Cohen's w (omega)

Calculates Cohen's w for a table of nominal variables.

Usage

cohenW(
  x,
  y = NULL,
  p = NULL,
  ci = FALSE,
  conf = 0.95,
  type = "perc",
  R = 1000,
  histogram = FALSE,
  digits = 4,
  reportIncomplete = FALSE,
  ...
)

Arguments

x: Either a two-way table or a two-way matrix. Can also be a vector of observations for one dimension of a two-way table.
y: If x is a vector, y is the vector of observations for the second dimension of a two-way table.
p: If x is a vector of observed counts, p can be given as a vector of theoretical probabilties, as in a chi-square goodness of fit test.
ci: If TRUE, returns confidence intervals by bootstrap. May be slow.
conf: The level for the confidence interval.
type: The type of confidence interval to use. Can be any of "norm", "basic", "perc", or "bca". Passed to boot.ci.
R: The number of replications to use for bootstrap.
histogram: If TRUE, produces a histogram of bootstrapped values.
digits: The number of significant digits in the output.
reportIncomplete: If FALSE (the default), NA will be reported in cases where there are instances of the calculation of the statistic failing during the bootstrap procedure. In the case of the goodness-of-fit scenario, setting this to TRUE will have no effect.
...: Additional arguments passed to chisq.test.

Value

A single statistic, Cohen's w. Or a small data frame consisting of Cohen's w, and the lower and upper confidence limits.

Details

Cohen's w is used as a measure of association between two nominal variables, or as an effect size for a chi-square test of association. For a 2 x 2 table, the absolute value of the phi statistic is the same as Cohen's w. The value of Cohen's w is not bound by 1 on the upper end.

Cohen's w is "naturally nondirectional". That is, the value will always be zero or positive. Because of this, if type="perc", the confidence interval will never cross zero. The confidence interval range should not be used for statistical inference. However, if type="norm", the confidence interval may cross zero.

When w is close to 0 or very large, or with small counts, the confidence intervals determined by this method may not be reliable, or the procedure may fail.

References

https://rcompanion.org/handbook/H_10.html

Cohen J. 1992. "A Power Primer". Psychological Bulletin 12(1): 155-159.

Cohen, J. 1988. Statistical Power Analysis for the Behavioral Sciences, 2nd Ed. Routledge.

Author

Salvatore Mangiafico, mangiafico@njaes.rutgers.edu

Examples

### Example with table
data(Anderson)
fisher.test(Anderson)
#> 
#> 	Fisher's Exact Test for Count Data
#> 
#> data:  Anderson
#> p-value = 0.000668
#> alternative hypothesis: two.sided
#> 
cohenW(Anderson)
#> Cohen w 
#>  0.4387 

### Example for goodness-of-fit
### Bird foraging example, Handbook of Biological Statistics
observed = c(70,   79,   3,    4)
expected = c(0.54, 0.40, 0.05, 0.01)
chisq.test(observed, p = expected)
#> Warning: Chi-squared approximation may be incorrect
#> 
#> 	Chi-squared test for given probabilities
#> 
#> data:  observed
#> X-squared = 13.593, df = 3, p-value = 0.003514
#> 
cohenW(observed, p = expected)
#> Cohen w 
#>  0.2952 

### Example with two vectors
Species = c(rep("Species1", 16), rep("Species2", 16))
Color   = c(rep(c("blue", "blue", "blue", "green"),4),
            rep(c("green", "green", "green", "blue"),4))
fisher.test(Species, Color)
#> 
#> 	Fisher's Exact Test for Count Data
#> 
#> data:  Species and Color
#> p-value = 0.01211
#> alternative hypothesis: true odds ratio is not equal to 1
#> 95 percent confidence interval:
#>   1.463598 60.596055
#> sample estimates:
#> odds ratio 
#>   8.279362 
#> 
cohenW(Species, Color)
#> Cohen w 
#>     0.5