Analysis of clustered survey data based on two‐stage informative sampling and associated two‐level models

This paper deals with making inference on parameters of a two‐level model matching the design hierarchy of a two‐stage sample. In a pioneering paper, Scott and Smith (Journal of the American Statistical Association, 1969, 64, 830–840) proposed a Bayesian model based or prediction approach to estimating a finite population mean under two‐stage cluster sampling. We provide a brief account of their pioneering work. We review two methods for the analysis of two‐level models based on matching two‐stage samples. Those methods are based on pseudo maximum likelihood and pseudo composite likelihood taking account of design weights. We then propose a new method for analysis of two‐level models based on a normal approximation to the estimated cluster effects and taking account of design weights. This method does not require cluster sizes to be constants or unrelated to cluster effects. We evaluate the relative performance of the three methods in a simulation study. Finally, we apply the methods to real data obtained from 2011 Nepal Demographic and Health Survey (NDHS).