Bayesian Logistic Regression with Lower and Upper Asymptotes

Fits Bayesian logistic regression models in which the success probability rises or falls between a lower and an upper asymptote instead of between 0 and 1, for binomial outcomes including binary data. The success probability equals the lower asymptote plus the difference between the two asymptotes multiplied by the logistic function of a linear predictor. The lower and upper asymptotes can be either estimated or fixed. With a single predictor such as log dose, this is the four-parameter logistic (Hill) model. Posterior computation is a full Gibbs sampler that requires no tuning.

boundedHill

boundedHill fits Bayesian bounded logistic regression models, also called four-parameter logistic or bounded Hill models, to binary and grouped-binomial data:

$$ \Pr(Y_i = 1 \mid x_i) = \delta_0 + (\delta_1-\delta_0) \operatorname{logit}^{-1}(x_i^\top\beta), \qquad 0 \leq \delta_0, \delta_1 \leq 1. $$

In ordinary logistic regression the response probability always goes from 0 to 1. Here the curve instead goes from a lower level delta0 to an upper level delta1. For example, a combination therapy can give some response even at a very low dose (delta0 > 0), and the response can level off below 100% at high doses (delta1 < 1). Each of delta0 and delta1 can be estimated with a Beta prior or fixed at a chosen value. Fixing delta0 = 0 and delta1 = 1 gives ordinary logistic regression.

Estimation uses a Gibbs sampler, so no proposal distributions need tuning. In each iteration, every observation is assigned to either the lower level or the upper level, delta0 and delta1 are drawn from Beta distributions, and the regression coefficients are drawn from a normal distribution using Pólya-Gamma augmentation (see ?bh_fit for references).

A note on labels

The same curve can be written in two ways: swapping delta0 and delta1 and changing the sign of beta gives exactly the same probabilities. The data alone therefore cannot tell which level is which. Giving delta0 a prior with a low mean and delta1 a prior with a high mean, as the default prior does, fixes the labels. Either way, the fitted probabilities from predict() are the same; only the labels of delta0, delta1 and beta are affected.

Installation

After the package is released on CRAN:

install.packages("boundedHill")

For a local source tree:

install.packages("boundedHill_0.1.0.tar.gz",
                 repos = NULL, type = "source")

Bernoulli regression

library("boundedHill")

set.seed(18)
n <- 160
dat <- data.frame(
  x = runif(n, -2.5, 2.5),
  group = factor(sample(c("A", "B"), n, replace = TRUE))
)
eta <- -0.4 + 1.2 * dat$x + 0.5 * (dat$group == "B")
dat$y <- rbounded_binom(eta, delta0 = 0.12, delta1 = 0.78)

fit <- bh_fit(
  y ~ x + group,
  data = dat,
  prior = bh_prior(
    delta0 = c(2, 10),
    delta1 = c(8, 2),
    beta_sd = 2
  ),
  control = bh_control(iter = 3000, warmup = 1500, chains = 4),
  seed = 2026
)

print(fit)
summary(fit)
bh_diagnose(fit)
plot(fit, type = "trace")
predict(fit, newdata = data.frame(
  x = c(-2, 0, 2),
  group = factor("A", levels = levels(dat$group))
))

Grouped binomial regression

Grouped responses are supplied without expanding individual trials:

grouped <- data.frame(
  x = seq(-2, 2, length.out = 7),
  success = c(3, 5, 9, 14, 17, 18, 19),
  failure = c(17, 15, 11, 6, 3, 2, 1)
)

fit_grouped <- bh_fit(
  cbind(success, failure) ~ x,
  data = grouped,
  prior = bh_prior(delta0 = c(2, 8), delta1 = c(8, 2)),
  seed = 45
)

Model family

The same engine covers several nested models:

# Standard Bayesian logistic regression
prior_logistic <- bh_prior(delta0 = 0, delta1 = 1)

# Unknown lower asymptote, upper asymptote fixed at one
prior_lower <- bh_prior(delta0 = c(1, 4), delta1 = 1)

# Both asymptotes unknown (Beta shapes)
prior_bounded <- bh_prior(delta0 = c(1, 4), delta1 = c(4, 1))

# The same, specified by prior mean and SD
prior_moments <- bh_prior(delta0 = bh_beta_moments(0.2, 0.15),
                          delta1 = bh_beta_moments(0.8, 0.15))

The default bh_prior() uses delta0 with mean 0.10 and delta1 with mean 0.85, both with SD 0.12, and independent N(0, 2^2) priors for the regression coefficients.

Public interface

References

Lee, M., Lee, C. H., and Kim, R. S. A novel Bayesian phase 1/2 design incorporating the Hill equation. Manuscript in preparation.

Lee, M., Lee, C. H., and Kim, R. S. boundedHill: Bayesian logistic regression with lower and upper asymptotes in R. Manuscript in preparation.