Lab Notebook

This document supplements the article Binomial Models.

Following the article, define link function for generalized linear model based on the logistic function as $\begin{equation} p_v = \max \left( \mathrm{logit}^{-1}(x_i\, \beta), \, \frac{1}{2} \right) \end{equation}$ where $p_v$ is the expected fraction of read count of the alternative allele. Compared to the article, $b_v$ has been replaced by $\beta$ but this has no impact on the main point of this post. Here we consider a single explanatory variable $x$ age, so $\beta=(\beta_0,\beta_1)$, where $\beta_0$ is the “intercept” term and $\beta_1$ mediates the effect of $x$ on $p_v$.

We implement the link function as logistic.upper as follows:

logistic.upper <- function(x, b0, b1) {
    l <- 1 / (1 + exp(-(b0 + x * b1)))
    l[l < 1/2] <- 1/2
    return(l)
}

Fix $p_1=0.9$ and, as discussed in the article, set $\beta_0 = \mathrm{logit}(p_1)$. Let $\beta_1=0.02$ corresponding to a $50$ year exponential time constant. Then $p_v$ starts with value $0.9$ at birth and decays to $1/2$ a bit later than a century.