Y$ the numbers tied on $X$ and $Y$, $n_0=n(n-1)/2$, and $m$ the smaller table dimension. conDisPairs() returns these pair counts as C, D, Ties_X, and Ties_Y and describes their calculation.

The measures are defined as follows:

Goodman–Kruskal $\gamma$ (gkGamma):: $$\gamma = \frac{P-Q}{P+Q}$$
Kendall $\tau_a$ (kendallTauA):: $$\tau_a = \frac{P-Q}{n_0}$$
Kendall $\tau_b$ (kendallTauB):: $$ \tau_b = \frac{P-Q} {\sqrt{(n_0-T_X)(n_0-T_Y)}} $$
Stuart $\tau_c$ (stuartTauC):: $$ \tau_c = \frac{2m(P-Q)} {n^2(m-1)} $$
Somers $D_{X\mid Y}$ (somersDelta):: $$ D_{X\mid Y} = \frac{P-Q} {n_0-T_Y} $$

Gamma and Kendall's coefficients are symmetric.

Somers' $D$ is directional: for tables, direction="row" returns $D_{R|C}$ and direction="column" returns $D_{C|R}$. In vector mode, ordAssocs() returns $D_{X|Y}$; reverse x and y for the other direction. somersDelta() performs this reversal when direction="column".

The c-statistic is used for a binary outcome and consistently ordered predictions and is related to Somers D as $C_{stat}=(D+1)/2$. See cStat() for direct estimation of the c-statistic from predicted values and a binary response.

References

Agresti, A. (2002) Categorical Data Analysis. Wiley, pp. 57–59.

Brown, M. B. and Benedetti, J. K. (1977). Sampling behavior of tests for correlation in two-way contingency tables. JASA, 72, 309–315.

Goodman, L. A. and Kruskal, W. H. (1954, 1963). Measures of association for cross classifications. JASA, 49, 732–764; 58, 310–364.

Kendall, M. (1955) Rank Correlation Methods. Charles Griffin.

Somers, R. H. (1962). A new asymmetric measure of association for ordinal variables. American Sociological Review, 27, 799–811.

Examples

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

ordAssocs(tab, conf.level = 0.95)
#> $gamma
#>       est       lci       uci 
#> 0.5313123 0.3479918 0.7146328 
#> 
#> $tauA
#>       est       lci       uci 
#> 0.2068323 0.1281336 0.2855310 
#> 
#> $tauB
#>       est       lci       uci 
#> 0.3372567 0.2114030 0.4631105 
#> 
#> $tauC
#>       est       lci       uci 
#> 0.4110953 0.2546754 0.5675151 
#> 
#> $somers
#>       est       lci       uci 
#> 0.2569444 0.1591986 0.3546903 
#> 
#> $cstat
#>       est       lci       uci 
#> 0.6284722 0.5795993 0.6773451 
#> 
kendallTauB(tab, conf.level = 0.95)
#>       est       lci       uci 
#> 0.3372567 0.2114030 0.4631105 
somersDelta(tab, direction = "column")
#> [1] 0.442672

# Vector example
x <- c(1,2,2,3,3,3,4,5)
y <- c(1,3,2,1,5,3,4,5)

kendallTauA(x, y, conf.level=0.95)
#>       est       lci       uci 
#> 0.5357143 0.0648461 1.0000000 
somersDelta(x, y, direction = "column", conf.level=0.95)
#>       est       lci       uci 
#> 0.6250000 0.2776229 0.9723771