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The situation is a little complex.
Our goal is to study some soil properties in a large area.
According to different climate conditions, this area can be divided into several sub-regions.
And each sub-region is composed of several counties.
Now, the problem is instead of choosing some counties in each sub-region randomly, we choose those "typical" or "representative" ones according to some experts' opinions.
After that, soil samples are collected randomly in each selected county.

Is this still a random sample? If not, what kind of sampling method it should belong to, typical case sampling or expert sampling?
Could these samples represent the whole area? Is it still possible to make statistical inferences from the samples to the population?

Would someone please give some thoughts on this?

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    $\begingroup$ While this is not social research you could benefit from survey methodology literature. Google for "survey sampling", you should find multiple books and links for different sampling schema's for obtaining representative sampling designs. See: stats.stackexchange.com/questions/30715/… and stats.stackexchange.com/questions/11018/… for additional resources. $\endgroup$ – Tim Jan 11 '15 at 13:47
  • $\begingroup$ The tenses in your question are ambiguous. Exactly what is the stage of the study at the present time: field work completed; counties selected, but no field work done; or something else? $\endgroup$ – Steve Samuels Jan 12 '15 at 2:00
  • $\begingroup$ @SteveSamuels By now, field work completed. $\endgroup$ – gh2017554 Jan 12 '15 at 4:19
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This sample of counties was "stratified" by region, but the selection of counties was not random. The name for this kind of sample is "purposive".

Lohr (1999, p 463) has the following evaluation of purposive samples, based on an example discussed by Jerzy Neyman (1934). Neyman's paper is perhaps the foundation document of survey sampling theory.

Here is the quote from her book:

Neyman's paper pretty much finished off the idea that results from purposive samples could be generalized to the population. He presented an example of the purposive sample taken by Gini and Galvani in the late 1920's. Gini and Galvani chose 29 districts that gave the averages of all 214 districts in the 1921 Italian census, on a dozen variables. But Neyman showed that all statistics other than the average values of the controls showed a violent contrast between the sample and the whole population.

In your case, I assume that the experts had some criteria on which they based their selection. Can you describe them and how they defined "typical"? Like Neyman, if you have external information on the same criteria for all counties, you can compare the distribution of the purposive sample and the entire population.

The pity is that you could have created a valid random sample based on the criteria that the experts used. It would probably have involved stratifying on their criteria in the selection of counties and of areas within counties. Tille (2006) discusses some advanced sampling algorithms that can balance a sample on multiple criteria. (These are implemented in R.) But you could have created a valid random sample counties even without using Tille's methods.

References

Sharon Lohr, 1999, Sampling. Design and Analysis, Duxbury.

Jerzy Neyman, 1934. On the two different methods of the representative method: The method of stratified sampling and the method of purposive selection. J. Royal Statistical Society 197: 558-60

Tillé, Yves. 2006. Sampling algorithms. New York: Springer.

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  • $\begingroup$ Thank you for your answer.The "typical" counties were chosen in this way: First, the major landforms, soil types and land use practices were identified in each sub-region. Then counties which have major landforms, soil types and land use practices were selected. From the selected counties, some were chosen as the "typical" ones based on experts' knowledge to represent the whole sub-region. $\endgroup$ – gh2017554 Jan 13 '15 at 5:08
  • $\begingroup$ The process up to the selection was exemplary. The problem is how the experts defined "typical"; did they describe the algorithm by which they made their choices? In my experience (and in Neyman's), such selections present a distribution very different from the actual one. $\endgroup$ – Steve Samuels Jan 13 '15 at 17:03
  • $\begingroup$ I strongly suggest that you compare the selected sample to what is known about the entire population and report the differences. Perhaps you can use this information to post-stratify the results in order to better represent the population. This might reduce bias, though standard errors will still be problematic, as they assume random sampling. $\endgroup$ – Steve Samuels Jan 13 '15 at 17:04

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