  In repeated measures factorial designs involving clustered units, parametric
methods such as linear mixed effects models are used to handle within subject
correlations. However, assumptions of these parametric models such as
continuity and normality are usually hard to come by in many cases. The
homoscedasticity assumption is rather hard to verify in practice. Furthermore,
these assumptions may not even be realistic when data are measured in a
non-metric scale as commonly happens, for example, in Quality of Life outcomes.
In this article, nonparametric effect-size measures for clustered data in
factorial designs with pre-post measurements will be introduced. The
effect-size measures provide intuitively-interpretable and informative
probabilistic comparisons of treatment and time effects. The dependence among
observations within a cluster can be arbitrary across treatment groups. The
effect-size estimators along with their asymptotic properties for computing
confidence intervals and performing hypothesis tests will be discussed.
ANOVA-type statistics with $\chi^2$ approximation that retain some of the
optimal asymptotic behaviors in small samples are investigated. Within each
treatment group, we allow some clusters to involve observations measured on
both pre and post intervention periods (referred to as complete clusters),
while others to contain observations from either pre or post intervention
period only (referred to as incomplete clusters). Our methods are shown to be,
particularly effective in the presence of multiple forms of clustering. The
developed nonparametric methods are illustrated with data from a three-arm
Randomized Trial of Indoor Wood Smoke reduction. The study considered two
active treatments to improve asthma symptoms of kids living in homes that use
wood stove for heating.
