Single and joint inclusion probabilities are generally available and known for complex survey designs up to the point where survey weights are modified due to nonresponse and population controls. Best practice by sophisticated survey practitioners generally includes weight modifications, first by calibration, ratio adjustment or raking to correct for nonresponse, next by further steps to impose population survey controls; and also, often, by final steps involving weight truncation or cell-collapsing to constrain the modified weights, usually so that the largest and smallest weights do not differ by more than a designated multiplicative factor. These adjustments are sometimes made in successive stages, the order of which may differ from one survey to another. In this article, generalized-raking calibration methodology is adapted to allow all of these adjustments, and possibly additional nonlinear constraints, to be accomplished in a single stage, after which linearization-based large-sample variance formulas are available.

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