Inverse Estimation/Calibration Functions

Functions to facilitate inverse estimation (e.g., calibration) in linear, generalized linear, nonlinear, and (linear) mixed-effects models. A generic function is also provided for plotting fitted regression models with or without confidence/prediction bands that may be of use to the general user.

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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:

  • lm - linear models (multiple predictor variables allowed)
  • glm - generalized linear models (multiple predictor variables allowed)
  • nls - nonlinear least-squares models
  • lme - linear mixed-effects models (fit using the nlme package)


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 specific 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")
# Inverse estimation
invest(binom_fit, y0 = 0.5)   # median lethal dose
invest(binom_fit, y0 = 0.9)   # 90% lethal dose
invest(binom_fit, y0 = 0.99)  # 99% lethal dose
# estimate    lower    upper 
#   1.7788   1.7702   1.7862

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

Including a factor variable

Multiple predictor variables are allowed for objects of class lm and glm. For instance, the example from ?MASS::dose.p can be re-created as follows:

# Load package, assuming it is already installed
# Data
ldose <- rep(0:5, 2)
numdead <- c(1, 4, 9, 13, 18, 20, 0, 2, 6, 10, 12, 16)
sex <- factor(rep(c("M", "F"), c(6, 6)))
SF <- cbind(numdead, numalive = 20 - numdead)
budworm <- data.frame(ldose, numdead, sex, SF)
# Logistic regression
budworm.lg0 <- glm(SF ~ sex + ldose - 1, family = binomial, data = budworm)
# Using dose.p function from package MASS
dose.p(budworm.lg0, cf = c(1, 3), p = 1/4)
#               Dose        SE
# p = 0.25: 2.231265 0.2499089
# Using invest function from package investr
invest(budworm.lg0, y0 = 1/4, 
       interval = "Wald", = "ldose", 
       newdata = data.frame(sex = "F"))
# estimate    lower    upper       se 
#   2.2313   1.7415   2.7211   0.2499

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

Three new replicates of the response (309, 296, 419) at an unknown concentration of interest ($x_0$) are measured. It is desired to estimate $x_0$.

# 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


NEWS for investr package

Changes for version 1.0

  • Added functions.
  • Fixed roxygen documentation.

Changes for version 1.0.1

  • A few minor bug fixes.
  • Slightly better documentation.

Changes for version 1.1.0

  • invest now accepts objects of class 'lme' (experimental).
  • A few minor bug fixes and code improvements.
  • Added more tests.

Changes for version 1.1.1

  • Updated citation file.
  • Minor code changes.
  • plotFit should now plot models with transformed responses correctly.
  • Fixed error causing invest to fail because of a missing data argument.
  • Added more tests.

Changes for version 1.1.2

  • Changed tests to satisfy CRAN check.

Changes for version 1.2.0

  • Cleaned up examples.
  • Added bootstrap option to invest.

Changes for version 1.2.1

  • Cleaned up documentation.
  • Added AnyNA function for those using older versions of R.

Changes for version 1.3.0

  • invest now accepts objects of class 'glm' (experimental).
  • Functions calibrate and invest now return an object of class "invest".

Changes for version 1.3.1

  • Multiple predictor variables are allowed for "lm" and "glm" objects.
  • All non-base package functions are now imported.
  • The generic function predFit is now exported. This function is used by investr to obtain predictions, and hence, inverse predictions. For example, predFit can be used to obtain prediction intervals for nonlinear least-squares fits (i.e., models of class "nls").
  • Improved tests and test coverage.
  • plotFit gained methods for "rlm" and "lqs" objects from package MASS.

Reference manual

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1.4.0 by Brandon M. Greenwell, 6 years ago

Browse source code at

Authors: Brandon M. Greenwell

Documentation:   PDF Manual  

Task views: Chemometrics and Computational Physics

GPL (>= 2) license

Imports graphics, nlme, stats, utils

Depends on base

Suggests boot, datasets, knitr, MASS, testthat

Imported by basicTrendline.

Depended on by enveomics.R.

Suggested by chemCal.

See at CRAN