Stratified sampling without sampling At school we talked about stratified sampling regarding scientific studies.
I thought of a similar case:
Let's assume you have 4 questions for a certain number of people. Each of those people should only given one randomly chosen question. That leads to 4 groups of people, specified by the question they were asked.
Let's assume next that you want all of those groups to have the same age on average (you know the age of each person). I would sort them by age, split in 4 groups and then assign them randomly to another 4 groups one by one. So each of those 4 target groups have the same number of people from the source groups.
So you can divide a certain amount of entities with a known property to a certain amount of groups, all having nearly the same average property value.
My question:
Is this some kind of stratified sampling (granded, you're not doing the 'sample'-part), or is it a fully different procedure, and if so does it have a name?
 A: What you are asking about (splitting questionnaire into parts, and only administering each part to a random subsample of all survey participants) is matrix sampling. See https://www.bls.gov/osmr/abstract/st/st070070.htm. Stratification before randomizing the version of the questionnaire is definitely a wise thing to do.
A: The gold standard way to do this sort of random allocation when you are allocating people or groups of people (units) to treatment arms proceeds like this.
1 - form the units to be randomised into strata based on the relevant variables (in your case you mention just one). Make the sizes of the stratum a multiple of the number of arms to which you wish to allocate (in your case this is four) but not all the same size, so some 8, some 12, some 16 and so on..
2 - now for each stratum draw a sample without replacement so you get exactly equal numbers of each arm (so 2 of each for stratum size 8, 3 for 12, 4 for 16). Search for permuted blocks if that needs more detail.
The reason for having a variable block size is to stop people being able to guess which is going to come next as after seven have been allocated from a block of size eight they know which is left if they have been counting. In your application it may not be so necessary to conceal the allocation from the units so you can use the same size stratum throughout. You could even use size four which is what I think you suggest in your question.
