---
title: "Equivalence Testing for Normal Outcomes"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Equivalence for Normal Outcomes}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
options(rmarkdown.html_vignette.check_title = FALSE)
```


```{r setup}
library(ssutil)
```

## 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:

```{r equivalence-example}
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
)
```

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:  $96.98\%^3 = 91.21\%$.