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I am a statistics student trying to find data suitable for a particular method. From what I have read so far, it is of interest to determine if variants within a chromosome are associated with a phenotype. I would like to understand this more and if possible, find an example of this application. In particular, what test statistic is used when associating each chromosome variant to a phenotype? Also, is it possible to gain access to examples of SNP Chip data fitting this context?

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There are a very large number of such methods. I would recommend googling "GWAS" or "genome wide association analysis" and you will find many many results.

One relatively simple example would be this plant genotype-phenotype dataset, associated with this publication. Some basic summary description of the GWAS dataset can be found here. By now the methods in this paper are very old and naive seeming, but it gives an idea for how people approached this problem in the golden age of GWAS. The data are small and simple and full of strong associations. Additionally, the phenotype measurements are based on isogenic inbred lines, which are more or less homozygous, making the analysis much easier, interpretation a lot simpler.

Another point in favor of using the plant data is that there is actually some strong genotype-phenotype association with a really strong mechanistic basis, like the various flowering time associations. For better or for worse, such unambiguous associations are often absent in e.g. human phenotypes of interest.

For some background on the general problem of controlling for population structure in GWAS, see the old EMMA paper. A much more recent example of current statistical methodology that has been making the rounds on Twitter can be found here.

For very large catalogs of GWAS data in humans, I can point you here or here. The UK Biobank dataset in particular is very commonly used and quite high quality.

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