Draper & Smith sections (3.2, 9.6) address prediction interval issues, but
I'm also concerned with the linear fit itself. The Model II regression
literature makes it abundantly clear that OLS regression of x on y
frequently yields a different line than of y on x. The example below is not
so extreme, but those given e.g. by Ludbrook, J. (2012) certainly are. Rui
notes the logical problem of imputing an unknown x using a calibration
curve where the x values are without error. Regression x on y doesn't help
that.  But on a practical level, I definitely recall (years ago) using
predict.lm and newdata to predict x terms. I wish I remembered how.


require(stats)
#Make an illustrative data set
set.seed(seed = 1111)
dta <- data.frame(
    area = c(
        rnorm(6, mean = 4875, sd = 400),
        rnorm(6, mean = 8172, sd = 800),
        rnorm(6, mean = 18065, sd = 1200),
        rnorm(6, mean = 34555, sd = 2000)),
    concn = rep(c(25, 50, 125, 250), each = 6))
model <- lm(area ~ concn, data = dta)
inv.model <- lm(concn ~ area, data = dta)
plot(area ~ concn, data = dta)
abline(model)
inv.new = cbind.data.frame(area = c(1600, 34000))
inv.pred <- predict(inv.model, newdata = inv.new)
lines(x = inv.pred, y = unlist(inv.new), col = "red")

_____________________________
Ludbrook, J. (2012). "A primer for biomedical scientists on how to execute
Model II linear regression analysis." Clinical and Experimental
Pharmacology and Physiology 39(4): 329-335.

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