Optimal Sample Allocation in the Presence of Non-response
Assume stratified simple random sampling design. Population is split in strata. We have fixed sample size . The aim is to find optimal sample allocation , where and is minimised.
There is some population information available. The population size by strata: , population standard deviation of variable: , and expected response rate in each stratum: .
Optimal sample allocation in the presence of non-response can be computed as .
I do not have a proof but I have done some numerical calculations to test the optimality of the allocation. The R code of the calculations is available here.
The presented allocation is equal to the well known Neyman allocation  in case we have the same response expectation in each stratum.
 Neyman, Jerzy (1937). Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 236 (767): 333–380