Calculates the Common Language Effect Size (CLES)

Calculates the Common Language Effect Size (CLES) for two variables. The CLES function converts the effect size to a probability that a unit/subject will have a larger measurement than another unit/subject. See notebook for further details.

Usage

cles(data, group_variables, paired = FALSE, ci = FALSE, ...)

Arguments

data: dataframe; Data should be in wide format
group_variables: character vector or list with quoted names of the variables to be compared.
paired: boolean; Indicates whether variables are correlated as in a repeated measures design. Default is FALSE.
ci: boolean; Indicates whether bootstrap confidence intervals should be calculated. Default is FALSE.
...: Additional arguments that should be passed to get_boot_ci()

Value

When 'ci = FALSE', this function returns a scalar value estimate of the CLES. When 'ci = TRUE', this function returns a dataframe with the following columns:

ci_type: The method of calculating the bootstrap confidence intervals.
conf: The confidence level for the bootstrap confidence intervals,
.lower: The lower value of the bootstrap confidence interval.
.estimate: The CLES point estimate.
.upper: The upper value of the bootstrap confidence interval.

Details

This measure is also referred to as the Probability of Superiority. The conversion of effect size to a probability or percentage is supposed to be easier for the laymen to interpret. Interpretation:

Between-Subjects: The probability that a randomly sampled person from one group will have a higher observed measurement than a randomly sampled person from the other group.
Within-Subjects: The probability that an individual has a higher value on one measurement than the other.

Between-Subjects Formula: $$\tilde d = \frac{|M_1 - M_2|}{\sqrt{p_1\text{SD}_1^2 + p_2\text{SD}_2^2}}\\ Z = \frac{\tilde d}{\sqrt{2}}$$

$M_i$: The mean of the i^th group
$p_i$: The proportion of the sample size of the i^th group
$Z$: The z-score which is in turn used to produce the probability.

Within-Subjects Formula: $$Z = \frac{|M_1 - M_2|}{\sqrt{\operatorname{SD}_1^2 + \operatorname{SD}_2^2 - 2 \times r \times \operatorname{SD}_1 \times \operatorname{SD}_2}}$$

$M_i$: The mean of the i^th group
$r$: Pearson correlation between the two variables
$Z$: The z-score which is in turn used to produce the probability.

References

McGraw, K. O., & Wong, S. P. (1992). A common language effect size statistic. Psychological Bulletin, 111(2), 361–365. https://doi.org/10.1037/0033-2909.111.2.361

Examples


movie_dat <- dplyr::tibble(
   movie1 = c(9.00, 7.00, 8.00, 9.00, 8.00, 9.00, 9.00, 10.00, 9.00, 9.00),
   movie2 = c(9.00, 6.00, 7.00, 8.00, 7.00, 9.00, 8.00, 8.00, 8.00, 7.00)
)

# between-subjects design
cles(data = movie_dat,
     group_variables = list("movie1", "movie2"))
#> [1] 0.7870181

# within-subjects design and bootstrap CIs
cles(data = movie_dat,
     group_variables = list("movie1", "movie2"),
     paired = TRUE,
     ci = TRUE,
     R = 10000,
     type = c("bca", "perc"))
#>   ci_type conf .lower .estimate .upper
#> 1 percent 0.95 0.8080 0.9331928 0.9997
#> 2     bca 0.95 0.7602 0.9331928 0.9964