The statistical analysis was based on binomial likelihoods with a logit link function.
We use non-parametric Gaussian processes to model temporal correlation in gene expression and combine that with negative binomial likelihood for the count data.
So all the algebra that we have to do is when we multiply the binomial likelihood times the beta prior, we need to say, well, that p base, the one with y as its exponent, is going to be multiplied by this one.
Fourth, we used a normal approximation to the likelihood, instead of the exact binomial likelihood, in the modeling for meta-analysis.
Assumptions relating to heterogeneity and consistency were assessed using methods described by Song et al. 19 Differences in follow-up periods between studies were addressed in the analysis of the proportion of patients with improved vision, by fitting a binomial likelihood model with a complementary log-log link function which treated study length as an additional rate parameter.
Specifying a vague a priori beta distribution on the commuting probability from municipality i to j (p ij), and a binomial likelihood for c ij with binomial denominator equal to the population of i at the 2001 census (pop i), the resulting posterior distribution for p ij was a beta distribution with parameters 1 + c ij and 1 + pop i – c ij.
Given the prior, the method combines the data of a particular position with Binomial likelihood to produce a posterior allele frequency distribution.
The former would also support the use of the binomial likelihood to model our data.
Details of the estimation methods are given in Web Appendix 3. In brief, each study arm provided data with a binomial likelihood, which estimates a probability parameter, such as, representing the study-specific transition probability that a person in state i occupies state j, t years later.
Smoothness of expression dynamics is enforced by the GP prior, and agreement of expression dynamics with the read count data is quantified using the negative binomial likelihood.
