Compute the rank-biserial correlation (r_rb) and Cliff's delta
(δ) 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
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, ... )
A numeric or ordered vector, or a character name of one in
An optional data frame containing the variables.
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.
Confidence Interval (CI) level
a character string specifying the alternative hypothesis;
Controls the type of CI returned:
Toggle warnings and messages on or off.
Arguments passed to or from other methods. When
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
-1 complete dominance of the second sample (all values of the second
sample are larger than all the values of the first sample) to
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.
A data frame with the effect size
r_rank_biserial and its CI
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 intervals for the rank-biserial correlation (and Cliff's delta) are estimated using the normal approximation (via Fisher's transformation).
"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 - α)% confidence
interval contains all of the parameter values for which p > α
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 α 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).
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:
Other rank-based effect sizes:
data(mtcars) mtcars$am <- factor(mtcars$am) mtcars$cyl <- factor(mtcars$cyl) # Two Independent Samples ---------- (rb <- rank_biserial(mpg ~ am, data = mtcars)) # 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) print(rb, append_CLES = TRUE) # One Sample ---------- rank_biserial(wt ~ 1, data = mtcars, mu = 3) # same as: # rank_biserial("wt", data = mtcars, mu = 3) # rank_biserial(mtcars$wt, mu = 3) # Paired Samples ---------- dat <- data.frame( Cond1 = c(1.83, 0.5, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.3), Cond2 = c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29) ) (rb <- rank_biserial(Pair(Cond1, Cond2) ~ 1, data = dat, paired = TRUE)) # same as: # rank_biserial(dat$Cond1, dat$Cond2, paired = TRUE) interpret_rank_biserial(0.78) interpret(rb, rules = "funder2019")
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