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: