In many fields of science, repeated measurements of a response variable are taken on each subject over time to assess the changes in response. The cumbersome aspect in analyzing such data is that there are relationships between the measurements in the subject over time. There are two major policies in terms of overcoming or taking the relationships into account.
First, one can reduce the vector of responses of each subject to a single value by a descriptive statistic and apply standard univariate approaches to test the effects related to the corresponding summary measure. The use of the summary measure approach (SMA) was suggested by Wishart [1
] for the first time. Several strategies based on the least squares regression slope and mean of response over time were recommended to evaluate the differences between the groups [2
]. Moreover, the utility of Kendall's τb
as a summary measure of within-subjects trend in psychiatric longitudinal studies, where the key assumptions of parametric methods are not held, was investigated [7
Second, one can use methods which take the covariances between the measurements into account. Two common and traditional approaches for normally distributed responses are repeated measures ANOVA and MANOVA. In order to avoid inflating type I error rate, the denominator degrees of freedom of the F
statistics in the repeated measures ANOVA approach should be adjusted under departures from a restrictive assumption on covariance structures, namely sphericity. But there is no obvious advantage in using the adjusted F
tests against the multivariate tests, and generally the adjustments should be avoided [9
]. In contrast, the repeated measures MANOVA approach makes no assumption regarding covariance structure and hence, it is sometimes known as unstructured multivariate approach (UMA). The only key advantage of the repeated measures ANOVA approach over the UMA is that it can still be implemented in the case where the number of measurements is greater than the sample size.
The linear mixed model (LMM) is more advanced and flexible since it allows dealing with subjects which have incomplete measurements and are unequally spaced in the time period. But the performance of the LMM in testing the effects is highly dependent on the choice of appropriate covariance structure for errors [11
]. On the other hand, the choice of a parsimonious covariance structure in a small sample design can lead to more efficient inferences concerning the fixed-effects parameters. This aspect makes it inconvenient and unreliable, especially for those who are not familiar with the fundamental principles of mixed models.
Although SMA is a simple, robust and sometimes only applicable tool for the analysis of repeated measures studies, there exists no obvious performance comparison on using the SMA vs. other competitors. Moreover, the application of the SMA has been mostly based on using one summary statistic to assess only the total group difference.
The present study includes repeated measures data in which the pattern of the response profile can be described by a linear trend and the responses measured in a continuous scale. The main objectives of this study are:
a) To describe techniques to test time (within-subjects), group (between-subjects) and group × time interaction effects on the basis of two common summary measures, i.e. least square regression slope and mean of response over time.
b) To compare the performance of the SMA, LMM and UMA in the analysis of simulated data from a LMM framework under different types of covariance structures. The approach is also illustrated and compared with the competitors using two real data sets.
In our simulations, there is a focus on situations where the LMM may provide extremely unsatisfactory performance such as misspecification of the covariance structure for errors, small and moderate sample sizes, and relatively a large number of measurements.