Bootstrapped TOST with Log Transformed t-tests

Performs equivalence testing using the Two One-Sided Tests (TOST) procedure with bootstrapped log-transformed t-tests. This approach is particularly useful for ratio-scale data where the equivalence bounds are expressed as ratios (e.g., bioequivalence studies).

Usage

boot_log_TOST(x, ...)

# Default S3 method
boot_log_TOST(
  x,
  y = NULL,
  hypothesis = c("EQU", "MET"),
  paired = FALSE,
  var.equal = FALSE,
  eqb = 1.25,
  alpha = 0.05,
  null = 1,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999,
  ...
)

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

Arguments

x: a (non-empty) numeric vector of positive data values on a ratio scale.
...: further arguments to be passed to or from methods.
y: an optional (non-empty) numeric vector of positive data values on a ratio scale.
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.
eqb: Equivalence bound expressed as a ratio. Can provide 1 value (e.g., 1.25 for bounds of 0.8 and 1.25) or 2 specific values that represent the lower and upper equivalence bounds (e.g., c(0.8, 1.25)).
alpha: alpha level (default = 0.05).
null: the ratio value under the null hypothesis (default = 1).
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).
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 means ratio calculation.
- Items include: d (means ratio 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.

Details

The function implements a bootstrap method for log-transformed TOST as recommended by He et al. (2022) and corresponds to the proposal in Chapter 16 of Efron and Tibshirani (1994). This is approximately equivalent to the percentile bootstrap method mentioned by He et al. (2014).

For two-sample tests, the test is of \(\bar log(x) - \bar log(y)\) which corresponds to testing the ratio of geometric means. For paired samples, the test is of difference scores on the log scale, \(z = log(x) - log(y) = log(x/y)\), which also corresponds to a ratio test.

The bootstrap procedure follows these steps:

Log-transform the data
Perform resampling with replacement to generate 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
Back-transform for the ratio of means

Note that all input data must be positive (ratio scale with a true zero) since log transformation is applied. The function will stop with an error if any negative values are detected.

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

Purpose

Use this function when:

Your data is on a ratio scale (all values must be positive)
You want to establish equivalence based on the ratio of means rather than their difference
Traditional parametric methods may not be appropriate due to skewed distributions
You need to analyze bioequivalence data where bounds are expressed as ratios

References

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

He, Y., Deng, Y., You, C., & Zhou, X. H. (2022). Equivalence tests for ratio of means in bioequivalence studies under crossover design. Statistical Methods in Medical Research, 09622802221093721.

Food and Drug Administration (2014). Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs — General Considerations. Center for Drug Evaluation and Research. Docket: FDA-2014-D-0204. https://www.fda.gov/regulatory-information/search-fda-guidance-documents/bioavailability-and-bioequivalence-studies-submitted-ndas-or-inds-general-considerations

Examples

# Example 1: Two-Sample Test for Bioequivalence
# Generate ratio scale data (e.g., drug concentrations)
test_group <- rlnorm(30, meanlog = 3.5, sdlog = 0.4)
ref_group <- rlnorm(30, meanlog = 3.6, sdlog = 0.4)

# FDA standard bioequivalence bounds (80% to 125%)
result <- boot_log_TOST(x = test_group,
                        y = ref_group,
                        eqb = 1.25,  # Creates bounds of 0.8 and 1.25
                        R = 999)     # Reduce for demonstration

# Example 2: Paired Sample Test
# Generate paired ratio scale data
n <- 20
baseline <- rlnorm(n, meanlog = 4, sdlog = 0.3)
followup <- baseline * rlnorm(n, meanlog = 0.05, sdlog = 0.2)

# Test with asymmetric bounds
result <- boot_log_TOST(x = followup,
                        y = baseline,
                        paired = TRUE,
                        eqb = c(0.85, 1.20),
                        boot_ci = "perc")