On the power of the test for cluster bias

Cluster bias refers to measurement bias with respect to the clustering variable in
multilevel data. The absence of cluster bias implies absence of bias with respect to any
cluster-level (level 2) variable. The variables that possibly cause the bias do not have to be
measured to test for cluster bias. Therefore, the test for cluster bias serves as a global test
of measurement bias with respect to any level 2 variable. However, the validity of the
global test depends on the Type I and Type II error rates of the test. We compare the
performance of the test for cluster bias with the restricted factor analysis (RFA) test,
which can be used if the variable that leads to measurement bias is measured. It appeared
that the RFA test has considerably more power than the test for cluster bias. However,
the false positive rates of the test for cluster bias were generally around the expected
values, while the RFA test showed unacceptably high false positive rates in some
conditions.Weconclude that if no significant cluster bias is found, still significant bias with
respect to a level 2 violator can be detected with an RFA model. Although the test for
cluster bias is less powerful, an advantage of the test is that the cause of the bias does not
need to be measured, or even known.