Robustness of parameter and standard error estimates against ignoring a contextual effect of a subject level covariate in cluster randomized trials

In experimental research, it is not uncommon to
assign clusters to conditions.When analysing the data of such
cluster-randomized trials, a multilevel analysis should be
applied in order to take into account the dependency of firstlevel
units (i.e., subjects) within a second-level unit (i.e., a
cluster). Moreover, the multilevel analysis can handle covariates
on both levels. If a first-level covariate is involved,
usually the within-cluster effect of this covariate will be
estimated, implicitly assuming the contextual effect to be
equal. However, this assumption may be violated. The focus
of the present simulation study is the effects of ignoring the
inequality of the within-cluster and contextual covariate
effects on parameter and standard error estimates of the
treatment effect, which is the parameter of main interest in
experimental research. We found that ignoring the inequality
of the within-cluster and contextual effects does not affect the
estimation of the treatment effect or its standard errors.
However, estimates of the variance components, as well as
standard errors of the constant, were found to be biased.