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).
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://paul-buerkner.github.io/brms/reference/autocor-terms.html
and ?brms.mmrm::brm_formula.
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.
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.