2.3: Statistical Analysis
To analyze the relationship between epistatic effect and separation distance, we conducted a likelihood ratio test that compares a null model (where separation distance is unrelated to epistasis) against an alternative model (where epistasis decays with increasing separation). More precisely, we defined the null model to be that epistasis values are sampled from a normal distribution that is independent of the separation between residues. For the alternative model, epistasis values are sampled from a normal (same mean as the null case) with a standard deviation that decays exponentially as a function of separation according to a e𝛼 r where r is the separation between residue site alpha-carbons (EQ 3) and a and 𝛼 are the curve’s parameters estimated by maximum likelihood for the dataset. This maximum likelihood was determined by a grid-search method, considering all possible a and 𝛼, taking the resulting model with the largest likelihood. The likelihood ratio is given by the ratio of the log of the two likelihoods of the data under the two models: