Cohen's w (omega)

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

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.

See also

cramerV

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