Defining PSU in "Sampled with Replacement" Cluster Samples I am trying to decide on an approach to estimate design effect for a multi stage cluster survey.  The clusters were selected with probability proportional to size sampling WITH replacement.  The primary sampling units (districts) are large enough that some were sampled multiple times.  When defining the PSU variable should the unique districts within the sample be used, or a cluster variable representing the true number of clusters sampled?
For example:
A district gets sampled twice, should the PSU variable described, be one or two individual clusters?
Thanks!
 A: Update:
I'm sorry to say that my 2014 answer was wrong. In Stata or elsewhere, specify the original primary sampling unit as the PSU.
2018-06-04 I Now believe that my original answer is correct.
The relevant sample size for estimating the design effect (and other precision-related measures) is the number of draws, not the number of unique PSUs selected. If a PSU was drawn twice, it counts as two draws. Note that it is the total number of draws that is fixed by a with-replacement design. 
Terminology can get ambiguous when sampling is with replacement. I think it safest to reserve the terms "primary sampling unit", "PSU", "cluster", "first-stage unit" to denote the fixed geographic unit that is selected at the first stage.  What distinguishes the data from different draws of the same PSU? Each contains a different set of second stage sampling units (SSUs). Moreover, each second stage sample must be drawn independently of the other second stage samples, even those from the same PSU.
So when a sampling analysis program like Stata asks you to specify a "psu", substitute the name of the variable that designates the draw.
