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

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~~;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:

$C_{s,t}=\mu+\beta_{1}\times B_{s}+\beta_{2}\times V_{t}+\beta_{3}\times T_{s}+\beta_{4}\times V_{t}\times T_{s}+\gamma\times s(t)+\varepsilon_{s,t}$

Full and Dummized Structure

The interaction are preferred to be expanded:

$C_{s,t}=\mu+\beta_{1}\times B_{s}+\beta_{2}\times V_{t}+\beta_{3}\times T_{s}+\beta_{41}\times(V_{t}=24)\times T_{s}+\beta_{42}\times(V_{t}=42)\times T_{s}+\gamma\times s(t)+\varepsilon_{s,t}$ $C_{s,t}=\mu+\beta_{1}\times B_{s}+\beta_{2}\times V_{t}+\beta_{3}\times T_{s}+\beta_{41}\times(V_{t}=21)\times T_{s}+\beta_{42}\times(V_{t}=42)\times T_{s}+\gamma\times s(t)+\varepsilon_{s,t}$

At this step, if you notice, T_s can be simplified by factoring out(additive term for β₂, β₃, β₄₁ and β₄₂). However, V_t has to be retained as part of matrix s(t).

Simplified and Dummized Structure

~~Then, factoring~~Factoring out T_s

$C_{s,t}=\mu+\beta_{1}\times B_{s}+\beta_{2}\times V_{t}+\beta'_{41}\times(V_{t}=24)\times T_{s}+\beta'_{42}\times(V_{t}=42)\times T_{s}+\gamma\times s(t)+\varepsilon_{s,t}$ $C_{s,t}=\mu+\beta_{1}\times B_{s}+\beta_{2}\times V_{t}+\beta'_{41}\times(V_{t}=21)\times T_{s}+\beta'_{42}\times(V_{t}=42)\times T_{s}+\gamma\times s(t)+\varepsilon_{s,t}$

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

$\widehat{C_{21}}=\mu+\beta_{1}\times \overline{B_{s}}$

The change from baseline at 21 days for Test

$\widehat{C_{21}}=\mu+\beta_{1}\times \overline{B_{s}}+\beta'_{41}$

The change from baseline at 42 days for Reference

$\widehat{C_{42}}=\mu+\beta_{1}\times \overline{B_{s}}+\beta_{2}$

The change from baseline at 42 days for Test

$\widehat{C_{42}}=\mu+\beta_{1}\times \overline{B_{s}}+\beta_{2}+\beta'_{42}$

Then, the difference between Reference and Test at 42 days

$\widehat{D_{42}}=\beta'_{42}$

Baseline-contrained Longitudial

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}=~~24)~~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}

\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}