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I am using stratified sampling and neyman's optimal allocation to compute the best sample size for each stratum. neyman's optimal allocation is given by the formula,

$$n_h = n \frac{N_h * S_h}{\sum_i N_i * S_i}$$

where $$\sum_h n_h = n$$ $$\sum_i N_i = N$$

and $n$ is the total sample size, $n_h$ is the sample size for stratum h, $S_h$ is the standard deviation for stratum h, $N_h$ is the population size for stratum h, and $N$ is the total population size.

My question/concern is, "isn't there a catch 22 here? i need $S_h$ to compute $n_h$, but to get $S_h$, i would have already done a previous sampling, thereby determined a previous $n_h$, so that i could estimate $S_h$."

Anybody out there ever done this type of sample size estimation please shed light on what is done in the real world.

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Neyman Allocation (or modifications of NA) is often used in practice. Yes, you are right, we never know $S_h$ when doing the calculation of sample allocation. But we can estimate $S_h$ or use some approximation of $S_h$.

Assume $S_h(y)$ is computed for a variable $\textbf{y}$. $S_h$ can be estimated from the previous survey, if $\textbf{y}$ was observed in the previous survey.

There could be another variable $\textbf{z}$ which is correlated with $\textbf{y}$. $\textbf{z}$ could be available from other survey or some auxiliary data source (register, census). Then you can use $S_h(z)$ or $s_h(z)$ (an estimate) as approximate for $S_h(y)$.

It could be possible to guess $S_h(y)$, for example in case of binary $\textbf{y}$.

Keep in mind - you will never achieve optimal allocation in practice, so allocation close to the optimal could be good enough.

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