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I need some advice on what sort of statistical analysis I want to run.

I ran an experiment that started with equal abundances of 20 strains of bacteria. This experiment had 4 treatments of interest, replicated with 15 populations per treatment. This experiment ran for multiple months, with populations growing fairly large. At the end, from each population I isolated out 3 individual bacteria, and identified to which of the 20 original bacterial strains they belonged. With 15 replicate populations per treatment, this gives 45 strains IDed per treatment, and 180 strains IDed in total.

What I’m trying to figure out is how to determine if these 4 different treatments ended up selecting for different strains among the 20 starting strains. Qualitatively I can see there are differences; some treatments have way more of one strain than others. Though statistically I’m having a hard time conceptualizing on what approach I should use. It feels like I should use some sort of binomial or chisquare approach as these represent effectively random draws from the population. But with multiple possible strains to choose from I'm not sure where to go.

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    $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Commented Sep 4 at 20:02
  • $\begingroup$ Are each of the strains equally likely to be isolated if the mixture has equal numbers of each strain? You need some sort of experimental demonstration of that because otherwise you cannot be confident of "effectively random draws". $\endgroup$
    – Michael Lew
    Commented Sep 4 at 21:09
  • $\begingroup$ Out of curiosity, when you take 3 individual bacteria out of each population, how many bacteria would you guess-timate were in each population? 10's? 1000's? $\endgroup$
    – jginestet
    Commented Sep 5 at 1:21

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You certainly could use a $\chi^2$ test. It is a valid method for your type of problem. You have a multinomial distribution (20 possible strains, mutually exclusive). You say that you are "trying to figure out how to determine if these 4 different treatments ended up selecting for different strains". For this you could use a 4x20 contingency matrix, and a $\chi^2$ test. It will tell you if the "after treatment" distributions are the same for all 4 treatments (the treatments selected for the same strains), or are different (they selected differentially). See e.g. here (Berkeley stat)
After that, you can use 6 20x2 contingency matrices to look at all the pairs of treatments, and see which are different, or not. You may have to use a multiple comparison correction here; Sidak?

However... Out of a sample size of 45 (per treatment), assuming absolutely no effect of the treatments, you would expect 2.25 bacteria per strain. And that is where you may run into issues. The observed counts will be quite low (maybe 0 through 4-5?). A normal approximation (which is what the $\chi^2$ test is) is not really valid with such low counts. In addition, you would need a large effect to actually get a significant test result (observing 1 A and 5 B bacteria is not significant, and is not such a large deviation from observing 2.25, and 2.25 -if one could observe 0.25 of a bacteria). So, 45 feels like a very small sample size, given the 20 different strains (if you had had, say, 3 or 4 strains, the situation would have been better).
You could also run a Fisher-exact test; that will deal with the "normal approximation" issue. But not too many software packages support Fisher exact for tables as large as 20x4 (R should, particularly that the counts will be low). But that leaves the low power/large effect issue. For this you need to identify more bacteria...

So, try a $\chi^2$ or Fisher test, just to see the order of magnitude of the p-values, but do not be surprised if the result is not significant.

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