A Note on Model Specification: Mixed Model for Repeated Measurement (MMRM)

Mixed Model for Repeated Measurement (MMRM) 
 Full and Naive Structure 
 Let 
 
 C s, t be the Change from Baseline in Patient s at Visit t ; 
 BL s be the Observation of Patient s at Baseline t=0 ; 
 V t be the Visit for C t , for example, 21 or 42 days; 
 T be the planned treatment group for Patient s , for example, Reference vs Test; 
 s(t) be the variance-covariance matrix of Patient s nesting Visit t ; 
 and a error term present the residual. 
 
 The model can be written as: 
 
 
 
 Full and Dummized Structure 
 The interaction are preferred to be expanded: 
 
 
 
 
At this step, if you notice, T s can be simplified by factoring out(additive term for β 2 , β 3 , β 41 and β 42 ). However, V t has to be retained as part of matrix s(t).
 Simplified and Dummized Structure 
 Factoring out T s 
 
 
 
 Linear Combination 
 Let B̅ be the population mean at Baseline. 
 Once model converage, β are optimized. Estimations can be obtained through linear combination of terms. 
 The change from baseline at 21 days for Reference 
 
 
 
 The change from baseline at 21 days for Test 
 
 
 
 The change from baseline at 42 days for Reference 
 
 
 
 The change from baseline at 42 days for Test 
 
 
 
 Then, the difference between Reference and Test at 42 days 
 
 
 
 Baseline-contrained Longitudial Data Analysis (cLDA) 
 To be updated. 
 Appendix 
 
 CodeCogs 
 C_{s,t} = \mu + \beta_{1}\times B_{s} + \beta_{2}\times V_{t} + \beta_{3}\times T_{s} + \beta_{4}\times V_{t} \times T_{s} + \gamma\times s(t) + \varepsilon_{s,t} 
 C_{s,t} = \mu + \beta_{1}\times B_{s} + \beta_{2}\times V_{t} + \beta_{3}\times T_{s} + \beta_{41}\times (V_{t}=21)\times T_{s} + \beta_{42}\times (V_{t}=42)\times T_{s} + \gamma\times s(t) + \varepsilon_{s,t} 
 C_{s,t} = \mu + \beta_{1}\times B_{s} + \beta_{2}\times V_{t} + \beta'_{41}\times (V_{t}=21)\times T_{s} + \beta'_{42}\times (V_{t}=42)\times T_{s} + \gamma\times s(t) + \varepsilon_{s,t} 
 \widehat{C_{21}} = \mu + \beta_{1}\times \overline{B_{s}} + \beta_{2}\times V_{t} + \beta'_{41}\times (V_{t}=21)\times T_{s} + \beta'_{42}\times (V_{t}=42)\times T_{s}