Post-Stratification Variance Help

I have some data that I post-stratified. Looking at equations in
Cochran (1977) Sampling Techniques (p.135) and Levy and Lemeshow (1991)
Sampling of Populations (p.137), they give the following for the
formula of variance for the mean in a post-stratified sample :

Var(x-bar post-strat) = ((N-n)/N) * (1/n) * sum[ (Nh/N) * VARhx] +
((N-n)/N) * (1/n^2) * sum[VARhx*((N-Nh)/N)]

{Notice, almost everything is given in terms of overall sample (n) and
pop (N) size}

The formula for variance for the mean in a pre-stratified sample is:

Var(x-bar pre-strat) = ((Nh-nh)/Nh) * (1/N^2) * sum[ (Nh^2) * VARhx/nh]


{Notice, almost everything is given in terms of stratum sample (nh) and
pop (Nh) size}

I have an example below, where I compare the variance computed if the
sample was pre or post-stratified. When the sample sizes are equal,
the pre-stratified variances (within strata) are always smaller than
the post-stratified. Further, the overall variance of the mean is
smaller as well.

However, when the sample sizes are different, the pre-stratified
variances (within strata) are sometimes larger than the post-stratified
variances (within), and further, the total pre-stratified variances can
be larger than the total post-stratified variance.

Strata Var n N Pre-Strat Post-Strat
1 4 2 20 0.2 0.27
2 4 2 20 0.2 0.27
3 4 2 20 0.2 0.27
SUM 6 60 0.6 0.8


Strata Var n N Pre-Strat Post-Strat
1 4 10 20 0.02 0.083
2 4 2 20 0.2 0.083
3 4 2 20 0.2 0.083
SUM 14 60 0.42 0.25

This confuses me. I thought we should be penalized for
post-stratification.

The issue is, in pre-stratified, the increase in sample size in a
single stratum improves only that stratum's variance estimate, whereas
in post-stratified, it improves all estimates.

Is this right?

Thanks,

Warren

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