Bootstrapped Standardized Mean Difference (SMD) Calculation

Calculates standardized mean differences (SMDs) with bootstrap confidence intervals. This function provides more robust confidence intervals for Cohen's d, Hedges' g, and other SMD measures through resampling methods.

Usage

boot_smd_calc(
  x,
  ...,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999
)

# Default S3 method
boot_smd_calc(
  x,
  y = NULL,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  mu = 0,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999,
  ...
)

# S3 method for class 'formula'
boot_smd_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.
var.equal: a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
alpha: alpha level (default = 0.05)
bias_correction: Apply Hedges' correction for bias (default is TRUE).
rm_correction: Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass: Option to calculate Glass's delta instead of Cohen's d style SMD ('glass1' uses first group's SD, 'glass2' uses second group's SD).
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.
mu: null value to adjust the calculation. If non-zero, the function calculates x-y-mu (default = 0).
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 indicating what should happen when the data contain NAs.

Value

A data frame containing the following information:

estimate: The SMD 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 standardized mean differences. It is an extension of the smd_calc() function that uses resampling to provide more robust confidence intervals, especially for small sample sizes or when data violate assumptions of parametric methods.

The function implements the following bootstrap approach:

Calculate the raw SMD and its standard error using the original data
Create R bootstrap samples by resampling with replacement from the original data
Calculate the SMD and its standard error for each bootstrap sample
Calculate confidence intervals using the specified method

Three bootstrap confidence interval methods are available:

Studentized bootstrap ("stud"): Accounts for the variability in standard error estimates. Usually provides the most accurate coverage probability and is set as the default.
Basic bootstrap ("basic"): Uses the empirical distribution of bootstrap estimates. Simple approach that works well for symmetric distributions.
Percentile bootstrap ("perc"): Uses percentiles of the bootstrap distribution directly. More robust to skewness in the bootstrap distribution.

The function supports various SMD variants:

Classic standardized mean difference (bias_correction = FALSE)
Bias-corrected version (bias_correction = TRUE)
Glass's delta: Uses only one group's standard deviation as the denominator (glass = "glass1" or "glass2")
Repeated measures d: Accounts for correlation in paired designs (rm_correction = TRUE)

The function supports three study designs:

One-sample design: Standardizes the difference between the sample mean and zero (or other specified value)
Two-sample independent design: Standardizes the difference between two group means
Paired samples design: Standardizes the mean difference between paired observations

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

Purpose

Use this function when:

You need more robust confidence intervals for standardized mean differences
You want to account for non-normality or heterogeneity in your effect size estimates
Sample sizes are small or standard error approximations may be unreliable
You prefer resampling-based confidence intervals over parametric approximations
You need to quantify uncertainty in SMD estimates more accurately

Examples

# Example 1: Independent groups comparison with studentized bootstrap CI
set.seed(123)
group1 <- rnorm(30, mean = 100, sd = 15)
group2 <- rnorm(30, mean = 110, sd = 18)

# Use fewer bootstrap replicates for a quick example
result <- boot_smd_calc(x = group1, y = group2,
                      boot_ci = "stud",
                      R = 999)

# Example 2: Using formula notation with basic bootstrap and Hedges' g
df <- data.frame(
  value = c(group1, group2),
  group = factor(rep(c("A", "B"), each = 30))
)
result <- boot_smd_calc(formula = value ~ group,
                      data = df,
                      boot_ci = "basic",
                      bias_correction = TRUE,
                      R = 999)

# Example 3: Paired samples with percentile bootstrap
set.seed(456)
before <- rnorm(30)
after <- rnorm(30)
result <- boot_smd_calc(x = before,
                      y = after,
                      paired = TRUE,
                      boot_ci = "perc",
                      R = 999)

# Example 4: Glass's delta with homogeneous variances
set.seed(456)
control <- rnorm(25, mean = 50, sd = 10)
treatment <- rnorm(25, mean = 60, sd = 10)
result <- boot_smd_calc(x = control,
                      y = treatment,
                      glass = "glass1",
                      boot_ci = "stud",
                      R = 999)