For practitioners: do you need weighted linear regression?

In my statistics course, similarly to I guess all other teachers concerned in calibration, I teach that for instrumental analyses weighted linear regression should be used. Why? Non-weighted linear regression (the one we can use with LINEST, SLOPE and INTERCEPT in excel) assumes same signal precision (repeatability standard deviation in other words) for all concentrations in the calibration range. For instrumental analyses however the relative standard deviation of the signal is usually (there are some specific instruments where this does not apply) nearly constant over concentrations used.

Veronica Meyer1 published in LC/GC a good simulation aiming to show how much results are influenced by either using or not using weighting in linear regression.

Their simulations effectively demonstrate that advantages of weighting are observed only if all following four things happen simultaneously:

1. Absolute repeatability standard deviation is not constant over given concentration range.

2. The calibration range is very wide.

3. Calibration points are distributed equally over the calibration range.

4. Sample result is at the lower end of the calibration range.

For example if calibration points 2, 1000, 2000, 3000 and 5000 units were used for calibration and the sample with actual concentration of 2.0 units was measured unweighted regression yielded answer of 8.3 but weighted resulted in 1.95 units. This simulation only included random errors.

On the other hand if a narrower calibration range – around one order of magnitude – would be used there is no significant difference in using or not using weighting.

So what to do if you are in the lab doing your actual analyses? I’d suggest you to prepare at least 5 point approximately equally spaced standard solutions for each order of magnitude your method needs to work in. For example if you samples concentrations may range from 10 – 1000 ppb I’d suggest following solutions: 10, 25, 50, 75, 100, 250, 500, 750 and 1000 ppb.

After analysing these solutions I would break up this calibration into two parts 1) 10-100 ppb and 2) 100-1000 ppb. This way you can assure that the highest concentrations on the calibration graph do not influence the accuracy of the samples in the lower end.

Good calibration!

1 V. Meyer Weighted Linear Least-Squares Fit — A Need? Monte Carlo Simulation Gives the Answer, LC/GC 28 (2015) 204-210.