# Weighting for stratified random sample with non-proportionally allocated sample

If I have a national survey data that used stratified random sample design, and the proportion of each stratum (e.g., state) sampled is not identical to the proportion of the population (e.g., proportions of the actual state populations) [I am assuming that the stratification was not done based on the proportion of the actual state population, but based on something else. i.e., strata were not states], do I need to use a weight to account for disproportionate selection if I want to make comparisons between states?

I would appreciate if you could provide a good reference for that as well.

A stratified design effectively means that separate surveys are designed within each stratum - units selected within one stratum are independent of all selections within other strata. Estimates of total are made within each stratum, and then combined to come up with the estimate of total across the population:

$$\hat{Y} = \sum_{\text{strata}}\hat{Y}_{\text{stratum}}\\ = \sum_{\text{strata}}\sum_{\text{units}}y_iw_i$$

The design weight for a unit in the survey should only weight the unit within the stratum that the unit belongs to. There is no need to modify design weights, assuming you have them.

I recommend reading Model Assisted Survey Sampling (Sarndel, Swenson, Wretman) or Practical Sampling Techniques (K Som)

• What if the stratification was not done based on the proportions of the state populations and thus the proportion of the each state in the sample is disproportionate to the actual state population sizes, but I still want to make comparisons between the states? – K.Y Jul 31 '17 at 5:53
• Do you mean the ratio of sample sizes between strata isn't the same as the ratio of population sizes? This doesn't matter, as long as you have the design weights for each unit. If one stratum has a higher sampling fraction than another, then this means that its units will have lower design weights. – RoryT Jul 31 '17 at 6:17
• Yes, you are right. "The ratio of sample sizes between strata isn't the same as the ratio of population sizes." And, the problem is..I don't have the design weights for each unit. So, my question is how should I weight the data in that situation? If I am supposed to construct some weights and apply them, what would be the best approach to do that? – K.Y Jul 31 '17 at 7:07
• Well, the best thing would be to calculate the design weights. Do you know how units were sampled within each stratum? – RoryT Jul 31 '17 at 7:29
• What I know is that they did random sampling within each stratum, but what I don't know is what kind of strata they used. In their manual, they only indicate that they used stratified random sampling, but they don't provide any information as to what kind of strata they used. I am assuming that they didn't use states as strata since the ratio of sample sizes between states is not consistent with the ratio of the actual state population sizes at all, and they are not in approximately equal sizes either. – K.Y Aug 3 '17 at 0:29