Bootstrapped Standardized Effect Size (SES) Calculation

Calculates non-SMD standardized effect sizes with bootstrap confidence intervals. This function provides more robust confidence intervals for rank-based and probability-based effect size measures through resampling methods.

Usage

boot_ses_calc(
  x,
  ...,
  paired = FALSE,
  ses = "rb",
  alpha = 0.05,
  boot_ci = c("basic", "stud", "perc"),
  R = 1999
)

# Default S3 method
boot_ses_calc(
  x,
  y = NULL,
  paired = FALSE,
  ses = c("rb", "odds", "logodds", "cstat"),
  alpha = 0.05,
  boot_ci = c("basic", "stud", "perc"),
  R = 1999,
  ...
)

# S3 method for class 'formula'
boot_ses_calc(formula, data, subset, na.action, ...)

Arguments

x: a (non-empty) numeric vector of data values.
...: further arguments to be passed to or from methods.
paired: a logical indicating whether you want a paired t-test.
ses: a character string specifying the effect size measure to calculate: - "rb": rank-biserial correlation (default) - "odds": Wilcoxon-Mann-Whitney odds - "logodds": Wilcoxon-Mann-Whitney log-odds - "cstat": concordance statistic (C-statistic/AUC)
alpha: alpha level (default = 0.05)
boot_ci: method for bootstrap confidence interval calculation: "stud" (studentized, default), "basic" (basic bootstrap), or "perc" (percentile bootstrap).
R: number of bootstrap replications (default = 1999).
y: an optional (non-empty) numeric vector of data values.
formula: a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data: an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset: an optional vector specifying a subset of observations to be used.
na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Value

A data frame containing the following information:

estimate: The effect size estimate calculated from the original data
bias: Estimated bias (difference between original estimate and median of bootstrap estimates)
SE: Standard error estimated from the bootstrap distribution
lower.ci: Lower bound of the bootstrap confidence interval
upper.ci: Upper bound of the bootstrap confidence interval
conf.level: Confidence level (1-alpha)
boot_ci: The bootstrap confidence interval method used

Details

This function calculates bootstrapped confidence intervals for rank-based and probability-based effect size measures. It is an extension of the ses_calc() function that uses resampling to provide more robust confidence intervals, especially for small sample sizes.

The function implements the following bootstrap approach:

Calculate the raw effect size using the original data
Create R bootstrap samples by resampling with replacement from the original data
Calculate the effect size for each bootstrap sample
Apply the Fisher z-transformation to stabilize variance for rank-biserial correlation values
Calculate confidence intervals using the specified method
Back-transform the confidence intervals to the original scale
Convert to the requested effect size measure (if not rank-biserial)

Three bootstrap confidence interval methods are available:

Basic bootstrap ("basic"): Uses the empirical distribution of bootstrap estimates
Studentized bootstrap ("stud"): Accounts for the variability in standard error estimates
Percentile bootstrap ("perc"): Uses percentiles of the bootstrap distribution directly

The function supports three study designs:

One-sample design: Compares a single sample to a specified value
Two-sample independent design: Compares two independent groups
Paired samples design: Compares paired observations

Note that extreme values (perfect separation between groups) can produce infinite values during the bootstrapping process. The function will issue a warning if this occurs, as it may affect the accuracy of the confidence intervals.

For detailed information on calculation methods, see vignette("robustTOST").

Purpose

Use this function when:

You need more robust confidence intervals for non-parametric effect sizes
You prefer resampling-based confidence intervals over asymptotic approximations
You need to quantify uncertainty in rank-based effect sizes more accurately

Examples

# Example 1: Independent groups comparison with basic bootstrap CI
set.seed(123)
group1 <- c(1.2, 2.3, 3.1, 4.6, 5.2, 6.7)
group2 <- c(3.5, 4.8, 5.6, 6.9, 7.2, 8.5)

# Use fewer bootstrap replicates for a quick example
result <- boot_ses_calc(x = group1, y = group2,
                        ses = "rb",
                        boot_ci = "basic",
                        R = 99)
#> Bootstrapped results contain extreme results (i.e., no overlap), caution advised interpreting the with confidence intervals.

# Example 2: Using formula notation to calculate concordance statistic
data(mtcars)
result <- boot_ses_calc(formula = mpg ~ am,
                        data = mtcars,
                        ses = "cstat",
                        boot_ci = "perc",
                        R = 99)
#> Bootstrapped results contain extreme results (i.e., no overlap), caution advised interpreting the with confidence intervals.

# Example 3: Paired samples with studentized bootstrap CI
data(sleep)
with(sleep, boot_ses_calc(x = extra[group == 1],
                          y = extra[group == 2],
                          paired = TRUE,
                          ses = "rb",
                          boot_ci = "stud",
                          R = 99))
#> Bootstrapped results contain extreme results (i.e., no overlap), caution advised interpreting the with confidence intervals.
#>    estimate bias        SE  lower.ci  upper.ci conf.level boot_ci
#> 1 0.9818182    0 0.0353526 0.9818182 0.9978825       0.95    stud

# Example 4: Comparing different bootstrap CI methods
if (FALSE) { # \dontrun{
# Basic bootstrap
basic_ci <- boot_ses_calc(x = group1, y = group2, boot_ci = "basic")

# Percentile bootstrap
perc_ci <- boot_ses_calc(x = group1, y = group2, boot_ci = "perc")

# Studentized bootstrap
stud_ci <- boot_ses_calc(x = group1, y = group2, boot_ci = "stud")

# Compare the results
rbind(basic_ci, perc_ci, stud_ci)
} # }