knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )
The main function of the 'gfilogisreg' package is gfilogisreg
. It simulates
the fiducial distribution of the parameters of a logistic regression model.
To illustrate it, we will consider a logistic dose-response model for inference on the median lethal dose. The median lethal dose (LD50) is the amount of a substance, such as a drug, that is expected to kill half of its users.
The results of LD50 experiments can be modeled using the relation $$ \textrm{logit}(p_i) = \beta_1(x_i - \mu) $$ where $p_i$ is the probability of death at the dose administration $x_i$, and $\mu$ is the median lethal dose, i.e. the dosage at which the probability of death is $0.5$. The $x_i$ are known while $\beta_1$ and $\mu$ are fixed effects that are unknown.
This relation can be written in the form $$ \textrm{logit}(p_i) = \beta_0 + \beta_1 x_i $$ with $\mu = -\beta_0 / \beta_1$.
We will perform the fiducial inference in this model with the following data:
dat <- data.frame( x = c( -2, -2, -2, -2, -2, -1, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2 ), y = c( 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 ) )
Let's go:
library(gfilogisreg) set.seed(666L) fidsamples <- gfilogisreg(y ~ x, data = dat, N = 500L)
Here are the fiducial estimates and $95\%$-confidence intervals of the parameters $\beta_0$ and $\beta_1$:
gfiSummary(fidsamples)
The fiducial estimates are close to the maximum likelihood estimates:
glm(y ~ x, data = dat, family = binomial())
Now let us draw the fiducial $95\%$-confidence interval about our parameter of interest $\mu$:
gfiConfInt(~ -`(Intercept)`/x, fidsamples)
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