---
title: "Bioequivalence and Crossover Analysis with sasLM"
author: "Kyun-Seop Bae"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Bioequivalence and Crossover Analysis with sasLM}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(comment = NA)
options(width = 100)
library(sasLM)
```

## The standard 2x2 crossover analysis

Average bioequivalence with a 2x2 crossover design is conventionally analyzed with
SAS PROC GLM:

```
PROC GLM DATA=be;
  CLASS SEQ SUBJ PRD TRT;
  MODEL LNCMAX = SEQ SUBJ(SEQ) PRD TRT;
  RANDOM SUBJ(SEQ) / TEST;
  LSMEANS TRT / CL ALPHA=0.1;
  ESTIMATE 'T - R' TRT -1 1 / CL ALPHA=0.1;
RUN;
```

The package ships `BEdata`, real data from a 2x2 bioequivalence study with three
hospitalization groups. The same analysis with `sasLM`:

```{r}
BEdata = af(BEdata, c("ADM", "SEQ", "PRD", "TRT", "SUBJ")) # columns as factors
formula1 = log(CMAX) ~ SEQ/SUBJ + PRD + TRT
GLM(formula1, BEdata)
```

`SEQ/SUBJ` denotes subjects nested within sequence, equivalent to
`SEQ SUBJ(SEQ)` of SAS. The Type I, II, and III tables above match SAS PROC GLM
for this unbalanced data set (91 subjects).

## Testing SEQ against the correct error term

The sequence effect must be tested against the subject-within-sequence mean square,
not the residual. This is what the RANDOM / TEST statement of SAS does:

```{r}
RanTest(formula1, BEdata, Random="SUBJ")
```

## The 90% confidence interval of the geometric mean ratio

The two-one-sided-tests (TOST) procedure at the 5% level is operationally the
90% confidence interval of T/R in the log scale:

```{r}
ci0 = CIest(formula1, BEdata, "TRT", c(-1, 1), conf.level=0.90)
ci0
exp(ci0[, c("Estimate", "Lower CL", "Upper CL")]) # GMR and its 90% CI
```

The exponentiated estimate and confidence limits are the geometric mean ratio (GMR)
and its 90% confidence interval. Bioequivalence is concluded when the interval is
contained in [0.80, 1.25].

## Least squares means of treatments

```{r}
LSM(formula1, BEdata, "TRT", conf.level=0.90)
```

## Notes

* For unbalanced designs with missing periods, a mixed effects model
  (SAS PROC MIXED or the 'nlme' package) is generally preferred; see the example
  in `?sasLM` for the equivalent `nlme::lme` call.
* The author's 'BE' package builds the complete bioequivalence workflow
  (including sample size and outlier diagnostics) on top of 'sasLM'.