---
title: "Model"
bibliography: '`r system.file("bibliography.bib", package = "brms.mmrm")`'
csl: '`r system.file("asa.csl", package = "brms.mmrm")`'
output:
  rmarkdown::html_vignette:
    toc: true
    number_sections: true
vignette: >
  %\VignetteIndexEntry{Model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

The `brms.mmrm` R package implements a mixed model of repeated measures (MMRM), a popular and flexible model to analyze continuous longitudinal outcomes (@mallinckrodt2008, @mallinckrodt2017, @bamdd). `brms.mmrm` focuses on marginal MMRMs for randomized controlled parallel studies with discrete time points, where each patient shares the same set of time points. Whereas the [`mmrm`](https://openpharma.github.io/mmrm/) package is frequentist, `brms.mmrm` fits models in Bayesian fashion using [`brms`](https://paul-buerkner.github.io/brms/) [@burkner2017].

# Model

Let $y_1, \ldots, y_N$ be independent data points observed for individual patients in a clinical trial. Each $y_n$ is a numeric vector of length $T$, where $T$ is the number of discrete time points in the dataset (e.g. patient visits in the study protocol). We model $y_n$ as follows:

$$
\begin{aligned}
y_n \sim \text{Multivariate-Normal}\left ( \text{mean} = X_n b, \ \text{variance} = \Sigma_n \right )
\end{aligned}
$$
Above, $X_n$ is the fixed effect model matrix of patient $n$, and its specific makeup is determined by arguments such as `intercept` and `group` in `brm_formula()`. $b$ is a constant-length vector of fixed effect parameters.

The MMRM in `brms.mmrm` is a [distributional model](https://paul-buerkner.github.io/brms/articles/brms_distreg.html), which means it uses a linear regression structure for both the mean and the variance of the multivariate normal likelihood. In particular, the $T \times T$ symmetric positive-definite residual covariance matrix $\Sigma_n$ of patient $n$ decomposes as follows:

$$
\begin{aligned}
\Sigma_n &= \text{diag}(\sigma_n) \cdot \Lambda \cdot \text{diag}(\sigma_n) \\
\sigma_n &= \text{exp} \left ( Z_n b_\sigma \right)
\end{aligned}
$$

Above, $\sigma_n$ is a vector of $T$ time-specific scalar standard deviations, and $\text{diag}(\sigma_n)$ is a diagonal $T \times T$ matrix. $Z_n$ is a patient-specific matrix which controls how the [distributional parameters](https://paul-buerkner.github.io/brms/articles/brms_distreg.html) $b_\sigma$ map to the more intuitive standard deviation vector $\sigma_n$. The specific makeup of $Z_n$ is determined by the `sigma` argument of `brm_formula()`, which in turn is produced by `brm_formula_sigma()`.

$\Lambda$ is a symmetric positive-definite correlation matrix with diagonal elements equal to 1 and off-diagonal elements between -1 and 1. The structure of $\Lambda$ depends on the `correlation` argument of `brm_formula()`, which could describe an unstructured parameterization, ARMA, compound symmetry, etc. These alternative structures and priors are available directly through `brms`. For specific details, please consult <https://paul-buerkner.github.io/brms/reference/autocor-terms.html> and `?brms.mmrm::brm_formula`.

# Priors

 The scalar components of $b$ are modeled as independent with user-defined priors specified through the `prior` argument of `brm_model()`. The hyperparameters of these priors are constant. The default priors are improper uniform for non-intercept terms and a data-dependent Student-t distribution for the intercept. The variance-related distributional parameters $b_\sigma$ are given similar priors
 
For the correlation matrix $\Lambda$, the default prior in `brms.mmrm` is the [LKJ correlation distribution](https://mc-stan.org/docs/functions-reference/correlation_matrix_distributions.html#lkj-correlation) with shape parameter equal to 1. This choice of prior is only valid for unstructured correlation matrices. Other correlation structures, such ARMA, will parameterize $\Lambda$ and allow users to set priors on those new specialized parameters.

# Sampling

`brms.mmrm`, through [`brms`](https://paul-buerkner.github.io/brms/), fits the model to the data using the Markov chain Monte Carlo (MCMC) capabilities of [Stan](https://mc-stan.org/) [@stan2023]. Please read <https://mc-stan.org/users/documentation/> for more details on the methodology of [Stan](https://mc-stan.org/). The result of MCMC is a collection of draws from the full joint posterior distribution of the parameters given the data. Individual draws of scalar parameters such as $\beta_3$ are considered draws from the marginal posterior distribution of e.g. $\beta_3$ given the data.

# Imputation of missing outcomes

Under the missing at random (MAR) assumptions, MMRMs do not require imputation (@bamdd). However, if the outcomes in your data are not missing at random, or if you are targeting an alternative estimand, then you may need to impute missing outcomes. `brms.mmrm` can leverage either of the two alternative solutions described at <https://paul-buerkner.github.io/brms/articles/brms_missings.html>. Please see the [usage vignette](https://openpharma.github.io/brms.mmrm/articles/usage.html) for details on the implementation and interface.

# References
