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
::install_github("bgreenwell/investr") devtools
```

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")
::install.views("ChemPhys") ctv
```

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
<- glm(cbind(y, n-y) ~ ldose, data = beetle,
beetle.glm 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)
#> estimate lower upper
#> 1.778753 1.770211 1.786178
# 90% lethal dose
invest(beetle.glm, y0 = 0.9)
#> estimate lower upper
#> 1.833221 1.825117 1.843068
# 99% lethal dose
invest(beetle.glm, y0 = 0.99)
#> estimate lower upper
#> 1.864669 1.853607 1.879133
```

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

```
invest(beetle.glm, y0 = 0.5, interval = "Wald")
#> estimate lower upper se
#> 1.7787530 1.7709004 1.7866057 0.0040065
# The MASS package function dose.p can be used too
::dose.p(beetle.glm, p = 0.5)
MASS#> Dose SE
#> p = 0.5: 1.778753 0.00400654
```

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
<- rep(0:5, 2)
ldose <- c(1, 4, 9, 13, 18, 20, 0, 2, 6, 10, 12, 16)
numdead <- factor(rep(c("M", "F"), c(6, 6)))
sex <- cbind(numdead, numalive = 20 - numdead)
SF <- data.frame(ldose, numdead, sex, SF)
budworm
# Logistic regression
<- glm(SF ~ sex + ldose - 1, family = binomial, data = budworm)
budworm.glm
# Using dose.p function from package MASS
dose.p(budworm.glm, cf = c(1, 3), p = 1/4)
#> Dose SE
#> p = 0.25: 2.231265 0.2499089
# Using invest function from package investr
invest(budworm.glm, y0 = 1/4,
interval = "Wald",
x0.name = "ldose",
newdata = data.frame(sex = "F"))
#> estimate lower upper se
#> 2.2312647 1.7414522 2.7210771 0.2499089
```

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
<- nls(weight ~ theta1/(1 + exp(theta2 + theta3 * log(conc))),
nas.nls 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")
#> estimate lower upper
#> 2.263854 1.772244 2.969355
# Wald method
invest(nas.nls, y0 = c(309, 296, 419), interval = "Wald")
#> estimate lower upper se
#> 2.2638535 1.6888856 2.8388214 0.2847023
```

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)
<- invest(nas.nls, y0 = c(309, 296, 419), interval = "percentile",
boo nsim = 9999, seed = 101)
# print bootstrap summary
boo #> estimate lower upper se bias
#> 2.2638535 1.7889885 2.9380360 0.2946540 0.0281456
plot(boo) # plot results
```