# Sample size for two-staged cluster sampling

I have a question about the R-package samplesize4surveys.

I want to calculate sample size for tests of difference in proportions in a two-step clustered sampling strategy (students in schools) that meets certain power/confidence criteria.

I understand that the required sample size depends on the intraclass-correlation (ICC), the population totals (of schools and students) as well as the test that is to be done and desired power and confidence levels.

The function ss2s4p in the R-package samplesize4surveys computes such sample size estimates (see here). It takes, as arguments, the population totals, the approximate level of the proportion, the ICC (rho), the effect size in terms of minimal difference that is to be detected, the confidence level and the maximum number of final units to be selected per cluster.

Its output is descriped as follows:

This function returns a grid of possible sample sizes. The first column represent the design effect, the second column is the number of clusters to be selected, the third column is the number of units to be selected inside the clusters, and finally, the last column indicates the full sample size induced by this particular strategy.

It seems to me that this function in this package answers the question that I have.

However, the results seem very puzzling to me.

I would have expected to see that the number of clusters to be selected multiplied by the number of units to be selected inside the clusters is equal to the full sample size induced by this particular strategy.

However, this is not the case in most examples provided in the help page of the function itself.

For example,

ss2s4p(N=100000, P=0.5, delta=0.05, M=500, to=40, rho=0.1)

gives the following grid of sampling strategies

   Deff  nI  m  n2s
1   1.0 500  1 1514
2   1.1 500  2 1663
3   1.2 500  3 1811
4   1.3 490  4 1959
5   1.4 422  5 2106
6   1.5 376  6 2253
7   1.6 343  7 2400
8   1.7 319  8 2546
9   1.8 300  9 2692
10  1.9 284 10 2837
11  2.0 272 11 2982
12  2.1 261 12 3126
13  2.2 252 13 3270
14  2.3 244 14 3414
15  2.4 238 15 3557
16  2.5 232 16 3700
17  2.6 226 17 3842
18  2.7 222 18 3984
19  2.8 218 19 4125
20  2.9 214 20 4266
21  3.0 210 21 4407
22  3.1 207 22 4547
23  3.2 204 23 4687
24  3.3 202 24 4827
25  3.4 199 25 4965
26  3.5 197 26 5104
27  3.6 195 27 5242
28  3.7 193 28 5380
29  3.8 191 29 5517
30  3.9 189 30 5654
31  4.0 187 31 5791
32  4.1 186 32 5927
33  4.2 184 33 6063
34  4.3 183 34 6198
35  4.4 181 35 6333
36  4.5 180 36 6468
37  4.6 179 37 6602
38  4.7 178 38 6736
39  4.8 177 39 6869
40  4.9 176 40 7003


But if I sample 1 unit from all 500 clusters, then I expect my full sample size induced by this strategy to be 500 and not 1514. It however makes sense that the design effect is 1 in this case, as, obviously I do not need to worry about ICC if I only sample one individual per cluster.

Perhaps something different is meant by unit other than individuals within a cluster?

I appreciate any insights.

The package calculates required number of sample units by DEFF, and offers solutions ranging from m = 1 to to number of units within a cluster. In line 1 of the results displayed, 1514 units cannot be sampled with m = 1 units per cluster, since there are only 500 clusters altogether.
In short, you need to disregard the rows of the results where nI * m < n2s.