# [no subject]

```Suppose all N clusters in the population contain M individuals.  Our design is
a random sample of n clusters, with a subsample of m individuals from each
selected cluster.  Let ss1 be the sample variance of the n cluster means, and
ss2 the average of the sample variances from the n clusters.  Further let SS1
and SS2 be the respective population values.  The FPC's for the two stages are
f1 = n/N and f2 = m/M, respectively.

We are looking at three different variance estimators here:

Stata without FPC (recommended):

v0 = ss1/n

Stata with FPC (not recommended):

v1 = (1-f1)*ss1/n

Unbiased 2 Stage clustered estimator (not yet implemented in Stata):

v2 = (1-f1)*ss1/n + f1*(1-f2)*ss2/(m*n)

The expected values of the estimators are:

E(v0) = SS1/n + (1-f2)*SS2/(m*n)

E(v1) = (1-f1)*SS1/n + (1-f1)*(1-f2)*SS2/(m*n)

E(v2) = (1-f1)*SS1/n + (1-f2)*SS2/(m*n)

The bias of v0 is:

E(v0) - E(v2) = f1*SS1/n > 0

whereas the bias of v1 is:

E(v0) - E(v2) = -f1*(1-f2)*SS2/(m*n) < 0

Reference:

Cochran, W. G. 1977.  Sampling Techniques. Wiley: New York.

--Jeff