Compute the rank-biserial correlation ($$r_{rb}$$) and Cliff's delta ($$\delta$$) effect sizes for non-parametric (rank sum) differences. These effect sizes of dominance are closely related to the Common Language Effect Sizes. Pair with any reported stats::wilcox.test().

## Usage

rank_biserial(
x,
y = NULL,
data = NULL,
mu = 0,
paired = FALSE,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)

cliffs_delta(
x,
y = NULL,
data = NULL,
mu = 0,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)

## Arguments

x, y

A numeric or ordered vector, or a character name of one in data. Any missing values (NAs) are dropped from the resulting vector. x can also be a formula (see stats::wilcox.test()), in which case y is ignored.

data

An optional data frame containing the variables.

mu

a number indicating the value around which (a-)symmetry (for one-sample or paired samples) or shift (for independent samples) is to be estimated. See stats::wilcox.test.

paired

If TRUE, the values of x and y are considered as paired. This produces an effect size that is equivalent to the one-sample effect size on x - y.

ci

Confidence Interval (CI) level

alternative

a character string specifying the alternative hypothesis; Controls the type of CI returned: "two.sided" (default, two-sided CI), "greater" or "less" (one-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs.

verbose

Toggle warnings and messages on or off.

...

Arguments passed to or from other methods. When x is a formula, these can be subset and na.action.

## Value

A data frame with the effect size r_rank_biserial and its CI (CI_low and CI_high).

## Details

The rank-biserial correlation is appropriate for non-parametric tests of differences - both for the one sample or paired samples case, that would normally be tested with Wilcoxon's Signed Rank Test (giving the matched-pairs rank-biserial correlation) and for two independent samples case, that would normally be tested with Mann-Whitney's U Test (giving Glass' rank-biserial correlation). See stats::wilcox.test. In both cases, the correlation represents the difference between the proportion of favorable and unfavorable pairs / signed ranks (Kerby, 2014). Values range from -1 complete dominance of the second sample (all values of the second sample are larger than all the values of the first sample) to +1 complete dominance of the fist sample (all values of the second sample are smaller than all the values of the first sample).

Cliff's delta is an alias to the rank-biserial correlation in the two sample case.

## Ties

When tied values occur, they are each given the average of the ranks that would have been given had no ties occurred. This results in an effect size of reduced magnitude. A correction has been applied for Kendall's W.

## Confidence (Compatibility) Intervals (CIs)

Confidence intervals for the rank-biserial correlation (and Cliff's delta) are estimated using the normal approximation (via Fisher's transformation).

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - $$\alpha$$)% confidence interval contains all of the parameter values for which p > $$\alpha$$ for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen $$\alpha$$ level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

## References

• Cureton, E. E. (1956). Rank-biserial correlation. Psychometrika, 21(3), 287-290.

• Glass, G. V. (1965). A ranking variable analogue of biserial correlation: Implications for short-cut item analysis. Journal of Educational Measurement, 2(1), 91-95.

• Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11-IT.

• King, B. M., & Minium, E. W. (2008). Statistical reasoning in the behavioral sciences. John Wiley & Sons Inc.

• Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological bulletin, 114(3), 494.

• Tomczak, M., & Tomczak, E. (2014). The need to report effect size estimates revisited. An overview of some recommended measures of effect size.

Other standardized differences: cohens_d(), mahalanobis_d(), means_ratio(), p_superiority()

Other rank-based effect sizes: p_superiority(), rank_epsilon_squared()

## Examples

# \donttest{
data(mtcars)
mtcars$am <- factor(mtcars$am)
mtcars$cyl <- factor(mtcars$cyl)

# Two Independent Samples ----------
(rb <- rank_biserial(mpg ~ am, data = mtcars))
#> r (rank biserial) |         95% CI
#> ----------------------------------
#> -0.66             | [-0.84, -0.36]
# Same as:
# rank_biserial("mpg", "am", data = mtcars)
# rank_biserial(mtcars$mpg[mtcars$am=="0"], mtcars$mpg[mtcars$am=="1"])

# More options:
rank_biserial(mpg ~ am, data = mtcars, mu = -5)
#> r (rank biserial) |        95% CI
#> ---------------------------------
#> -0.21             | [-0.56, 0.20]
#>
#> - Deviation from a difference of -5.
print(rb, append_CLES = TRUE)
#> r (rank biserial) |         95% CI
#> ----------------------------------
#> -0.66             | [-0.84, -0.36]
#>
#>
#> ## Common Language Effect Sizes:
#> Pr(superiority) |       95% CI
#> ------------------------------
#> 0.17            | [0.08, 0.32]
#>
#> - Non-parametric CLES

# One Sample ----------
rank_biserial(wt ~ 1, data = mtcars, mu = 3)
#> r (rank biserial) |        95% CI
#> ---------------------------------
#> 0.21              | [-0.18, 0.54]
#>
#> - Deviation from a difference of 3.
# same as:
# rank_biserial("wt", data = mtcars, mu = 3)
# rank_biserial(mtcars$wt, mu = 3) # Paired Samples ---------- dat <- data.frame( Cond1 = c(1.83, 0.5, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.3), Cond2 = c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29) ) (rb <- rank_biserial(Pair(Cond1, Cond2) ~ 1, data = dat, paired = TRUE)) #> r (rank biserial) | 95% CI #> -------------------------------- #> 0.78 | [0.30, 0.94] # same as: # rank_biserial(dat$Cond1, dat\$Cond2, paired = TRUE)

interpret_rank_biserial(0.78)
#> [1] "very large"
#> (Rules: funder2019)
#>
interpret(rb, rules = "funder2019")
#> r (rank biserial) |       95% CI | Interpretation
#> -------------------------------------------------
#> 0.78              | [0.30, 0.94] |     very large
#>
#> - Interpretation rule: funder2019
# }