Equivalence Testing for Normal Outcomes

Introduction

This vignette illustrates how the sim_power_equivalence_normal() function can be used to estimate the empirical power of an equivalence test under the assumption of normally distributed outcomes. The method simulates repeated trials and determines whether all pairwise confidence intervals for differences in group means fall within user-specified equivalence limits.

Example: Equivalence Among Vaccine Lots

An evaluation is conducted to determine whether three manufacturing lots of a vaccine produce equivalent immune responses. The outcome is the antibody concentration measured on the \(log_{10}\) scale, assumed to follow a normal distribution with a standard deviation of 0.4.
Equivalence is declared if the confidence intervals for the ratio of all pairwise comparisons fall entirely within the range [2/3, 3/2].

Since the analysis is conducted on the \(log_{10}\) scale, the equivalence limits are transformed to log10(2/3) and log10(3/2).

A total of 1,000 trials is simulated with 172 subjects per group and a 95% confidence level:

set.seed(12345)
sim_power_equivalence_normal(
  ngroups = 3,
  npergroup = 172,
  sd = 0.4,
  llimit = log10(2/3),
  ulimit = log10(3/2),
  nsim = 1000,
  t_level = 0.95
)
#> Empirical Power Result
#> ----------------------- 
#> Power:       0.9130
#> 95% CI:      [0.8938, 0.9297]
#> Simulations: 1000

The result shows the proportion of simulations in which all pairwise comparisons satisfy the equivalence criterion.

A power calculation in nQuery® shows that a Two One-Sided Equivalence Test (TOST) for a two-group design, with an alpha level of 0.025, 172 participants per group, a standard deviation of 0.4, and equivalence limits of log10(2/3) and log10(3/2), has a power of 96.98%. When extended to three comparisons, assuming independence, the joint probability of all three satisfying the equivalence condition is approximately: \(96.98\%^3 = 91.21\%\).