In stratified random sampling, we divide the population into mutually exclusive strata. We sample from each stratum separately, using simple random sampling.
Stratified random sampling is useful for two reasons. First, it allows us to ensure that at least in terms of the sample strata, our sample is representative. This means sub-populations are represented in the sample in exactly the same proportion as they appear in the population.
Stratifying only increases efficiency if the strata differ strongly from each other relative to the differences within each strata.
Imagine we want to sample the quality of cat food produced on an assembly line. The line produces cat food made with fish and cat food made with beef. Suppose the average quality of beef cat food is higher than that of fish cat food. Also, the quality varies relatively little when we consider each type of food separately. Under these circumstances, we will obtain a more accurate estimate of the populations food quality if we stratify on food type. This is because quality is related to food type. Even a small overrepresentation of one food type can distort our overall estimate of food quality.
Suppose our stratum of fish cat food is relatively small or is known to strongly vary in quality. In both cases, our estimate of the quality of fish cat food might be much less likely to be accurate than that of beef cat food. It might be worth it to take a bigger sample of fish cat food so we have a better chance of getting an accurate estimate. Of course, this means overrepresenting fish cat food.
We can correct for this overrepresentation by weighing the separate estimates of fish and beef cat food according to their stratum sizes before averaging them into an overall estimate of food quality. This way, the sample value is representative, efficient and more likely to be accurate.
Multi-cluster stage sampling
A solution is to randomly sample in stages by first selecting clusters of elements. Say, we want to sample math performance in the population of all Dutch students currently in their third year of secondary education.
1st: We start by forming a sampling frame of all school districts. This is the first stage, where students are clustered in districts. We randomly select a very small sample of school districts. We can use stratification to make sure we include districts in urban and rural areas.
2nd In the second stage, we randomly select schools from the previously selected districts. Students are now clustered in schools.
3rd In the third stage, third year math classes are randomly sampled from the previously selected schools.
4th We can even include a fourth stage where students are randomly sampled from the previously selected classes.
