Calibration Weighting for Nonresponse that is Not Missing at Random Allowing More Calibration than Response-Model Variables
Calibration weighting can be used to remove bias when unit nonresponse is a function of one or more survey variables. This is done by allowing the model variables in the weight-adjustment function to differ from the variables in the calibration equation. An extension of calibration weighting allows there to be more calibration variables than model variables. Rather than equating the two sides of a calibration equation, the difference between the sides is minimized in some sense. This paper discusses different ways of doing that, assuming one knows the correct form and arguments of the response function. A promising solution results from an alternative version of the calibration equation that essentially reduces the number of calibration equations to be equal to the number of model variables. A helpful insight into choosing calibration variables for given model variables follows.
Kott, P. S., & Liao, D. (2017). Calibration Weighting for Nonresponse that is Not Missing at Random: Allowing More Calibration than Response-Model Variables. Journal of Survey Statistics and Methodology, 5(2), 159-174. https://doi.org/10.1093/jssam/smx003