8.6 Repeated Measures in Mixed Models

Repeated measures data arise when multiple observations are collected from the same subjects over time or under different conditions. This introduces correlation between observations from the same subject, which must be accounted for in the statistical model.

Mixed-effects models are particularly effective for repeated measures because they allow us to model both fixed effects (e.g., treatment, time) and random effects (e.g., subject-specific variability).

The general form of a mixed-effects model for repeated measures is:

$$Y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha \beta)_{ij} + \delta_{i(k)} + \epsilon_{ijk}$$

Where:

• $Y_{ijk}$ = Response for the $i$-th group, $j$-th time point, and $k$-th subject.

• $\mu$ = Overall mean.

• $\alpha_i$ = Fixed effect of the $i$-th group (e.g., treatment group).

• $\beta_j$ = Fixed effect of the $j$-th time point (repeated measure effect).

• $(\alpha \beta)_{ij}$ = Interaction effect between group and time (fixed).

• $\delta_{i(k)} \sim N(0, \sigma^2_\delta)$ = Random effect of the $k$-th subject within the $i$-th group (captures subject-specific deviations).

• $\epsilon_{ijk} \sim N(0, \sigma^2)$ = Residual error (independent across observations).

Here, $i = 1, \dots, n_A$ (number of groups), $j = 1, \dots, n_B$ (number of repeated measures), and $k = 1, \dots, n_i$ (number of subjects in group $i$).

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The variance-covariance matrix of the repeated observations for the $k$-th subject in the $i$-th group is given by:

$$\mathbf{Y}_{ik} = \begin{pmatrix} Y_{i1k} \\ Y_{i2k} \\ \vdots \\ Y_{in_Bk} \end{pmatrix}$$

1. Compound Symmetry (CS) Structure

Under the compound symmetry assumption (common in random-intercepts models), the covariance matrix is:

$$\mathbf{\Sigma}_{\text{subject}} = \begin{pmatrix} \sigma^2_\delta + \sigma^2 & \sigma^2_\delta & \cdots & \sigma^2_\delta \\ \sigma^2_\delta & \sigma^2_\delta + \sigma^2 & \cdots & \sigma^2_\delta \\ \vdots & \vdots & \ddots & \vdots \\ \sigma^2_\delta & \sigma^2_\delta & \cdots & \sigma^2_\delta + \sigma^2 \end{pmatrix}$$

This matrix can be rewritten as:

$$\mathbf{\Sigma}_{\text{subject}} = (\sigma^2_\delta + \sigma^2) \begin{pmatrix} 1 & \rho & \cdots & \rho \\ \rho & 1 & \cdots & \rho \\ \vdots & \vdots & \ddots & \vdots \\ \rho & \rho & \cdots & 1 \end{pmatrix}$$

Where:

• $\sigma^2_\delta$ = Variance due to subject-specific random effects.

• $\sigma^2$ = Residual variance.

• $\rho = \frac{\sigma^2_\delta}{\sigma^2_\delta + \sigma^2}$ = Intra-class correlation coefficient (ICC).

Key Points:

• Compound Symmetry Structure is the product of a scalar and a correlation matrix.

• The correlation between any two repeated measures from the same subject is constant ($\rho$).

• This structure assumes equal correlation across time points, which may not hold if measurements are collected over time.

Refer to Random-Intercepts Model for a detailed discussion of compound symmetry.

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2. Autoregressive (AR(1)) Structure

If repeated measures are collected over time, it may be more appropriate to assume an autoregressive correlation structure, where correlations decay as the time gap increases.

The AR(1) variance-covariance matrix is:

$$\mathbf{\Sigma}_{\text{subject}} = \sigma^2 \begin{pmatrix} 1 & \rho & \rho^2 & \cdots & \rho^{n_B-1} \\ \rho & 1 & \rho & \cdots & \rho^{n_B-2} \\ \rho^2 & \rho & 1 & \cdots & \rho^{n_B-3} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \rho^{n_B-1} & \rho^{n_B-2} & \rho^{n_B-3} & \cdots & 1 \end{pmatrix}$$

Where:

• $\sigma^2$ = Residual variance.

• $\rho$ = Autoregressive parameter ($|\rho| < 1$), representing the correlation between consecutive time points.

Key Characteristics:

• Correlations decrease exponentially as the time lag increases.

• Appropriate for longitudinal data where temporal proximity influences correlation.

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In matrix notation, the mixed model can be written as:

$$\mathbf{Y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}$$

Where:

• $\mathbf{Y}$ = Vector of observed responses.

• $\mathbf{X}$ = Design matrix for fixed effects (e.g., group, time, interaction).

• $\boldsymbol{\beta}$ = Vector of fixed-effect coefficients.

• $\boldsymbol{\epsilon} \sim N(0, \sigma^2 \mathbf{\Sigma})$ = Random error vector.

• $\mathbf{\Sigma}$ = Variance-covariance matrix of residuals:

  • Block diagonal structure if the covariance structure is identical for each subject.
  • Within each block (subject), the structure can be compound symmetry, AR(1), or another suitable structure depending on the data.

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Choosing the Right Covariance Structure

• Compound Symmetry:
  • Suitable when correlations are constant across repeated measures (e.g., in randomized controlled trials).
  • Simple and interpretable but may be too restrictive for longitudinal data.
• Autoregressive (AR(1)):
  • Best when measurements are taken over equally spaced time intervals and correlations decay over time.
  • Assumes stronger correlation for adjacent time points.
• Unstructured (UN):
  • Allows different variances and covariances for each time point.
  • Provides maximum flexibility but requires more parameters and larger sample sizes.

Model selection criteria (AIC, BIC, likelihood ratio tests) can help determine the most appropriate covariance structure.

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