Maximum Likelihood Under Response Biased Sampling

Informative sampling occurs when the probability of inclusion in sample depends onthe value of the survey response variable. Response or size biased sampling is aparticular case of informative sampling where the inclusion probability is proportionalto the value of this variable. In this paper we describe a general model for responsebiased sampling, which we call array sampling, and develop maximum likelihood andestimating equation theory appropriate to this situation. The Missing InformationPrinciple (MIP) (Orchard and Woodbury, 1972) yields one (indirect) approach tolikelihood based survey inference (Breckling et al 1994). Some have questioned itsapplicability in the case of informative sampling, because of the way it conditions onthe given sample. In this paper we describe a direct approach and show that it and theMIP-based approach lead to identical results under array sampling. Comparison ismade to the weighted likelihood based approach described in Krieger andPfeffermann (1992). Extensions to the theory are also explored.