I'm calibrating a piece of lab instrumentation. I create solutions of known concentration ($x$) and measure my instrument response ($y$). On unknown samples, I measure the response and use the regression line to predict the actual concentration (reverse regression).
I know the equation for $PI(x)$, the prediction interval given some value of $x$:
$$PI(x) = t \cdot Syx \cdot \sqrt{ \frac{1}{q} + \frac{1}{n} + \frac{(x - \bar{x})^2}{Sxx} }$$
And the prediction interval bands are plotted as:
$$Y(x) = mx + b \pm PI(x)$$
where:
$Syx$ is sqrt of (sum of squared residuals divided by degrees of freedom)
$Sxx$ is $\sum (x_i - \bar{x})^2$
$q$ is the number of replicate runs
$n$ in the number of points in the calibration
$m$ and $b$ are the regression parameters
$t$ is the inverse $t$ value at whatever significance level you are interested in
If I want the uncertainty of $x$ given a measurement of $y$, would I use the inverse of the $PI(x)$? It seems like the Prediction Interval given a value of $x$ is what range of $\bar{y}$ you would expect to see for future analyses of one or more samples of known $x$. The inverse of the prediction interval given $x$ are the roots of a very large complicated quadratic equation. Using the inverse of the above $PI$ equation would find the $x$ value that has the given y as an upper $PI$ bound, and the $x$ that has the same $y$ as a lower bound. The two intervals (left versus right) will be slightly different.
A colleague was asking about a passage in a textbook of his (Quantitative Chemical Analysis by Daniel Harris), where it stated that this uncertainty estimation is instead:
$$\Delta x = t \frac{Syx}{m} \sqrt{\frac{1}{q} + \frac{1}{n} + \frac{(y - \bar{y})^2}{\frac{m^2}{Sxx}}}$$
It appears that this in the same as the first equation, where the appropriate $y$ values have been substituted for $x$. Which of these is correct? Using actual data, the results are similar. The textbook value is about the average of the two that are calculated from the inverse. However, when using $x$ values near the extents of the calibration line, or for poorly fitted data, the differences between the two are vastly different.
