Repeated measures in time have historically been handled as either a multivariate analysis or as a univariate split-plot in time. The focus in this course is limited to only the latter.

A split-plot in time approach looks at each subject (experimental unit) as a main plot which is then split into sub-plots (time periods). Historically, the default assumption in split-plot in time data analyses has been that the correlations among responses at different time points are the same for all treatment levels (compound symmetry). However, depending on the study and nature of data, other correlation structures can be more appropriate (e.g. autoregressive lag 1).

Most of the current software facilitates the inclusion of different correlation structures which now allows for repeated measures to accommodate the presence of different correlated structures in residuals.

If we look at the ANOVA mixed model in general terms, we have:

response = fixed effects + random effects + errors

In the case of repeated measures with measures taken at \(p\) time points, the covariance structure of the errors can be expressed as a matrix. The diagonal elements of this matrix are the error variances at each time point. The off-diagonals are the covariances between successive time points. In general, the variance-covariance matrix can be expressed as follows:

\(\Sigma_i=\begin{bmatrix}
\sigma_{1}^2 & \sigma_{12} & \ldots & \sigma_{1p}\\
\sigma_{21} & \sigma^2_{2} & &\sigma_{2p}\\
\vdots & & \ddots & \vdots\\
\sigma_{p1} & \sigma_{p2} & \ldots & \sigma^2_{p}
\end{bmatrix}\)

The structure shown above does not assume any specific properties of the variances and covariances and is called an unstructured covariance structure. Note that there are \(p\) variances and \(p(p-1)/2\) covariances, which adds to \(p(p+1)/2\) unknown quantities to define this matrix. So, even for a small number of time points, a substantial number of parameters will have to be estimated. Therefore, in practice, specific structures are imposed to reduce the number of distinct parameters that need to be estimated, which will be discussed in Section 11.3.

Variance Components (VC)

\(\begin{bmatrix} \sigma_1^2 & 0 & 0 & 0 \\ 0 & \sigma_1^2 & 0 & 0 \\ 0 & 0 &\sigma_1^2 & 0 \\ 0 & 0 & 0 & \sigma_1^2 \end{bmatrix}\)

The variance component structure (VC) is the simplest, where the correlations of errors within a subject are presumed to be 0. This structure is the default setting in proc mixed, but is not a reasonable choice for most repeated measures designs. It is included in the exploration process to get a sense of the effect of fitting other structures.

Compound Symmetry

\(\sigma^2 \begin{bmatrix} 1.0 & \rho & \rho & \rho \\ \rho & 1.0 & \rho & \rho \\ \rho & \rho & 1.0 & \rho \\ \rho & \rho & \rho & 1.0 \end{bmatrix} = \begin{bmatrix} \sigma_b^2+\sigma_e^2 & \sigma_b^2 & \sigma_b^2 & \sigma_b^2 \\ \sigma_b^2& \sigma_b^2+\sigma_e^2 & \sigma_b^2 & \sigma_b^2 \\ \sigma_b^2 & \sigma_b^2 & \sigma_b^2+\sigma_e^2 & \sigma_b^2 \\ \sigma_b^2 & \sigma_b^2 & \sigma_b^2 & \sigma_b^2+\sigma_e^2 \end{bmatrix}\)

The simplest covariance structure that includes within-subject correlated errors is compound symmetry (CS). Here we see correlated errors between time points within subjects, and note that these correlations are presumed to be the same for each set of times, regardless of how distant in time the repeated measures are made.

First Order Autoregressive AR(1)

\(\sigma^2 \begin{bmatrix} 1.0 & \rho & \rho^2 & \rho^3 \\ \rho & 1.0 & \rho & \rho^2 \\ \rho^2 & \rho & 1.0 & \rho \\ \rho^3 & \rho^2 & \rho & 1.0 \end{bmatrix}\)

The autoregressive (Lag 1) structure considers correlations to be highest between adjacent times, and a systematically decreasing correlation with increasing distance between time points. For one subject, the error correlation between time 1 and time 2 would be \(\rho^{t_2-t_1}\). Between time 1 and time 3 the correlation would be less, and equal to \(\rho^{t_3-t_1}\). Between time 1 and 4, the correlation is lesser, as \(\rho^{t_4-t_1}\), and so on. Note that this structure is only applicable for evenly spaced time intervals for the repeated measure; so that consecutive correlations are \(\rho\) raised to powers of 1, 2, 3, etc.  

Spatial Power

\(\sigma^2 \begin{bmatrix} 1.0 & \rho^{\frac{|t_1-t_2|}{|t_1-t_2|}} & \rho^{\frac{|t_1-t_3|}{|t_1-t_2|}} & \rho^{\frac{|t_1-t_4|}{|t_1-t_2|}} \\ \rho^{\frac{|t_1-t_2|}{|t_1-t_2|}} & 1.0 & \rho^{\frac{|t_2-t_3|}{|t_1-t_2|}} & \rho^{\frac{|t_2-t_4|}{|t_1-t_2|}} \\ \rho^{\frac{|t_1-t_3|}{|t_1-t_2|}} & \rho^{\frac{|t_2-t_3|}{|t_1-t_2|}} & 1.0 & \rho^{\frac{|t_3-t_4|}{|t_1-t_2|}} \\ \rho^{\frac{|t_1-t_4|}{|t_1-t_2|}} & \rho^{\frac{|t_2-t_4|}{|t_1-t_2|}} & \rho^{\frac{|t_3-t_4|}{|t_1-t_2|}} & 1.0 \end{bmatrix}\)

When time intervals are not evenly spaced, a covariance structure equivalent to the AR(1) is the spatial power (SP(POW)). The concept is the same as the AR(1) but instead of raising the correlation to powers of 1, 2, 3, etc, the correlation coefficient is raised to a power that is the actual difference in times (e.g. \(t_2-t_1\) for the correlation between time 1 and time 2). This method requires quantitative values for the time variable in the data so that the exponents in the SP(POW) structure can be calculated. If an analysis is run wherein the repeated measures are equally spaced in time, the AR(1) and SP(POW) structures yield identical results.

Unstructured Covariance

\( \begin{bmatrix} \sigma_1^2 & \sigma_{12} & \sigma_{13} & \sigma_{14} \\ \sigma_{12} & \sigma_2^2 & \sigma_{23} & \sigma_{24} \\ \sigma_{13} & \sigma_{23} & \sigma_3^2 & \sigma_{34}\\ \sigma_{14} & \sigma_{24} & \sigma_{34} & \sigma_4^2 \end{bmatrix}\)

The unstructured covariance structure (UN) is the most complex because it is estimating unique correlations for each pair of time points. As there are many parameters (all distinct correlations), the estimates most times will not be computable. SAS for instance may return an error message indicating that there are too many parameters to estimate with the data.

Choosing the Best Covariance Structure

The fit statistics used for model selection can also be utilized in choosing the best covariance matrix. The model selections most commonly supported by software are -2 Res Log Likelihood, Akaike’s information criterion - corrected (AICC), and Bayesian Information Criteria (BIC). These statistics are functions of the log likelihood and can be compared across different models, as well as different covariance structures provided the effects are the same in each model. The smaller the criterion statistics value is, the better the model is, and if the criterion values are close, the simpler model is preferred.

BIC tends to choose simpler models compared to AICC. Choosing a model that is too simple however inflates the Type I error rate. Therefore, if controlling Type I error is of importance, AICC may be the better criterion. On the other hand, if loss of power is of more concern, BIC might be preferable (Guerin and Stroup, 2000).

In addition to using the above fit statistics, graphical approaches are also available. See Graphical Approach for more details. In some situations, combining information from both approaches to make the final choice may also prove to be beneficial.

This lesson introduced us to the topic of repeated measures designs. The focus was on repeated measures in time where each experimental unit is assigned to exactly one treatment level the response is observed over several time periods. The responses from the same experimental unit observed over time can be correlated and the model assumption of independent observations is no longer valid. Therefore, an appropriate covariance structure should be imposed to account for the correlated nature of the response, and the best is chosen based on fit statistics. 

Other scenarios can result in repeated measures, not necessarily over time. In a repeated measure design, the important feature is that multiple measurements are being made on the same experimental unit. Another case of this is the cross-over design wherein the treatments themselves are switched on the same experimental unit during the course of the experiment. This will be the topic of the next lesson.