# Confidence interval limits out of range

I have constructed a calibration curve to measure a pollutant concentration in water. My calibration standards ranged from 2 to 20 mg/L. I used a blank (0 mg/L).

I have obtained a predicted concentration in a sample (2.5 mg/L) near the lowest concentration used for calibration (2.0 mg/L). The lower limit of of the confidence interval for the prediction (1.62-2.53) is out of the tested range. Can this concentration be considered reliable? Should a confidence interval be within the range of X values used for calibration?

An additional (or complementary) question, would be: are there any differences between the exposed question and a value out of range with an upper limit of the confidence interval within the range? For example, a point value of 1.70 with the same confidence interval. Would'n it be extrapolation in both cases?

Should a confidence interval be within the range of X values used for calibration?

Not necessarily...

Can this concentration be considered reliable?

Yes, with a caveat that you are making an assumption that your model fits reality outside of your calibration range.

A model is not reality, but an approximation that helps us understand reality. Your calibration curve has an underlying assumed model distribution. The simplest in common use is a linear model, $$Y_i=b_0+b_1X_i+\epsilon_i$$ where $$\epsilon_i$$ is the "error", that is the deviation of the measured value from the model. The model is built by minimizing the error terms (or more precisely, minimizing the sum of their squares). The error in the input data leads to uncertainty in the resulting model.

In plain English, you don't know that your lowest Y measurement is abnormally high due to random noise (or due to the imperfect fit of your model w.r.t. reality). Your confidence intervals reflect that uncertainty.

Back to your result: if you can reasonably assume that your model fits reality outside of your calibration range, then you can reasonably accept your result. If you aren't sure, you can re-calibrate using a wider range of inputs to your curve.

• Thank you @DocBuckets. I don't usually extrapolate, but I have read that it can be reasonable in some cases... It depends on how confident we are in our model and on how far is the extrapolated value from our curve, I guess. Your answer suggested me an additional, related question. I have added it to the original one, but I think your reply is valid for this question, too.Thank you again.
– mmv
Commented May 7, 2020 at 18:03
• It is only extrapolation if your predicted value is outside of your calibration range. If the predicted value falls within your calibration range, the confidence interval falling outside of the calibration range is more a reflection of the uncertainty of your calibration standards to fit your model, either due to random noise or that your model is an imperfect representation of what's really going on. For example, if you model an exponential curve as a linear one, the error bands of your calibration curve will be relatively wide even if your measurements were perfect and without random error. Commented May 10, 2020 at 17:11