Say I have a database with
10000 protein sequences of the same species in it. I was to study if mutations at positions
j are correlated. I can decompose each sequence into strings of binary variables
11 - i is mutated, j is mutated 10 - i is mutated, j is not mutated 01 - i is not mutated, j is mutated 00 - i is not mutated, j is not mutated
I can sum these up and place them into a contingency table
j 1 | 0 --------- 1 | a | b | i------------ 0 | c | d | ---------
If I normalize this contingency table by
N=a+b+c+d, I get joint frequencies
j 1 | 0 ------------------- 1 | f(1,1) | f(1,0) | i---------------------- 0 | f(0,1) | f(0,0) | -------------------
Are these joint proportions (e.g.
f(1,1)) probabilities? If not, under what assumptions can I say that these joint proportions approximate the probabilities of two mutations occuring together in the population? If I was given a database of 50,000 sequences and I made two contingency tables from random sample of 10,000, I assume the two tables, and thus the proportions calculated from them, would be different. So I am asking, under what circumstances are the joint proportions from the contingency table an appropriate approximation of the probabilities of seeing these events in the true population?
I know that I can use similarity coefficients like Jaccard, Dice, or Ochiai to determine whether the positions are correlated, but I am confused as to the difference between probability distributions and the normalized contingency table cells.
To me, the normalized contingency table cells look like bivariate marginal distributions (if my protein is 100 amino acids long, then I have marginalized the total joint probability distribution over the other 98 amino acids). Then the normalized row/column sums look like univariate marginal distributions. Is this wrong? Is it that these normalized contingency table cells are maximum likelihood estimates to the joint probability distributions?