Standardized Mean Difference (SMD) Calculation

Calculates standardized mean difference (SMD) effect sizes and their confidence intervals from raw data. This function focuses solely on effect size estimation without performing hypothesis tests.

Usage

smd_calc(
  x,
  ...,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z")
)

# Default S3 method
smd_calc(
  x,
  y = NULL,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  mu = 0,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z"),
  ...
)

# S3 method for class 'formula'
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: An option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, 'glass1' will use the first group's SD as the denominator whereas 'glass2' will use the 2nd group's SD.
smd_ci: Method for calculating SMD confidence intervals. Methods include 'goulet', 'noncentral t' (nct), 'central t' (t), and 'normal method' (z).
y: an optional (non-empty) numeric vector of data values.
mu: a number indicating the true value of the mean for the two-tailed test (or difference in means if you are performing a two sample test).
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 standardized mean difference estimate (Cohen's d, Hedges' g, or Glass's delta)
SE: Standard error of the estimate
lower.ci: Lower bound of the confidence interval
upper.ci: Upper bound of the confidence interval
conf.level: Confidence level (1-alpha)

Details

This function calculates standardized mean differences (SMD) for various 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

The function supports multiple SMD variants:

Cohen's d: Classic standardized mean difference (bias_correction = FALSE)
Hedges' g: Bias-corrected version of Cohen's d (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)

Different confidence interval calculation methods are available:

"nct": Uses the noncentral t-distribution (most accurate in most cases)
"goulet": Uses the Goulet-Pelletier method
"t": Uses the central t-distribution
"z": Uses the normal distribution

Note that unlike the t_TOST and related functions, smd_calc only calculates effect sizes and their confidence intervals without performing hypothesis tests.

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

Purpose

Use this function when:

You need to calculate standardized effect sizes (Cohen's d, Hedges' g, Glass's delta)
You want confidence intervals for your effect size estimates
You need effect sizes for meta-analysis or reporting
You want to compare effect sizes across different studies or measures
You don't need the hypothesis testing components of the TOST functions

Examples

# Example 1: Independent groups comparison (Cohen's d)
set.seed(123)
group1 <- rnorm(30, mean = 100, sd = 15)
group2 <- rnorm(30, mean = 110, sd = 18)
smd_calc(x = group1, y = group2, bias_correction = FALSE)
#>                 estimate        SE  lower.ci  upper.ci conf.level
#> Cohen's d(av) -0.9355902 0.2766132 -1.465782 -0.398128       0.95

# Example 2: Independent groups with formula notation (Hedges' g)
df <- data.frame(
  value = c(group1, group2),
  group = factor(rep(c("A", "B"), each = 30))
)
smd_calc(formula = value ~ group, data = df)
#>                  estimate        SE  lower.ci   upper.ci conf.level
#> Hedges's g(av) -0.9234253 0.2762605 -1.446724 -0.3929514       0.95

# Example 3: Paired samples with repeated measures correction
before <- c(5.1, 4.8, 6.2, 5.7, 6.0, 5.5, 4.9, 5.8)
after <- c(5.6, 5.2, 6.7, 6.1, 6.5, 5.8, 5.3, 6.2)
smd_calc(x = before, y = after, paired = TRUE, rm_correction = TRUE)
#>                  estimate        SE  lower.ci   upper.ci conf.level
#> Hedges's g(rm) -0.6600134 0.2456809 -1.345188 0.06017759       0.95

# Example 4: Glass's delta (using only first group's SD)
smd_calc(x = group1, y = group2, glass = "glass1")
#>                    estimate        SE  lower.ci   upper.ci conf.level
#> Glass's delta(g) -0.9210091 0.2917046 -1.427478 -0.3877403       0.95