Simple Emax model fit with Stan
In rstanemax: Emax Model Analysis with 'Stan'

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

library(rstanemax)
library(dplyr)
library(ggplot2)
set.seed(12345)

This vignette provide an overview of the workflow of Emax model analysis using this package.

Typical workflow

Model run with `stan_emax` function

stan_emax() is the main function of this package to perform Emax model analysis on the data. This function requires minimum two input arguments - formula and data. In the formula argument, you will specify which columns of data will be used as exposure and response data, in a format similar to stats::lm() function, e.g. response ~ exposure.

data(exposure.response.sample)

fit.emax <- stan_emax(response ~ exposure, data = exposure.response.sample,
                      # the next line is only to make the example go fast enough
                      chains = 2, iter = 1000, seed = 12345)

fit.emax

plot() function shows the estimated Emax model curve with 95% credible intervals of parameters.

plot(fit.emax)

Output of plot() function (for stanemax object) is a ggplot object, so you can apply additional settings as you would do in ggplot2.
Here is an example of using log scale for x axis (note that exposure == 0 is hanging at the very left, making the curve a bit weird).

plot(fit.emax) + scale_x_log10() + expand_limits(x = 1)

Raw output from rstan is stored in the output variable, and you can access it with extract_stanfit() function.

class(extract_stanfit(fit.emax))

Prediction of response with new exposure data

posterior_predict() function allows users to predict the response using new exposure data. If newdata is not provided, the function returns the prediction on the exposures in original data. The default output is a matrix of posterior predictions, but you can also specify "dataframe" or "tibble" that contain posterior predictions in a long format. See help of rstanemax::posterior_predict() for the description of two predictions, respHat and response.

response.pred <- posterior_predict(fit.emax, newdata = c(0, 100, 1000), returnType = "tibble")

response.pred %>% select(mcmcid, exposure, respHat, response)

You can also get quantiles of predictions with posterior_predict_quantile() function.

resp.pred.quantile <- posterior_predict_quantile(fit.emax, newdata = seq(0, 5000, by = 100))
resp.pred.quantile

Input data can be obtained in a same format with extract_obs_mod_frame() function.

obs.formatted <- extract_obs_mod_frame(fit.emax)

These are particularly useful when you want to plot the estimated Emax curve.

ggplot(resp.pred.quantile, aes(exposure, ci_med)) +
  geom_line() + 
  geom_ribbon(aes(ymin=ci_low, ymax=ci_high), alpha = .5) +
  geom_ribbon(aes(ymin=pi_low, ymax=pi_high), alpha = .2) +
  geom_point(data = obs.formatted,
             aes(y = response)) +
  labs(y = "response")

Posterior draws of Emax model parameters can be extracted with extract_param() function.

posterior.fit.emax <- extract_param(fit.emax)
posterior.fit.emax

Fix parameter values in Emax model

You can fix parameter values in Emax model for Emax, E0 and/or gamma (Hill coefficient). See help of stan_emax() for the details. The default is to fix gamma at 1 and to estimate Emax and E0 from data.

Below is the example of estimating gamma from data.

data(exposure.response.sample)

fit.emax.sigmoidal <- stan_emax(response ~ exposure, data = exposure.response.sample, 
                                gamma.fix = NULL,
                                # the next line is only to make the example go fast enough
                                chains = 2, iter = 1000, seed = 12345)

fit.emax.sigmoidal

You can compare the difference of posterior predictions between two models (in this case they are very close to each other):

exposure_pred <- seq(min(exposure.response.sample$exposure),
                     max(exposure.response.sample$exposure),
                     length.out = 100)

pred1 <- 
  posterior_predict_quantile(fit.emax, exposure_pred) %>% 
  mutate(model = "Emax")
pred2 <- 
  posterior_predict_quantile(fit.emax.sigmoidal, exposure_pred) %>% 
  mutate(model = "Sigmoidal Emax")

pred <- bind_rows(pred1, pred2)


ggplot(pred, aes(exposure, ci_med, color = model, fill = model)) +
  geom_line() + 
  geom_ribbon(aes(ymin=ci_low, ymax=ci_high), alpha = .3) +
  geom_ribbon(aes(ymin=pi_low, ymax=pi_high), alpha = .1, color = NA) +
  geom_point(data=exposure.response.sample, aes(exposure, response), 
             color = "black", fill = NA, size=2) +
  labs(y = "response")

Set covariates

You can specify categorical covariates for Emax, EC50, and E0. See help of stan_emax() for the details.

data(exposure.response.sample.with.cov)

test.data <-
  mutate(exposure.response.sample.with.cov,
         SEX = ifelse(cov2 == "B0", "MALE", "FEMALE"))

fit.cov <- stan_emax(formula = resp ~ conc, data = test.data,
                     param.cov = list(emax = "SEX"),
                     # the next line is only to make the example go fast enough
                     chains = 2, iter = 1000, seed = 12345)

fit.cov
plot(fit.cov)

You can extract MCMC samples from raw stanfit and evaluate differences between groups.

fit.cov.posterior <- 
  extract_param(fit.cov)

emax.posterior <- 
  fit.cov.posterior %>% 
  select(mcmcid, SEX, emax) %>% 
  tidyr::pivot_wider(names_from = SEX, values_from = emax) %>% 
  mutate(delta = FEMALE - MALE)

ggplot2::qplot(delta, data = emax.posterior, bins = 30) +
  ggplot2::labs(x = "emax[FEMALE] - emax[MALE]")

# Credible interval of delta
quantile(emax.posterior$delta, probs = c(0.025, 0.05, 0.5, 0.95, 0.975))

# Posterior probability of emax[FEMALE] < emax[MALE]
sum(emax.posterior$delta < 0) / nrow(emax.posterior)