Calculate Stuart's \(\tau_{c}\) 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.

`StuartTauC(x, y = NULL, 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.

- 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

Stuart's \(\tau_{c}\) makes an adjustment for table size in addition to a correction for ties. \(\tau_{c}\) is
appropriate only when both variables lie on an ordinal scale.

It is estimated by

$$ \tau_{c} = \frac{2 m \cdot(P-Q)}{n^2 \cdot (m-1)}$$
where P equals the number of concordances and Q the number of discordances, n is the total amount of observations and m = min(R, C). The range of \(\tau_{c}\) is [-1, 1].

See http://support.sas.com/documentation/cdl/en/statugfreq/63124/PDF/default/statugfreq.pdf, pp. 1739 for the estimation of the asymptotic variance.

The use of Stuart's Tau-c versus Kendall's Tau-b is recommended when the two ordinal variables under consideration have different numbers of values, e.g. good, medium, bad versus high, low.

## 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.

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, 309-315.

Goodman, L. A., & Kruskal, W. H. (1954) Measures of
association for cross classifications. *Journal of the
American Statistical Association*, 49, 732-764.

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.

## Author

Andri Signorell <andri@signorell.net>

## 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)))
StuartTauC(tab, conf.level=0.95)
#> tauc lwr.ci upr.ci
#> 0.4110953 0.2546754 0.5675151
```