knitr::opts_chunk$set( collapse = TRUE, comment = "#>", fig.align = "center", fig.path = "man/figures/README-" )
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 an 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 https://github.com/bgreenwell/investr. To report bugs or issues, contact the main author directly or submit them to https://github.com/bgreenwell/investr/issues.
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) # Alternatively, install the development version from GitHub devtools::install_github("bgreenwell/investr")
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 install.packages("ctv") ctv::install.views("ChemPhys")
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.
# Load required packages library(investr) # Binomial regression beetle.glm <- glm(cbind(y, n-y) ~ ldose, data = beetle, family = binomial(link = "cloglog")) plotFit(beetle.glm, lwd.fit = 2, cex = 1.2, pch = 21, bg = "lightskyblue", lwd = 2, xlab = "Log dose", ylab = "Probability") # Median lethal dose invest(beetle.glm, y0 = 0.5) # 90% lethal dose invest(beetle.glm, y0 = 0.9) # 99% lethal dose invest(beetle.glm, y0 = 0.99)
To obtain an estimate of the standard error, we can use the Wald method:
invest(beetle.glm, y0 = 0.5, interval = "Wald") # The MASS package function dose.p can be used too MASS::dose.p(beetle.glm, p = 0.5)
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 required packages library(MASS) # 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.glm <- glm(SF ~ sex + ldose - 1, family = binomial, data = budworm) # Using dose.p function from package MASS dose.p(budworm.glm, cf = c(1, 3), p = 1/4) # Using invest function from package investr invest(budworm.glm, y0 = 1/4, interval = "Wald", x0.name = "ldose", newdata = data.frame(sex = "F"))
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 for the nasturtium data nas.nls <- nls(weight ~ theta1/(1 + exp(theta2 + theta3 * log(conc))), start = list(theta1 = 1000, theta2 = -1, theta3 = 1), data = nasturtium) # Plot the fitted model plotFit(nas.nls, lwd.fit = 2)
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(nas.nls, y0 = c(309, 296, 419), interval = "inversion") # Wald method invest(nas.nls, y0 = c(309, 296, 419), interval = "Wald")
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(nas.nls, y0 = c(309, 296, 419), interval = "percentile", nsim = 9999, seed = 101) boo # print bootstrap summary plot(boo) # plot results
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