Last updated: 2020-06-09

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Rmd 2618f84 DongyueXie 2020-06-09 wflow_publish(“analysis/bayesglm.Rmd”)

apeglm

Let \(Y_{gi}\) denote the count of RNA-seq fragments assigned to gene g in sample i. We assume that \(Y_{gi}\) follows a NB distribution with mean \(\mu_{gi}\) and dispersion \(\alpha_g\).

The model is \(Y_{gi}\sim NB(\mu_{gi},\alpha_g)\), where \(\mu_{gi} = s_{gi}q_{gi}\) and \(\log(q_{gi})=x_i^T\beta_g\), \(\beta_g=(\beta_{g0},...,\beta_{gK}})\).

Prior on \(\beta\): \(\beta_{gk}\sim Cauchy(0,S_k)\).

First obtain MLE \(\hat\beta_{gk}\) and its standard error \(e_{gk}\) then set \(S_k = \sqrt{A}\) where A is an empircal estimate proposed in Efron and Morris, 1975 for normal distribution. Then the method uses Laplace approximation for posterior calculation.

bayesglm

The first step of our model is to standardize the input variables. Binary inputs are shifted to have a mean of 0 and to differ by 1 in their lower and upper conditions. Other inputs are shifted to have a mean of 0 and scaled to have a standard deviation of 0.5.

Assume prior independence of the coefficients as a default assumption, with the understanding that the model could be reparameterized if there are places where prior correlation is appropriate. For each coefficient, we assume a Student-t prior distribution with mean 0, degrees-of-freedom parameter ν, and scale s, with ν and s chosen to provide minimal prior information to constrain the coefficients to lie in a reasonable range.

We assign independent Cauchy prior distributions with center 0 and scale 2.5 to each of the coefficients in the logistic regression except the constant term. When combined with the standardization, this implies that the absolute difference in logit probability should be less then 5, when moving from one standard deviation below the mean, to one standard deviation above the mean, in any input variable.