Incorporate a quasi-experimental device into sensitivity analysis #22
Description
Certain patterns of hidden biases may be valid based on the chosen Gamma, but implausible when considering covariates observed for everyone in the population. For example, suppose w is a set of weights in a Gamma region, and let z be a covariate with known population mean and variance.
Then the weights imply the population mean of z is (1/M) w^t z. If this is far from the known population mean, we can reject these weights as being inconsistent with what we know about the population, even if they are in the Gamma region we are considering.
A quasi-experimental device incorporates this control outcome into the sensitivity analysis optimization problem:
minimize w^T y / w^T 1
subject to wl \preceq w \preceq wu
| (1/M) w^T z - \mu | <= \zeta
for a suitably chosen threshold \zeta.
A few notes:
- Ideally we would use the SIPW estimator in the control outcome, but we need to keep things linear for a convex optimization problem.
- At first blush, we might set \zeta to be \sqrt{\sigma^2/M} times 1.96, but this would be the variance of a simple random sample. We can multiply this by the design effect, but the design effect varies by the weights. We could use the largest design effect over the Gamma region, but that might neuter the control outcome.