On the two-phase framework for joint model and design-based inference

We establish a mathematical framework that formally validates the two-phase “superpopulation viewpoint” proposed by Hartley and Sielken (1975), by defining a product probability space which includes both the design space and the model space. We develop a general methodology that combines finite population sampling theory and classical theory of infinite population sampling to account for the underlying processes that produce the data. Key results in this article are: the sample estimator and the model statistic are asymptotically independent; if a sequence converges in design law, it also converges in the law of the product space; and the distribution theory of the sample estimating equation estimator around a super-population parameter. We also study the interplay between dependence and independence of random variables when viewed in the design space, the product space and the model space and apply it to show formally that under a “simple random sample without replacement” design, we can “ignore” the design and work on the realm of the model space, but that under “simple random sample with replacement” we cannot ignore the design.