---
title: "Non-Inferiority"
lang: "en"
output:
  rmarkdown::html_vignette:
    css: vignette.css
    toc: true
    toc_depth: 1
vignette: >
  %\VignetteIndexEntry{Non-Inferiority}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---
```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#"
)
```
<div class="top"><a class="toplink" href="#nav" title="&uArr; top">&nbsp;</a></div>
<div id="nav">
<ul>
<li>[Main Vignette](vignette.html)</li>
<li>[ABE](ABE.html "Average Bioequivalence")</li>
<li>[RSABE](RSABE.html "Reference-scaled Average Bioequivalence")</li>
<li><span title="» You are here «">Non-Inferiority</span></li>
<li>[Dose-Proportionality](DP.html)</li>
<li>[Power Analysis](PA.html)</li>
</ul>
</div>
<h3>Details and examples of other methods are accessible via the menu bar on top of the page and in the <a href="https://cran.r-project.org/package=PowerTOST/PowerTOST.pdf" title="PDF">online manual</a> of all functions.</h3>
```{r setup}
library(PowerTOST) # attach the library
```
# Defaults

| Parameter | Argument | Purpose | Default |
|-|--|------|--|
| $\small{\alpha}$ | `alpha` | Nominal level of the test | `0.025` |
| $\small{\pi}$ | `targetpower` | <span title="typically 0.80 – 0.90">Minimum desired power</span> | `0.80` |
| logscale | `logscale` | Analysis on log-transformed or original scale? | `TRUE` |
| margin | `margin` | Non-inferiority margin | see below |
| $\small{\theta_0}$ | `theta0` | ‘True’ or assumed T/R ratio | see below |
| *CV* | `CV` | CV | none |
| design | `design` | Planned design | `"2x2"` |
| imax | `imax` | Maximum number of iterations | `100` |
| print | `print` | Show information in the console? | `TRUE` |
| details | `details` | Show details of the sample size search? | `FALSE` |

Note that contrary to the other functions of the package a one-sided *t*-test (instead of <span title="Two One-Sided Tests">TOST</span>) is employed. Hence, $\small{\alpha}$ defaults to 0.025.\
Defaults depending on the argument `logscale`:

| Parameter | Argument | `logscale=TRUE` | `logscale=FALSE` |
|-|----|:--------:|:--------:|
| margin | `margin` | `0.80` | `–0.20` |
| $\small{\theta_0}$ | `theta0` | `0.95` | `+0.05` |

Arguments `targetpower`, `margin`, `theta0`, and `CV` have to be given as fractions, not in percent.\
The *CV* is generally the *within-* (intra-) subject coefficient of variation. Only for `design = "parallel"` it is the *total* (<span title="also known as">a.k.a.</span> pooled) *CV*.

Designs with one (parallel), two (conventional crossover and paired), and three or four periods (replicates) are supported.
```
#      design                        name   df
#  "parallel"           2 parallel groups  n-2
#       "2x2"               2x2 crossover  n-2
#     "2x2x2"             2x2x2 crossover  n-2
#     "2x2x3"   2x2x3 replicate crossover 2n-3
#     "2x2x4"   2x2x4 replicate crossover 3n-4
#     "2x4x4"   2x4x4 replicate crossover 3n-4
#     "2x3x3"   partial replicate (2x3x3) 2n-3
#     "2x4x2"            Balaam’s (2x4x2)  n-2
#    "2x2x2r" Liu’s 2x2x2 repeated x-over 3n-2
#    "paired"                paired means  n-1
```
The terminology of the `design` argument follows this pattern: `treatments x sequences x periods`. The conventional TR|RT (<span title="also known as">a.k.a.</span> AB|BA) design can be abbreviated as `"2x2"`. Some call the `"parallel"` design a ‘one-sequence’ design. The design `"paired"` has two periods but no sequences, *e.g.*, in studying linear pharmacokinetics a single dose is followed by multiple doses. A profile in steady state (T) is compared to the one after the single dose (R). Note that the underlying model assumes no period effects.

With `sampleN.noninf(..., details = FALSE, print = FALSE)` results are provided as a data frame<small> </small>^[R Documentation. *Data Frames.* 2020-10-26. [R-manual](https://stat.ethz.ch/R-manual/R-devel/library/base/html/data.frame.html).] with eight columns `Design`, `alpha`, `CV`, `theta0`, `Margin`, `Sample size`, `Achieved power`, and `Target power`. To access *e.g.*, the sample size use either `sampleN.noninf[1, 6]` or `sampleN.noninf[["Sample size"]]`. We suggest to use the latter in scripts for clarity.

<span class="hl">The estimated sample size gives always the *total* number of subjects (not subject/sequence in crossovers or subjects/group in parallel designs – like in some other software packages).</span>

# Non-Inferiority
If the supplied margin is < 1 (`logscale = TRUE`) or < 0 (`logscale = FALSE`), then it is assumed that *higher* response values are <u>better</u>. The hypotheses are with

* `logscale = TRUE`
$$\small{H_0:\theta_0 \leq \log({margin})\:vs\:H_1:\theta_0>\log({margin})}$$
    where $\small{\theta_0=\mu_\textrm{T}/\mu_\textrm{R}}$

* `logscale = FALSE`
$$\small{H_0:\theta_0 \leq {margin}\:vs\:H_1:\theta_0>{margin}}$$

    where $\small{\theta_0=\mu_T-\mu_R}$

## Example 1
Estimate the sample size for assumed intra-subject *CV* 0.25. Defaults `margin` 0.80 and $\small{\theta_{0}}$ 0.95 employed.
```{r example1a}
sampleN.noninf(CV = 0.25)
```
To get only the sample size:
```{r example1b}
sampleN.noninf(CV = 0.25, details = FALSE, print = FALSE)[["Sample size"]]
```
Note that the sample size is always rounded up to give balanced sequences (here a multiple of two). Since power is higher than our target, likely this was the case here. Let us assess that:\
Which power will we get with a sample size of 35?

```{r example1c}
power.noninf(CV = 0.25, n = 35)
```
Confirmed that with 35 subjects we will already reach the target power. That means also that one dropout will not compromise power.

# Non-Superiority
If the supplied margin is > 1 (`logscale = TRUE`) or > 0 (`logscale = FALSE`), then it is assumed that *lower* response values are <u>better</u>. The hypotheses are with

* `logscale = TRUE`
$$\small{H_{0}:\theta_0 \geq \log({margin})\:vs\:H_{1}:\theta_0<\log({margin})}$$
    where $\small{\theta_0=\mu_\textrm{T}/\mu_\textrm{R}}$

* `logscale = FALSE`
$$\small{H_{0}:\theta_0 \geq {margin}\:vs\:H_{1}:\theta_0<{margin}}$$

    where $\small{\theta_0=\mu_T-\mu_R}$

## Example 2
Estimate the sample size for assumed intra-subject *CV* 0.25.
```{r example2}
sampleN.noninf(CV = 0.25, margin = 1.25, theta0 = 1/0.95)
```
Same sample size like in [example 1](#example-1) since reciprocal values of both `margin` 0.80 and $\small{\theta_{0}}$ are specified.

# Bracketing Approach
Compare a new modified release formulation (regimen once a day) with an intermediate release formulation (twice a day).^[European Medicines Agency, Committee for Medicinal Products for Human Use. *Guideline on the pharmacokinetic and clinical evaluation of modified release dosage forms.* London. 20 November 2014. EMA/CPMP/EWP/280/96 Corr1. [online](https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-pharmacokinetic-and-clinical-evaluation-modified-release-dosage-forms_en.pdf).] *C*~min~ is the target metric for efficacy (non-inferiority) and *C*~max~ for safety (non-superiority). Margins are 0.80 for *C*~min~ and 1.25 for *C*~max~. *CV*s are 0.35 for *C*~min~ and 0.20 for *C*~max~; $\small{\theta_{0}}$ 0.95 for *C*~min~ and 1.05 for *C*~max~. Full replicate design due to the high variability of *C*~min~.\
Which PK metric leads the sample size? 
```{r bracket1}
res <- data.frame(design = "2x2x4", metric = c("Cmin", "Cmax"),
                  margin = c(0.80, 1.25), CV = c(0.35, 0.20),
                  theta0 = c(0.95, 1.05), n = NA, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0)
for (i in 1:2) {
  res[i, 6:7] <- sampleN.noninf(design = res$design[i],
                                margin = res$margin[i],
                                theta0 = res$theta0[i],
                                CV = res$CV[i],
                                details = FALSE,
                                print = FALSE)[6:7]
}
print(res, row.names = FALSE)
```
The sample size depends on *C*~min~. Hence, the study is ‘overpowered’ for *C*~max~.
```{r bracket2}
power.noninf(design = "2x2x4", margin = 1.25, CV = 0.20,
             theta0 = 1.05, n = 32)
```
Therefore, that gives us some ‘safety margin’ for *C*~max~.
```{r bracket3}
power.noninf(design = "2x2x4", margin = 1.25, CV = 0.25,
             theta0 = 1.10, n = 32) # higher CV, worse theta0
```
The bracketing approach does not necessarily give lower sample sizes than tests for equivalence. In this example we could aim at reference-scaling for the highly variable *C*~min~ and at conventional ABE for *C*~max~.

```{r bracket4}
res <- data.frame(design = "2x2x4", intended = c("ABEL", "ABE"),
                  metric = c("Cmin", "Cmax"), CV = c(0.35, 0.20),
                  theta0 = c(0.90, 1.05), n = NA, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0
res[1, 6:7] <- sampleN.scABEL(CV = res$CV[1], theta0 = res$theta0[1],
                              design = res$design[1], print = FALSE,
                              details = FALSE)[8:9]
res[2, 6:7] <- sampleN.TOST(CV = res$CV[2], theta0 = res$theta0[2],
                            design = res$design[2], print = FALSE,
                            details = FALSE)[7:8]
print(res, row.names = FALSE)
```
Which method is optimal is a case-to-case decision. Although in this example the bracketing approach seems to be the ‘winner’ (32 subjects instead of 34), we might fail if the *CV* of *C*~min~ is larger than assumed, whereas in reference-scaling we might still pass due to the expanded limits.
```{r bracket5}
n <- sampleN.scABEL(CV = 0.35, theta0 = 0.90, design = "2x2x4",
                    print = FALSE, details = FALSE)[["Sample size"]]
# CV and theta0 of both metrics worse than assumed
res <- data.frame(design = "2x2x4", intended = c("ABEL", "ABE"),
                  metric = c("Cmin", "Cmax"), CV = c(0.50, 0.25),
                  theta0 = c(0.88, 1.12), n = n, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0
res[1, 7] <- power.scABEL(CV = res$CV[1], theta0 = res$theta0[1],
                          design = res$design[1], n = n)
res[2, 7] <- power.TOST(CV = res$CV[2], theta0 = res$theta0[2],
                        design = res$design[2], n = n)
print(res, row.names = FALSE)
```
See also the vignettes [RSABE](RSABE.html "Reference-scaled Average Bioequivalence"), [ABE](ABE.html "Average Bioequivalence"), and [PA](PA.html "Power Analysis").

# Author{#author}
Detlew Labes

# License{#license}
<h4 class="author">[GPL-3](https://cran.r-project.org/web/licenses/GPL-3 "GNU General Public License, Version 3") `r Sys.Date()` Helmut Schütz</h4>