Fisher's Exact Test and the Hypergeometric Distribution

Key Takeaways

hello and welcome to a stat quickie today we're going to talk about Fisher's exact test and enrichment analysis but first let's eat some M&Ms I want to share some with my friends so I just take one handful and get seven blue and one red what does this say about the distribution of colors in the bag do I have more blues than normal lastly can I calculate a p-value from this delicious sample this bags supposed to have two servings and I think a serving of M&Ms is twenty M&Ms so there must be 40mm s in the bag I looked up the proportions of the different colors of M&Ms on the internet and this is what I found so on the right we have a histogram of an idealized bag of M&Ms I'm going to use the histogram of the ideal bag of M&Ms based on the proportions I got off the internet and my sample my handful of M&Ms to determine if my bag is special in this example I don't care about the order of how the M&Ms fell into my hand so let's consider every possible ordering of seven blue and one red as legit let's start by calculating the probability of getting seven blue M&Ms followed by a single red M&M the probability that the first M&M is blue equals eight divided by 48 because there are eight blue M&Ms divided by 40 because there are 40 M&Ms total now that I've got one M&M in my hand there are only seven blue M&Ms left in the bag the probability that the second M&M is blue equals 7 divided by 39 7 because there are now only 7 blue M&Ms in the bag divided by 39 because there are only 39 mm/s now there are only six blue M&Ms left in the bag the probability of getting a third blue M&M is 6 over 38 leaving five blue M&Ms left in the bag and by now you've probably grasped the pattern for how we determine the probabilities for getting a sequence of blue M&Ms once we have calculated the probabilities for getting 7 blue M&Ms in our hand we can now calculate the probability of getting 1 red M&M that's just 5 over 33 five because there are five red M&Ms divided by 33 because there are 33 M&Ms left in the bag at this point now just multiply all those probabilities together to get the probability of getting 7 blues followed by one red and that just equals a really small number that's rare but remember we don't care about order there's more work to do to get the probability of seven blues and one red in any order to calculate the probability of getting seven blues and one red we have to add together the probabilities of each possible ordering the good news is that the process of calculating the probabilities is the same as what we just did good thing we have computers because they'll do the work for us anyways the probability is still really small now what's the p-value if you'll remember from the p-value stat quest sometimes you can have very small probabilities but really large p-values and remember a p-value is the sum of the probabilities of all things equally rare or rarer this is all covered in the stat quest on p-values so that includes adding the probability of getting eight Blues in a row or seven oranges in one blue because that's equally rare and actually there are a lot of different ways you can come up with things that are equally rare or rarer too many to put on this stack quickie so we're just going to skip to the chase again good thing we have computers the p-value ends up being 0.01 so my bag is special hooray we just performed Fisher's exact test on the M&Ms enrichment for other things like does this list of genes have more involved in metabolism than normal it's done the exact same way and for any of you stat Questers that use our a programming language for doing statistics I provided the R code for doing the Fisher's exact test that we performed on the MMS in the description below hooray tune in next week for another stat quickie