# The ‘gfilogisreg’ package

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)
#>                  mean    median        lwr      upr
#> (Intercept) 0.5510683 0.5099083 -0.4386194 1.662866
#> x           0.9153775 0.8728642  0.2119688 1.944350
#> attr(,"confidence level")
#>  0.95

The fiducial estimates are close to the maximum likelihood estimates:

glm(y ~ x, data = dat, family = binomial())
#>
#> Call:  glm(formula = y ~ x, family = binomial(), data = dat)
#>
#> Coefficients:
#> (Intercept)            x
#>      0.5639       0.9192
#>
#> Degrees of Freedom: 24 Total (i.e. Null);  23 Residual
#> Null Deviance:       33.65
#> Residual Deviance: 26.22     AIC: 30.22

Now let us draw the fiducial $$95\%$$-confidence interval about our parameter of interest $$\mu$$:

gfiConfInt(~ -(Intercept)/x, fidsamples)
#>       2.5%      97.5%
#> -2.6565460  0.6644137