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)). brms.mmrm focuses on marginal MMRMs for
randomized controlled parallel studies with discrete time points, where
each subject 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
The MMRM in brms.mmrm is mathematically expressed as
follows. Subsequent sections define notation, data, parameters, and
priors.
\[ \begin{aligned} &y_i \stackrel{\text{ind}}{\sim} \text{Multivariate-Normal} \left (\text{mean} = X_{i} \beta, \ \text{covariance} = \text{diag}(\sigma) \cdot \Lambda \cdot \text{diag}(\sigma) \right ) \\ &\qquad \beta_p \stackrel{\text{ind}}{\sim} q_p() \\ &\qquad \Lambda \sim r() \\ &\qquad \tau = \log(\sigma) \\ &\qquad \qquad \tau_t \stackrel{\text{ind}}{\sim} s_t() \end{aligned} \]
Notation
- \(i\): positive integer index of a subject or patient in the study.
- \(t\): positive integer index of a discrete time point in the study.
- \(p\): positive integer index of a model coefficient parameter \(\beta_p\).
- \(\theta_j \stackrel{\text{ind}}{\sim} f()\): assuming \(j\) is a positive integer index that ranges from 1 to positive integer \(J\), this notation declares that parameters \(\theta_1, \ldots, \theta_J\) are independent and each follow the prior distribution with a common probability density function \(f()\).
- \(\theta_j \stackrel{\text{ind}}{\sim} f_j()\): same as above, except each \(\theta_j\) parameter follows its own separate prior distribution with probability density function \(p_j()\). \(p_i()\) and \(f_j()\) may differ if \(i \ne j\).
- \(\text{diag}(\theta)\): diagonal matrix with the scalar elements of vector \(\theta\) on the diagonal and off-diagonal elements equal to 0. The number of rows and number of columns in this matrix are both equal to the number of elements in vector \(\theta\).
Data
- \(y_i\): numeric vector of outcome observations for subject \(i\). The number of elements in the vector equals the number of discrete time points in the study, and for the purposes of this model specification, they are sorted chronologically with the first element of \(y_i\) observed first in the study. One or more elements of each \(y_i\) may be missing due to dropout, discontinuation, etc. The likelihood of the model assumes \(y_i\) is independent of \(y_j\) for \(i \ne j\).
-
\(X_i\): a matrix containing the
rows of the model matrix that correspond to subject \(i\). The rows of \(X_i\) correspond to the elements of the
vector \(y_i\) (equivalently, the
discrete time points of the study), and the columns of \(X_i\) correspond to the model coefficient
parameters \(\beta_p\). The composition
of \(X_i\) is determined by the
covariates in the input dataset and the choice of model formula supplied
by the user (via
brm_formula()).
Parameters
- \(\beta\): vector of model coefficients. \(\beta_p\) is scalar element \(p\) of \(\beta\).
- \(\Lambda\): matrix of pairwise correlations among each pair of time points within subjects. Outcomes measured from different subjects are assumed to be independent, and outcomes observed at different time points for the same subject are assumed to be correlated according to this matrix. \(\Lambda\) is positive-definite and symmetric, and the number of rows and columns is equal to the number of discrete time points of the study.
- \(\sigma\): vector of time-point-specific standard deviations of the residuals on the linear scale. \(\sigma_t\) is scalar element \(t\) of \(\sigma\).
- \(\tau\): same as \(\sigma\), but on the natural logarithmic scale. Each scalar element \(\tau_t\) of \(\tau\) is defined as the natural logarithm of \(\sigma_t\).
Priors
The priors on the parameters depend on the prior
argument of brm_model() and related functions. If priors
are not specified by the user, then the brms
package sets defaults. You can view the default priors using the get_prior()
function in brms. See
the brms for
information on how brms sets
default priors.
- \(q_p()\): univariate prior on the model coefficient \(\beta_p\).
- \(r()\): matrix prior on the within-subject residual correlation matrix parameter \(\Lambda\).
- \(s_t()\): univariate prior on the natural-log-standard-deviation parameter \(\tau_t\) of discrete time point \(t\).
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.
Inference
Inference in brms.mmrm, uses the estimated marginal
posterior distribution of the mean response at each combination of study
arm and time point. The emmeans
package (Lenth 2016) derives these
marginal posteriors while averaging over other covariates as nuisance
parameters. During this averaging process, the levels of categorical
nuisance parameters are weighted proportionally to their frequencies in
the dataset (with wt.nuis = "proportional" in
emmeans::ref_grid()).
The brm_marginal_draws() function, described in the
usage vignette, derives posterior draws of the marginals using posterior
draws of the model coefficients \(\beta_p\). Then, downstream functions like
brm_marginal_probabilities() compute numerical summaries of
these marginal draws.
Subgroup analysis
The model above supports subgroup analysis through the addition of a
categorical variable in the data to denote subgroup levels. To analyze
the subgroup, new fixed effects parameters \(\beta_p\) and columns of the model matrices
\(X_i\) are added to the model to
describe the additive effect of the subgroup and plausible two-way and
three-way interactions with treatment group, discrete time, and baseline
(if applicable). Marginal means are subgroup-specific, and model
comparison via the widely applicable information criterion (WAIC) and
expected log predictive density (ELPD) is implemented via R packages
loo and brms (Gabry et
al. (2019), Gelman and Hill (2007),
Vehtari et al. (2017), Vehtari et al. (2019)).