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I know that stratified random sampling is almost always preferred over simple random sampling, but I have also read that the Variance of sample mean (x-bar) from stratified random sampling could sometimes be larger than the Variance of x-bar under simple random sample. In those cases, SRS might a better method compared to stratified random sampling.

Logically, it seems to make sense, but I'm wondering if there's a more mathematical way to explain this, and maybe a (simple) real life example?

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    $\begingroup$ stratified random sampling is almost always preferred over simple random sampling who said that? Do you have any sources for these and other claims? $\endgroup$ Jul 17, 2019 at 11:06

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There are two factors at play here.

The first factor is that when you compute residuals to work out the variance, you subtract the stratum mean in stratified sampling and the sample mean in simple random sampling.

If the correlation within strata is positive (which is essentially always the case), this gives a reduction in variance. If the correlation within strata is exactly zero the stratification doesn't matter (stratifying on an irrelevant variable). If the correlation within strata is negative, stratification gives an increase in variance.

I can't think of any natural example where you'd get a negative correlation, but it is mathematically possible.

The second factor is the proportions allocated to each stratum. If you sample from each stratum in proportion to its size in the population, this doesn't apply and only the first factor matters. But stratified sampling often doesn't do this. For example, in a country with states or provinces, you might stratify on state/province and choose the same sample size from each one, in order to get good estimates within each one. As a consequence, you have unequal sampling probabilities for people in difference states/provinces. Estimation of the whole-population mean will be less efficient: you have traded some precision in whole-population estimation for precision in estimating means of smaller states/provinces.

In the US, for example, using equal sampling fractions across states is bad for estimating means specific to Iowa. We may oversample Iowa (relative to more populous states) so that means for Iowa are good enough for government use. As a result, estimates for the country as a whole are less accurate than if we had done simple random sampling.

So:

  • stratification good
  • oversampling of small strata bad (at a population level, but good at a stratum level)
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Stratified random sampling (STRS) is often preferred to simple random sampling (SRS) because it usually lead to better precision, if strata are chosed wisely. However, sometimes STRS may return higher variance with comparison to SRS. There is a mathematical explanation and it depends on the variance between and within the strata. You may refer to Sampling Techniques (Cochran, 1977) for more details.

Anyway, the STRS is not only chosen to reduce the variance, but also because it lets you represents all the strata in the population. For example, you may want to stratify over the gender, to be sure to include both males and females in your sample, or you could stratify on age classes, etc...

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