I have a quick question in combine surveys of different years. Suppose that we have a complex survey conducted each year for five year in a row and I want to combine the five surveys into one. Shall we treat years as strata or not?

I assume that probability of being sampled is small so one person cannot participate in surveys of two years. Also assume that weights are comparable.

Edit: Let's say (hypothetically) we want to study smoking in American schools. A survey divide the whole nation into 100 parts. First we randomly choose 10 clusters out of 100. For each cluster we select five schools. For each school we select up to two classes for each grade and interview each student in the class.

I expect that $\pi_i$ and $\pi_{ij}$ (both $i_{th}$ and $j_{th}$ person are selected) is easy to calculate (as least in theory). For instance, if they are in the same school but not in the same class. Then you can calculate $\pi_{ij}$ by multiplying the probability of (a) the cluster is selected (20/100, for simplicity) (b) the school is selected (5/# of school in that cluster) (c) Both of classes they belong to are selected

Nothing stops you from calculate 3-way or 4-way probabilities, while Marginal and pairwise probability is sufficient to calculate mean of a certain trait (i.e. proportion of smoking) and its SE.

We did the same survey for 5 years. You can assume that the sample is mostly independent, i.e. one can participate in the survey for at most once.

Some (elementary) thought:

Now I want to combine the 5 year data together. If I treat year as an additional strata, it says the following:

I have a grand survey, for each year (strata) I draw a certain number of samples.

Instead if I treat something (like gender) as ordinary variables it looks like this:

I have a survey in which I sample people from a population according to a procedure unrelated to gender. In the end, the (weighted) proportion of men and women should be representative of population while I have no control of the exact number.

What do you think of replacing 'years' with 'gender' above?

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    $\begingroup$ It helps to explain the mechanism used to select individuals for the surveys. Ultimately we would like to know (at least in theory), for each possible sequence of individuals in the survey, their chance of being the sequence that was actually used. Could you tell us what that was? $\endgroup$ – whuber Jun 5 '18 at 21:24

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