Compute the rank-biserial correlation ($$r_{rb}$$), Cliff's delta ($$\delta$$), rank epsilon squared ($$\varepsilon^2$$), and Kendall's W effect sizes for non-parametric (rank sum) tests.

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

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

rank_epsilon_squared(
x,
groups,
data = NULL,
ci = 0.95,
alternative = "greater",
iterations = 200,
...
)

kendalls_w(
x,
groups,
blocks,
data = NULL,
ci = 0.95,
alternative = "greater",
iterations = 200,
verbose = TRUE,
...
)

## Arguments

x Can be one of: A numeric vector, or a character name of one in data. A formula in to form of DV ~ groups (for rank_biserial() and rank_epsilon_squared()) or DV ~ groups | blocks (for kendalls_w(); See details for the blocks and groups terminology used here). A list of vectors (for rank_epsilon_squared()). A matrix of blocks x groups (for kendalls_w()). See details for the blocks and groups terminology used here. An optional numeric vector of data values to compare to x, or a character name of one in data. Ignored if x is not a vector. 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: "two.sided" (two-sided CI; default for rank-biserial correlation and Cliff's delta), "greater" (default for rank epsilon squared and Kendall's W) or "less" (one-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs. 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. Toggle warnings and messages on or off. Arguments passed to or from other methods. The number of bootstrap replicates for computing confidence intervals. Only applies when ci is not NULL. (Deprecated for rank_biserial()). A factor vector giving the group / block for the corresponding elements of x, or a character name of one in data. Ignored if x is not a vector.

## Value

A data frame with the effect size (r_rank_biserial, rank_epsilon_squared or Kendalls_W) 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 indicating that all values of the second sample are smaller than the first sample, to +1 indicating that all values of the second sample are larger than the first sample. (Cliff's delta is an alias to the rank-biserial correlation in the two sample case.)

The rank epsilon squared is appropriate for non-parametric tests of differences between 2 or more samples (a rank based ANOVA). See stats::kruskal.test. Values range from 0 to 1, with larger values indicating larger differences between groups.

Kendall's W is appropriate for non-parametric tests of differences between 2 or more dependent samples (a rank based rmANOVA), where each group (e.g., experimental condition) was measured for each block (e.g., subject). This measure is also common as a measure of reliability of the rankings of the groups between raters (blocks). See stats::friedman.test. Values range from 0 to 1, with larger values indicating larger differences between groups / higher agreement between raters.

### Ties

When tied values occur, they are each given the average of the ranks that would have been given had no ties occurred. No other corrections have been implemented yet.

## Confidence Intervals

Confidence intervals for the rank-biserial correlation (and Cliff's delta) are estimated using the normal approximation (via Fisher's transformation). Confidence intervals for rank Epsilon squared, and Kendall's W are estimated using the bootstrap method (using the {boot} package).

• 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.

• Kendall, M.G. (1948) Rank correlation methods. London: Griffin.

• 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 effect size indices: cohens_d(), effectsize(), eta_squared(), phi(), standardize_parameters()

## Examples

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

# Rank Biserial Correlation
# =========================

# Two Independent Samples ----------
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.

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

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

# Rank Epsilon Squared
# ====================

rank_epsilon_squared(mpg ~ cyl, data = mtcars)
#> Epsilon2 (rank) |       95% CI
#> ------------------------------
#> 0.83            | [0.78, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).

# Kendall's W
# ===========
dat <- data.frame(cond = c("A", "B", "A", "B", "A", "B"),
ID = c("L", "L", "M", "M", "H", "H"),
y = c(44.56, 28.22, 24, 28.78, 24.56, 18.78))
(W <- kendalls_w(y ~ cond | ID, data = dat, verbose = FALSE))
#> Kendall's W |       95% CI
#> --------------------------
#> 0.11        | [0.11, 1.00]
#>
#> - One-sided CIs: upper bound fixed at (1).

interpret_kendalls_w(0.11)
#>  "slight agreement"
#> (Rules: landis1977)
#>
interpret(W, rules = "landis1977")
#> Kendall's W |       95% CI |   Interpretation
#> ---------------------------------------------
#> 0.11        | [0.11, 1.00] | slight agreement
#>
#> - One-sided CIs: upper bound fixed at (1).
#> (Interpretation rule: landis1977)
# }