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I want to research the returns to education in my city as well as assess whether people who have studied abroad on average earn more than those who have studied within the country. The data required for this does not exist so I want to collect the data myself using questionnaires but the problem is that there isn't any sampling frame I can use to select the people I would distribute the questionnaires to. I imagine that distributing the questionnaires to the first n employed people i find will not be good for making inferences. So what can be done in situations like these where you do not have a sampling frame and there is nothing known about the population of interest?

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There is always a sampling frame. You have a few options.

  • One option is to consider all possible people you could reach using your sampling method and identify this as your target population. You could simply describe your sampling mechanism as a means of identifying your population.
  • Another option is to identify all possible people you could reach using your sampling method and assume that this population is representative of some other target population of interest. You could simply describe your sampling mechanism as a means of identifying your population and then clearly state the target population of interest. (You could assume just a parameter in the population you sampled from, such as the mean, is equivalent to the parameter in the target population.)
  • Yet another approach would be to identify all possible people you could reach using your sampling method and use inverse-probability-weights to make this population representative of some other target population of interest (assuming the weights are known). These same weights would be applied to your sample.
  • Lastly, you could consider all possible responses you would get in repeated experiments using only the people in your sample. Your population is not a broader population of people, it is a population of repeated experiments on the same people. This last approach is akin to flipping a coin. The population is the collection of all possible flips of the coin and your data is a sample from this population.

This sort of issue happens quite often even in randomized clinical trials. There is always the consideration of sampling bias $-$ that you are not sampling in a representative way from the target population you are interested in.

I suppose one last option is to consider your sample to be the entire target population and that there is no sampling variability, i.e. if you were to repeat this experiment many times you would get the same responses each time. What you might normally report as summary statistics would be your population-level quantities. There would be no inference since everything is known.

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  • $\begingroup$ Thanks for the very thorough response. The last approach you mentioned sounds like it's referring to bootstrapping. Is it similar? $\endgroup$
    – NoLifeKing
    Dec 4, 2021 at 11:49
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    $\begingroup$ Great question! The last approach I discuss is a conceptualization of what repeated experiments would represent when identifying a population and making probability statements. The bootstrap is a re-sampling technique applied to an observed data set. The resulting bootstrapped sampling distribution is an approximation that could represent any of the interpretations I mentioned above. You are sampling with replacement from your data set as if the data set is the population of interest. That population of interest is one of the four points I mention above. $\endgroup$ Dec 4, 2021 at 14:23

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