Dear prayad consult the book " sampling theory of survey with applications" by sukhatme et al (1984) page 322-324 for the best answer. There u will find a topic named allocation of sample sizes to the three stages where formula for optimum allocation for all the sample sizes you have asked for is given.

You may be able to do it by using Optimal Design software developed by Steve Raudenbush and collegues. I am not sure what you mean by unequal sizes. If you just mean that the sizes at each level are not the same, that is pretty straightforward in clustered designs. If you mean that within a level the groups will be of unequal sizes, you can use a harmonic mean to represent the size. The OD software assumes you are implementing a cluster randomized trial, but you can treat any effect size as if it is a treatment. Statistical power is most affected by the number of clusters at the top level. Go to http://sitemaker.umich.edu/group-based/optimal_design_software for more information and to download the free OD software. Bob

Robert

I am delighted with the Optimal Design software that you share with us, but I think that Prayad is asking for the sample size in a Sampling Design for "Estimation of Population Parameters" while the software is for "Experimental Design for Test of Hypotesis in Cluster Randomized Trials". I do not believe that this may be applied to that problem.

Dear Prayad

If you know the Kish design effect (K) for a similar sample, you could start by estimating which is your required sample size if you use a simple random sample for sampling the elements from the last stage (n).

Then multiplying that K by n (n = K * n), you will have how many items should have your cluster sample in total. Finally you should consider how you can get that number of elements in three stages like that of the similar sample considered.

If there is no information about K (the design effect), this K may be obtained with a pilot sample. When you get the results of the pilot you could estimate the Kish design effect and then adjust the samples as described above.

To estimate the Kish effect from the pilot sample you should estimate the intraclass correlation in each stage.

Thank you very much for your answers. I really appreciate your kindness.

Dear prayad consult the book " sampling theory of survey with applications" by sukhatme et al (1984) page 322-324 for the best answer. There u will find a topic named allocation of sample sizes to the three stages where formula for optimum allocation for all the sample sizes you have asked for is given.

Cluster sampling, as I understand it, is used for populations whose sizes and whose identities are not known. The process is to divide your area into a population of clusters, determine the cluster sample size, and randomly select your clusters. All the sub-population members of the selected clusters become your random samples.

Although multi-stage sampling is a design that is gaining ground, I have reservations as to its use, especially when size computation and random selection is applied in each stage because that will confuse the issue on confidence level, margin of error and variance (for formulae that estimate population mean), and confidence level, margin of error and proportion (for formulae that estimate population proportion).

Thank you for all answers.

Read a book titled " sampling theory of survey with applications" for sampling distribution.