Mixed models for repeted measurements
Resources:
mmrmpackage vignette- Mixed models with R
- GLMM FAQ
Mixed modeling in R
- Linear mixed models:
nlme::lme,lme4::mner,brms::brm - generalized linear mixed models (GLMM):
lme4::glmer,glmmTMB;brms::brmfor Bayesian - nonlinear mixed models:
nlme::nlme,lme4::nlmer;brms::brm
| equation | formula | meaning |
|---|---|---|
| \(B_0 + B_1X_i + e_i\) | no random effect | |
| \((B_0+b_{g,0} + B_1X_i + e_i)\) | x+(1|group) |
random group intercept |
(x|group) |
random slopt of x within group, with correlated intercept |
|
(1+x|group) |
||
(0+x | group) |
random slop of x within group, no variation in intercept |
|
(-1+x | group) |
||
(1| group) + (0+x|group) |
uncorrelated random intercept and random slopt within group |
Random intercept
Each group has different intercelpt, but the slope is the same
\[gpa = (b_0 + effect_{student} + b_{occasion} * occasion + e)\]
MMRM
MMRM has one distinct feature compared to other linear mixed models: subject-specific random effects are considered as residual effects (part of error correlation matrix).
Methodology
Basic linear mixed-effects model for a single level of grouping
\[ y_i = X_i \beta + Z_i b_i + \epsilon_i, i = 1, ..., n \] \[ b_i \sim N(0, \Psi), \epsilon \sim N(0, \sigma^2 I) \]
- \(\beta\) is p-dim vector of fixed effects
- \(b\) is q-dim vecor of random patient specific effects
- \(X_i\) of size \(n_i \times p\) and \(Z_i\) of size \(n_i \times q\) are regressor matrices relating observations to the fixed effects and random effects.
- \(\epsilon_i\) is \(n_i\)-dimensional within-subject error