investr: Inverse Estimation in R

Inverse estimation, also referred to as the calibration problem, is a classical and well-known problem in regression. In simple terms, it involves the use of an observed value of the response (or specified value of the mean response) to make inference on the corresponding unknown value of the explanatory variable.

A detailed introduction to investr has been published in The R Journal: "investr: An R Package for Inverse Estimation", You can track development at To report bugs or issues, contact the main author directly or submit them to

As of right now, investr supports (univariate) inverse estimation with objects of class:


The package is currently listed on CRAN and can easily be installed:

  # Install from CRAN
  install.packages("investr", dep = TRUE)

The package is also part of the ChemPhys task view, a collection of R packages useful for analyzing data from chemistry and physics experiments. These packages can all be installed at once (including investr) using the ctv package (Zeileis, 2005):

  # Install the ChemPhys task view


Dobson's Beetle Data

In binomial regression, the estimated lethal dose corresponding to a specif probability p of death is often referred to as LDp. invest obtains an estimate of LDp by inverting the fitted mean response on the link scale. Similarly, a confidence interval for LDp can be obtained by inverting a confidence interval for the mean response on the link scale.


# Dobson's beetle data

# Binomial regression
binom_fit <- glm(cbind(y, n-y) ~ ldose, data = beetle, 
                 family = binomial(link = "cloglog"))
plotFit(binom_fit, = 2, cex = 1.2, pch = 21, bg = "lightskyblue", 
        lwd = 2, xlab = "Log dose", ylab = "Probability")

# Estimate the 50% lethal dose
invest(binom_fit, y0 = 0.5)

# estimate    lower    upper 
#   1.7788   1.7702   1.7862
Alt text

Alt text

To obtain an estimate of the standard error, we can use the Wald method:

invest(binom_fit, y0 = 0.5, interval = "Wald")

# estimate    lower    upper       se 
#   1.7788   1.7709   1.7866   0.0040

# The MASS package function dose.p works too 
MASS::dose.p(binom_fit, p = 0.5)

#              Dose         SE
# p = 0.5: 1.778753 0.00400654

Bioassay on Nasturtium

The data here contain the actual concentrations of an agrochemical present in soil samples versus the weight of the plant after three weeks of growth. These data are stored in the data frame nasturtium and are loaded with the package. A simple log-logistic model describes the data well:

# Log-logistic model
log_fit <- nls(weight ~ theta1/(1 + exp(theta2 + theta3 * log(conc))),
               start = list(theta1 = 1000, theta2 = -1, theta3 = 1),
               data = nasturtium)
plotFit(log_fit, = 2)
Alt text

Alt text

Three new replicates of the response (309, 296, 419) at an unknown concentration of interest (x0) are measured. It is desired to estimate x0.

# Inversion method
invest(log_fit, y0 = c(309, 296, 419), interval = "inversion")

# estimate    lower    upper 
#   2.2639   1.7722   2.9694

# Wald method
invest(log_fit, y0 = c(309, 296, 419), interval = "Wald")  

# estimate    lower    upper       se 
#   2.2639   1.6889   2.8388   0.2847

The intervals both rely on large sample results and normality. In practice, the bootstrap may be more reliable:

# Bootstrap calibration intervals (may take a few seconds)
boo <- invest(log_fit, y0 = c(309, 296, 419), interval = "percentile", 
              nsim = 9999, seed = 101, progress = TRUE)
boo  # print bootstrap summary

# estimate    lower    upper       se     bias 
#   2.2639   1.7890   2.9380   0.2947   0.0281

plot(boo)  # plot results
Alt text

Alt text