Decomposing Variance in General Linear Mixed Models for Repeated Measurements : Understanding Between-Subject, Within-Subject, and Measurement Error Components

 In the linear mixed model:

$$\text{Var}(Y_i)=\underset{\text{Between-subject variance}}{\underbrace{Z_{i}G{Z}_i^{\mathrm{\prime }}}}+\underset{\text{Within-subject variance}}{\underbrace{R_i}}$$

• Between-subject variance ($Z_{i}G{Z}_i^{\mathrm{\prime }}$): Captures variability due to random effects, like subject-specific intercepts or slopes.
• Within-subject variance ($R_i$): Captures variability within a subject, which includes:
  • Measurement error
  • Other time-specific fluctuations

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📌 So where is measurement error?

Measurement error is part of the within-subject variance. If we assume:

$$R_i={\sigma }^{2}I$$

then all within-subject variability is attributed to independent measurement error with constant variance ${\sigma }^2$.

However, in more complex models, $R_i$ can include:

• Autocorrelation (e.g., AR(1) structure)
• Heteroscedasticity (changing variance over time)
• Measurement error

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✅ Summary

You can say:

Var($Y_i$) = between-subject variance + within-subject variance (including measurement error)