Superpopulation Model

A superpopulation model treats the finite population's $N$ values $y_1,\dots ,y_N$ as themselves a random realization from an underlying stochastic model, $y_i=f({\mathbf{x}}_i;\mathbf{\beta })+{\epsilon }_i$, governed by some superpopulation parameters $\mathbf{\beta }$ and error structure, rather than as fixed constants as under purely design-based inference. Under a superpopulation model, statistical properties of an estimator (bias, variance) are evaluated with respect to the model's own randomness, potentially averaged over the sampling design as well (model-assisted inference) or considered on its own (purely model-based inference); the superpopulation framework provides the formal theoretical justification for model-based small area estimation, model-assisted estimators like the generalized regression (GREG) estimator, and imputation methods for missing survey data, all of which lean on an explicit statistical model of the outcome variable rather than relying solely on the physical randomization built into the sampling design.

• The superpopulation framework is what makes model-based small area estimation methods, like the Fay-Herriot model, formally coherent, without treating the small area's own value as a random draw from some underlying model, there would be no statistical basis for 'borrowing strength' from related areas the way small area estimation does.
• Model-assisted estimation, a middle ground between pure design-based and pure model-based inference, uses a superpopulation model to guide the construction of a more efficient estimator (such as the GREG estimator) but still evaluates that estimator's core properties with respect to the sampling design, aiming to retain design-based robustness while gaining some of the efficiency of modeling.
• A key practical risk of superpopulation-model-based inference is that its validity depends on the assumed model actually being a reasonable description of how the population values arose, unlike design-based inference, whose validity rests only on the physical randomization of the sampling design and requires no assumption about the outcome variable's distribution at all.