Stage vs phase, again

As I have said before, the distinction between multistage and multiphase surveys is mostly important as a shibboleth for survey statisticians. The two=phase quantities \[\pi_i^*=P(i\text{ sampled}|\text{phase 1})P(i\in\text{phase 1})\] are not the same as the marginal sampling probabilities \[\pi_i=P(i\text{ sampled})\] but one uses them in exactly the same formulas, so the difference doesn’t matter all that much. It would still be somewhat interesting to know whether using \(1/\pi_i^*\) as w...