# Stratified random sampling when strata overlap

I am sure the title might be confusing, but here is what I am dealing with:

I am running a survey at a health care center. The health center has around 15k active visitors. There are 5 departments operating in the center

Patients may (and in fact) visit more than one department.

First, we calculated the sample size needed for a level of precision and confidence level (typically 5% and 95% respectively) for an overall percentage of satisfied patients.

The problem I am facing is how to distribute the resulting sample size among the 5 departments. I was hoping for a stratified random sampling but clearly, the strata are overlapping.

At first, I thought of the capacity of each department with respect to, let's say, number of examinations, but I am not so sure about that.

I have the number of unique patients that visited each department in a given period of time (i.e. 6 months) and also the number of examinations or visits to each department. Again, some patients visit more than one department

• Should I use some kind of stratified sampling?
• In what way?

Any help would be greatly appreciated

If you really have to divide the population into 5 strata, you need to make those strata mutually exclusive. You can achieve that by assigning the visitors to one department only, and that department can be the one they visited the most. Let's look at the following fake data set where we are asked to separate the population of 4 visitors who visited 2 departments into 2 strata, while the departments are not non-overlapping in terms of their visitors.

visitor | department | visitedTime
----------------------------------
1    |     a      |      5
1    |     b      |      4
2    |     a      |      0
2    |     b      |      2
3    |     a      |      9
3    |     b      |      1
4    |     a      |      0
4    |     b      |      5


We see some visitors visited only one department (2 and 4), and the other two visited both departments, causing the stratification to fail. We can collapse this data set to the most visited department per visitor, and get

visitor | mostVisited
---------------------
1    |     a
2    |     b
3    |     a
4    |     b


Departments are non-overlapping in the second table. From and interpretation stand point, I think it makes sense to ask the visitor about the department they visit the most. Therefore this strategy is perfectly rationalizable.

Now, you may have a highly unbalanced picture such that one department is always/never the most visited. Think about the reception, everybody has to visit it once per visit, causing that the most visited department. If you can omit such a department, you should, to make your life easy. If you can't, you can keep this department outside of stratification, i.e. sample a certain amount of visitors from it first, and then apply the "collapsed stratification" I just discussed to the remaining departments.

• Thanks for your answer @Fatih. This means I need to process the center's database first and work out frequency of visits before asking the patient about the most visited department. Also, there is a department that is crucial but is less visited, meaning the patients who visit it, usually visit another department more often. If i follow this process then i risk of not collecting an adequate sample from that department Nov 24, 2017 at 6:56
• I guess a more correct question to this issue is " how do I get a representative sample"... Nov 24, 2017 at 6:58
• My last paragraph intends to address such issues, @LefkiosPaikousis. I would keep such departments outside of stratification, i.e. sample a certain amount of visitors from them first, and then apply the strategy I discussed to the remaining departments. But I will keep thinking about your problem, I think it is interesting and very likely to be bumped into in real data! Nov 24, 2017 at 7:16

If you want the distribution of your sample to mirror that of the original population P, you can accomplish this even when the groups overlap. For N groups, G1...GN, divide the original population P into 2^N subsets labeled (g1, ..., gN), where gk is a boolean indicating inclusion/exclusion with respect to group Gk. You are guaranteed that none of these subsets overlap, and every sample belongs to exactly one of them. Then you apply stratified sampling to these disjoint subsets.

So, in your case, you have 5 groups, G1...G5. You would have 2^5 = 32 subsets. For each subset S, you have a vector of booleans (g1_S, g2_S, g3_S, g4_S, g5_S) associated with it, where g1_S indicates whether S is a subset of G1 or P - G1, g2_S indicates whether S is a subset of G2 or P - G2, g3_S indicates whether S is a subset of G3 or P - G3, etc. (Some of these 32 subsets may be empty. You can just drop them.) Applying stratification to these subsets will give you a sample set that mirrors the proportions of not only the groups, but their degree of overlap, in the original population.