Hi Conrad, this is an interesting question. I hope, the following can solve your problem (look at the second page): http://courses.wcupa.edu/rbove/Berenson/10th%20ed%20CD-ROM%20topics/section7_3.pdf Kind regards, Christian

Thank you for the link, this is helpful. I am just not sure if the application of this correction factor is appropriate in our situation. Thanks again for the link to the handout - that is very helpful!

Conrad -

The idea of the finite population correction factor is that if you are only concerned with the loss of accuracy due to sampling and thus having to estimate for the remainder of the population, that is, you are concerned with the error due to sampling, called the sampling error, then you estimate variance from the data you do collect in your sample, and assume that the data not collected would have the same variance. Then you estimate the impact on estimation for your finite population. You have no sampling error for the sample you collected, so the finite population correction factor causes you to only apply sampling error to the part you did not collect.

For a given stratum at a given stage, there is an inherent variability in responses you would get. From your sample, you estimate that. But you then only apply it to the part you did not collect, that you are estimating. Consider that if you collected a near census, say, randomly, 99 out of every 100 members of the finite population for that stratum, then there is little sampling error from what is left. On the other hand, if your sampling fraction is small, you may be able to ignore the finite population correction (fpc) factor.

This, as you noted, is associated with random sampling with equal probabilities of selection. It gets far more complicated when you look at unequal probabilities of selection, based on a 'size' measure, probability proportionate to size (PPS).

Consider this book, if you can borrow a copy:

Cochran, W.G.(1977), Sampling Techniques, 3rd ed., John Wiley & Sons.

On pages 259-261, section 9A.7, "The Horvitz-Thompson Estimator," you see estimation in one stage for a PPS sample. On page 261, estimator (9A.44) is the Yates-Grundy variance estimator for the Horvitz-Thompson PPS estimate. Note that instead of seeing n and N, as in simple random sampling, you see pi with subscripts. This is because of the uneven sample selection probabilities. In simple random sampling, the probability of selection within a stratum is n/N, with appropriate subscript, say h, to identify stratum, for every member of the stratum. But instead of n/N, we have a pi with yet an additional subscript (or two) to show individual probabilities of selection, based on individual sizes. (To go from PPS to simple random sampling, set pi=n/N.)

There are various estimates of variance for PPS sampling, some making more in the way of assumptions than others. PPS sampling and estimation can get very tricky. There is a famous, and cute little story, "Basu's Elephant Fable," that illustrates how easy it is to go astray.

Before looking at your application further, lets remember that we have been talking only about sampling error. However, nonsampling error (all kinds: individual measurements, frame errors, transcription, data processing, etc) can be a far bigger problem. That is why it is sometimes possible to obtain less overall error (less 'total survey error') from a sample, than from a census. Sometimes. Given resource restrictions, this can happen. So, always consider the quality of your data.

Back to your sample selection, I wondered why you used 'with replacement.' It seems less efficient. Also, I was not sure, from your description, where the size measure came in. I think that you picked villages with a PPS sample selection, and then used random sampling within a selected village? I was not sure where that number "30" came in. Regardless, you have a fairly complex design for variance estimation. Cochran (book above) mentions a three stage problem, but all stages had random sampling. My expertise is in single-stage model-based estimation, not so much in design-based sampling, but I recall back in the 1990s, that Statistics Canada developed programming to do design-based variance, I think, with any type of design-based sampling at each stage. This would also be the kind of thing that the US Census Bureau, and other such national agencies around the world would use, so perhaps one of them could point you to a good reference, if you don't find one. I think that free, or low cost software might be available for your purposes. Maybe. Also, the American Statistical Association's Survey Research Methods Section (SRMS) has a listserv where people discuss such problems.

Regarding a finite population correction factor for the stage at which you draw a random sample, you might look at the attached to see if it is of any help in figuring out what you need.

Whatever you do, you will need to test your results any way that you can, to make sure your results are reasonable. For example, if your final variance estimate is lower than it would have been from the PPS stage alone, you likely have an error.

Best wishes with your project.

Cheers - Jim

Article FINITE POPULATION CORRECTION (fpc) FACTOR

Thank you so much!  This is very helpful.