Skip to contents

Bootstrapped TOST with t-tests

boot_t_TOST.Rd

[Stable]

Performs equivalence testing using the Two One-Sided Tests (TOST) procedure with bootstrapped t-tests. This provides a robust alternative to traditional TOST when data may not meet all parametric assumptions.

Usage

boot_t_TOST(x, ...)

# Default S3 method
boot_t_TOST(
  x,
  y = NULL,
  hypothesis = "EQU",
  paired = FALSE,
  var.equal = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  eqbound_type = "raw",
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  mu = 0,
  R = 1999,
  boot_ci = c("stud", "basic", "perc"),
  ...
)

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

Arguments

x

a (non-empty) numeric vector of data values.

...

further arguments to be passed to or from methods.

y

an optional (non-empty) numeric vector of data values.

hypothesis

'EQU' for equivalence (default), or 'MET' for minimal effects test.

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.

eqb

Equivalence bound. Can provide 1 value (symmetric bound, negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.

low_eqbound

lower equivalence bounds (deprecated, use eqb instead).

high_eqbound

upper equivalence bounds (deprecated, use eqb instead).

eqbound_type

Type of equivalence bound. Can be 'SMD' for standardized mean difference (i.e., Cohen's d) or 'raw' for the mean difference. Default is 'raw'. Raw is strongly recommended as SMD bounds will produce biased results.

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).

mu

a number indicating the true value of the mean for the two-tailed test (default = 0).

R

number of bootstrap replications (default = 1999).

boot_ci

method for bootstrap confidence interval calculation: "stud" (studentized, default), "basic" (basic bootstrap), or "perc" (percentile bootstrap).

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

An S3 object of class "TOSTt" is returned containing the following slots:

  • "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

  • "eqb": A table of class "data.frame" containing equivalence bound settings.

  • "effsize": Table of class "data.frame" containing effect size estimates.

  • "hypothesis": String stating the hypothesis being tested.

  • "smd": List containing the results of the standardized mean difference calculations (e.g., Cohen's d).

    • Items include: d (estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation).

  • "alpha": Alpha level set for the analysis.

  • "method": Type of t-test.

  • "decision": List included text regarding the decisions for statistical inference.

  • "boot": List containing the bootstrap samples for SMD and raw effect sizes.

Details

The function implements a bootstrap method for TOST as described in Chapter 16 of Efron and Tibshirani (1994). This approach provides a robust alternative to traditional parametric TOST when data distributions may not meet standard assumptions.

The bootstrap procedure follows these steps:

  • Resample with replacement from the original data to create R bootstrap samples

  • For each bootstrap sample, calculate test statistics and effect sizes

  • Use the distribution of bootstrap results to compute p-values and confidence intervals

  • Combine results using the specified bootstrap confidence interval method

Three types of bootstrap confidence intervals are available:

  • Studentized ("stud"): Accounts for the variability in the standard error estimate

  • Basic/Empirical ("basic"): Uses the empirical distribution of bootstrap estimates

  • Percentile ("perc"): Uses percentiles of the bootstrap distribution

For two-sample tests, the test is of \(\bar x - \bar y\) (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein \(z = x - y\), and the test is of \(\bar z\) (mean of the difference scores). For one-sample tests, the test is of \(\bar x\) (mean of x).

For details on the calculations in this function see vignette("robustTOST").

Purpose

Use this function when:

  • You want more robust confidence intervals for your effect sizes

  • Sample sizes are small and parametric assumptions may not hold

  • You want to avoid relying on asymptotic approximations

References

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.

See also

Other Robust tests: boot_log_TOST(), boot_t_test(), brunner_munzel(), log_TOST(), wilcox_TOST()

Other TOST: boot_log_TOST(), simple_htest(), t_TOST(), tsum_TOST(), wilcox_TOST()

Examples

# Example 1: Two-Sample Test with Symmetric Bounds
set.seed(1234)
group1 <- rnorm(30, mean = 5, sd = 2)
group2 <- rnorm(30, mean = 5.5, sd = 2.2)

# Using symmetric bounds of ±1.5
result <- boot_t_TOST(x = group1,
                     y = group2,
                     eqb = 1.5,
                     R = 999)  # Using fewer replications for demonstration

# Example 2: Paired Sample Test with Percentile Bootstrap
set.seed(5678)
pre <- rnorm(25, mean = 100, sd = 15)
post <- pre + rnorm(25, mean = 3, sd = 10)

result <- boot_t_TOST(x = pre,
                     y = post,
                     paired = TRUE,
                     eqb = c(-5, 8),  # Asymmetric bounds
                     boot_ci = "perc")

# Example 3: One Sample Test
set.seed(9101)
scores <- rnorm(40, mean = 0.3, sd = 1)

# Testing if mean is equivalent to zero within ±0.5 units
result <- boot_t_TOST(x = scores,
                     eqb = 0.5,
                     boot_ci = "basic")