The `brms.mmrm`

R package implements a mixed model of
repeated measures (MMRM), a popular and flexible model to analyze
continuous longitudinal outcomes (Mallinckrodt et
al. (2008), Mallinckrodt and Lipkovich
(2017), Holzhauer and Weber
(2024)). `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`

package
is frequentist, `brms.mmrm`

fits models in Bayesian fashion
using `brms`

(Bürkner 2017).

## 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, 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
$b_\sigma$
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://paulbuerkner.com/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 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`

, fits the
model to the data using the Markov chain Monte Carlo (MCMC) capabilities
of Stan (Stan
Development Team 2023). Please read https://mc-stan.org/users/documentation/ for more
details on the methodology of Stan.
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 (Holzhauer and Weber (2024)).
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://paulbuerkner.com/brms/articles/brms_missings.html.
Please see the usage
vignette for details on the implementation and interface.

## References

*Journal of Statistical Software*, 80, 1–28. https://doi.org/10.18637/jss.v080.i01.

*Applied Modeling in Drug Development*, Novartis AG.

*Therapeutic Innovation and Regulatory Science*, 42, 303–319. https://doi.org/10.1177/009286150804200402.

*Analyzing longitudinal clinical trial data: A practical guide*, CRC Press, Taylor; Francis Group.

*Stan modeling language users guide and reference manual*.