I know that we can allocate observations from the total sample to each strata either using proportional allocation ($n_{h}/N_{h} = n/N$) or optimal allocation:

$$n_{h} = \left [ \frac{\frac{N_{h}S_{h}}{sqrt (c_{h})}}{\sum_{l=1}^{H}\frac{N_{l}S_{l}}{sqrt (c_{l})}{}} \right ]n \tag{1}$$

Where $N_{h}$ and $n_{h}$ are the population size and sample size in each stratum respectfully. $c_{h}$ is the cost of sampling one unit from stratum h, $S_{h}$ is the population standard deviation of stratum h and $n$ is the total sample size. What I am confused about is how do we actually choose $n$. The book I'm using (Lohr's SDA) says that

$$n = \frac{n_{0}}{1 + \frac{n_{0}}{N}\sum_{h = 1}^{H}\frac{N_{h}S_{h}^{2}}{Nv}} \tag{2}$$


$$n_{0} = \frac{z_{\alpha/2}^{2}v}{e^{2}} \tag{3}$$

($e$ is the margin of error) is the sample size if all strata fpcs could be ignored and

$$v = \sum_{h =1}^{H}\frac{n}{n_{h}} \left ( \frac{N_{h}S_{h}}{N} \right )^{2} \tag{4}$$

What confuses me is that $n$ is calculated using $v$ and $n_{0}$ which themselves are calculated using $n$. Could someone provide some clarification?


1 Answer 1


Equation 1 is a method of allocating samples to strata given a predefined total sample size, resulting in a relative sample size $n_h/n$. This can then be adjusted to determine the total sample size necessary to achieve a prespecified variance (Section 4.4.4).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.