where \(\mathcal{L}\), \(\theta_{1}\), \(\theta_{2}\) refers to the likelihood, log is the natural logarithm, and correspond to the null and alternative models respectively. Small values
of \(\Lambda\) indicate that the alternative model has more explanatory
power than the null. We first calculated the likelihood ratio for the
experimental data, \(\Lambda\)exp. In order to determine
statistical significance of \(\Lambda\)exp we then
obtained the distribution of Λ under the null through parametric
simulation. Specifically, we simulated datasets using the mean and
standard deviation of the experimental epistasis data. We then repeated
the fitting exercise used on the real dataset for the simulated dataset,
using the same separation data, and calculated \(\Lambda\). This process
was repeated 1000 times to obtain the distribution of \(\Lambda\) under
the null: \(\Lambda\)sim. The p-value for the test was
then calculated as the proportion of \(\Lambda\)sim less
than or equal to \(\Lambda\)exp.
Linear statistical models were used to determine the biophysical
features that are best able to explain the observed epistasis. The
absolute value of the epistasis, ϵ, was used as a response variable for
our model building. The choice to use the absolute value was necessary
to ensure a monotonic relationship between the features and the response
variable, as assumed when using linear models. . One could imagine
analyzing positive and negative epistasis separately; however, this was
not possible due to small sample sizes. All features described above
were considered in a standard model selection procedure, including all
pairwise interactions terms. For any features where we considered more
than one level of abstraction, only one level was included in any given
model. To evaluate model performance, the corrected Akaike information
criterion (AICc) was used. The corrected criterion was chosen over the
standard AIC due to the potential for overfitting models that contain a
large number of terms given a small amount of
data51.
Models were generated and tested using R
software52by considering all permutations of abstracted and non-abstracted
features. Model selection was performed using a modified form of stepAIC
from the
MASS53package to perform forward and backward selection based on AICc (further
verified by the AICc function of
AICcmodavg54and compared to standard AIC). Forward selection explores model space by
starting with a term-less model and systematically adding terms to find
the model with the best value for a given criterion. Conversely,
backward selection starts with the complete full-term model and removes
terms to find the best model. This model selection process was performed
twice with randomized input terms to avoid potential ordering bias
(terms treated differently based on their position in the initial list)
and the lowest AICc values were compared for consistency. Once we
verified that there was no ordering bias, the model with the lowest AICc
for both binding and folding was used for further analysis.
To rank the importance of features present in the final statistical
models for their effect on epistasis we compared R2values with and without each feature and it’s interactions. Features
with larger explanatory power of the observed epistasis will have a
larger change in R2 when removed.