Pearson's Contingency Coefficient

Computes Pearson's contingency coefficient for a contingency table. If x and y are supplied, the table is constructed first; see Association().

Usage

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

Arguments

x: either a contingency table, a two-column object (matrix, data.frame or list), or a vector of observations (together with y)
y: optional second vector. If x is not a vector, y must be NULL.
conf.level: confidence level of the interval. If set to NA (the default), only the point estimate is returned.
sides: character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().
correct: logical; whether Sakoda's adjusted Pearson's C should be returned; defaults to FALSE
...: further arguments. Named arguments known to normalizeToConfusion() or table() are used to build the table; R and type configure the bootstrap and are described under Details. Anything else is an error rather than a silent no-op.

Value

if conf.level = NA, a numeric scalar containing Pearson's contingency coefficient; otherwise a named numeric vector with elements:

est: point estimate of the contingency coefficient
lci: lower confidence interval bound
uci: upper confidence interval bound

Details

Pearson's contingency coefficient ranges from 0 to $\sqrt{(m-1)/m}$, where $m = \min(r, c)$. Consequently, its attainable maximum depends on the dimensions of the table.

Sakoda's correction divides the coefficient by this maximum: $$C_{\mathrm{S}} = \frac{C}{C_{\max}} = C\sqrt{\frac{m}{m-1}}.$$ The corrected coefficient therefore ranges from 0 to 1.

Since no generally accepted analytical interval is available, only bootstrap intervals are implemented. The interval is obtained from a multinomial bootstrap over the cells of the table. Two arguments configure it, both passed through \dots: R, the number of replicates (default 999), and type, one of "perc" (default) or "bca".

"perc" is the default deliberately. Under independence the parameter sits on the boundary of its range, where the sampling distribution of $C$ is not normal under any monotone transformation - which is precisely what BCa assumes. Both of its ingredients degrade there: the bias correction is read off the share of replicates below the estimate, which collapses when the estimate is at the edge of the bootstrap distribution, and the acceleration is a jackknife, which is not consistent for a functional that is not smooth. "bca" is the better choice for tables with a clearly non-zero association and a reasonable number of observations.

Confidence intervals are restricted to the attainable range. Measures such as cramerV may be preferable when inference is central.

For further information see ConfidenceIntervals().

References

Sakoda, J.M. (1977) Measures of Association for Multivariate Contingency Tables, Proceedings of the Social Statistics Section of the American Statistical Association (Part III), 777-780.

Efron, B., Tibshirani, R.J. (1993) An Introduction to the Bootstrap, Chapman & Hall, chapter 14.

Examples


tab <- apply(HairEyeColor, c(1, 2), sum)
contCoef(tab)
#> [1] 0.4351585

# just x and y
with(bedrock::untable(tab), contCoef(Hair, Eye))
#> [1] 0.4351585

set.seed(1)
contCoef(tab, conf.level = 0.95)
#>       est       lci       uci 
#> 0.4351585 0.3870546 0.4905756 

set.seed(1)
contCoef(tab, conf.level = 0.95, type = "bca", R = 999)
#>       est       lci       uci 
#> 0.4351585 0.3739549 0.4754878