  Multi-level modeling is an important approach for analyzing complex survey
data using multi-stage sampling. However, estimation of multi-level models can
be challenging when we combine several datasets with distinct hierarchies with
sampling weights. This paper presents a method for combining multiple datasets
with different hierarchical structures due to distinct informative sampling
designs for the same survey. To develop an approach with complete generality,
we propose to define a pseudo-cluster, a cluster containing only a singleton
observation, to unify the data structure and thereby enable estimation of
multi-level models incorporating sampling weights across the combined sample.
We justify incorporating sampling weights at each level of the hierarchical
model and in doing-so define a pseudo-likelihood estimation procedure.
Simulation studies are used to illustrate the effect of incorporating sampling
designs in this challenging multi-level modeling scenario. We demonstrate in
the simulation studies that considering a linear mixed model with sampling
weights provides unbiased estimates of model parameters and enhances the
estimation of the variance components of the random effects. The proposed
method is illustrated through a novel application from the National Survey of
Healthcare Organizations and Systems that sought to determine which
organizational characteristics or traits, as measured in the surveys, have the
strongest average relationship to the percentage of depression and anxiety
diagnoses in physician practices in the US.
