How do I propagate the error of a linear regression when projecting from Y to X? - r

Question

I'm trying to figure out how to propagate errors in the following case
I am calibrating a machine with a couple of standards (a, b, c) with
accepted values x. My machine measures y for these standards, with a
certain error (standard deviation of 1 in this example).
Then I measure replicates of a sample, yielding ynew. Now I want to
convert these values to the accepted measurement scale (the x-axis).
To do this, I can of course do some linear algebra and convert the slope and
intercept that I got from my standard measurements to a reversed slope and
intercept as follows
This works nicely to convert the input values, but how do I get proper estimates of the errors?
In R, I've tried the following:
library(broom) # for tidy lm
library(ggplot2) # for plotting
library(dplyr) # to allow piping
# find confidence value
cv <- function(x, level = 95) {
qt(1 - ((100 - level) / 100) / 2, df = length(x) - 1) * sd(x) / sqrt(length(x))
}
# find confidence interval
ci <- function(x, level = 95) {
xbar <- mean(x)
xci <- cv(x, level = level)
c(fit = xbar, lwr = xbar - xci, upr = xbar + xci)
}
set.seed(1337)
# create fake data
dat <- data.frame(id = rep(letters[1:3], 20),
x = rep(c(1, 7, 10), 20)) %>%
mutate(y = rnorm(n(), -20 + 1.25 * x, 1))
# generate linear model
mod <- lm(y ~ x, dat)
# tidy
mod_aug <- augment(mod)
# these are the new samples that my machine measures
ynew <- rnorm(10, max(dat$y) + 3)
# predict new x-value based on y-value that is outside of range
## predict(mod, newdata = data.frame(y = ynew), interval = "predict")
# Error in eval(predvars, data, env) : object 'x' not found
# or tidy
## augment(mod, newdata = data.frame(y = ynew))
# 50 row df that doesn't make sense
# found this function that should do the job, but it doesn't extrapolate
## approx(x = mod$fitted.values, y = dat$x, xout = ynew)$y
# [1] NA NA NA NA NA NA NA NA NA NA
# this one from Hmisc does allow for extrapolation
with_approx <- Hmisc::approxExtrap(x = mod_aug$.fitted, y = mod_aug$x, xout = ynew)$y
# but in case of lm, isn't using the slope and intercept of a model okay too?
with_itc_slp <- (- coef(mod)[1] / coef(mod)[2]) + (1 / coef(mod)[2] * ynew)
# this would be the 95% prediction interval of the model at the average
# sample position. Could also use "confidence" but this is more correct?
avg_prediction <- predict(mod,
newdata = data.frame(x = mean(with_itc_slp)),
interval = "prediction")
# plot it
ggplot(dat, aes(x = x, y = y, col = id)) +
geom_point() +
geom_hline(yintercept = ynew, col = "gray") +
geom_smooth(aes(group = 1), method = "lm", se = F, fullrange = T,
col = "lightblue") +
geom_smooth(aes(group = 1), method = "lm") +
# 95% CI of the new sample
annotate("pointrange", x = 1, y = mean(ynew),
ymin = ci(ynew)[2], ymax = ci(ynew)[3], col = "green") +
# 95% prediction interval of the linear model at the average transformed
# x-position
annotate("pointrange", x = mean(with_approx), y = mean(ynew),
ymin = avg_prediction[2], ymax = avg_prediction[3], col = "green") +
# transformed using approx
annotate("point", x = with_approx, y = ynew, size = 3, col = "blue",
shape = 1) +
# transformed using intercept and slope
annotate("point", x = with_itc_slp, y = ynew, size = 3, col = "red",
shape = 2) +
# it's pretty
coord_fixed()
resulting in this plot:
Now how do I go from these 95% CIs in the y-direction to transformed sample
x-values with a confidence interval in the x-direction?