# % coefficient of variation (%CV) for log-linear and log-log regression (calibrations)

I have a data set comprising the peak areas of an analyte measured in spiked calibration samples at 'known' and different concentrations levels. For each concentration level I have 5 replicates. My goal is to generate a calibration curve from this data and then, for each concentration level, determine the precision of the 'estimated concentration' expressed as percent coefficient of variation ($$\%CV$$).

If a linear relationship existed between the measured peak areas and 'known' concentrations, then for each concentration level I would simply calculate $$\%CV$$ as the standard deviation of the estimated concentration at a given 'known' concentration level and divide by the mean of the same estimated concentrations, before multiplying by 100. For my data set, however, I observe an inadequate linear (i.e. straight line) fit between measured peak areas and 'known' concentrations. Furthermore, there is heteroscedacity of the residuals when fitting a linear model.

To address the above, I've performed a $$\log_{10}$$ transformation of BOTH peak area and 'known' concentration. An adequate linear fit is observed. I would now like to calculate the precision (coefficient of variation, $$\%CV$$) of the estimated peak area based on this model.

According to the article cited below, the %CV for log-transformed data would be calculated as:

$$\%CV (\text{estimated concentration}) = 100\% \cdot \sqrt{10^{\ln(10)\theta^2_{\log}} -1}$$

Where (if I understood correctly): $$\theta^2_{\log}$$ is the variance of the $$\log$$-transformed data.

So, I would specifically like to know: is the formula proposed by Canchola, et al. appropriate in the case where BOTH the response (i.e. peak area) and predictor (i.e. 'known' concentration) variable have been transformed?

In my mind, seeing as I would consider the variable of the estimated concentration on the $$\log_{10}$$-transformed scale, the formula outlined by Canchola, et al. should be fine.

Finally: if I had only $$\log_{10}$$-transformed the peak areas and then estimated the concentration (i.e. log-linear relationship), would I need to use the Canchola, et al. equation?

Referenced article: Jesse A. Canchola, Shaowu Tang, Pari Hemyari, Ellen Paxinos, Ed Marins, "Correct use of percent coefficient of variation ($$\%CV$$) formula for $$\log$$-transformed data," MOJ Proteomics & Bioinformatics 2017, 6(4), 316-317 (DOI: 10.15406/mojpb.2017.06.00200).

• While the question may be ok for this site (and I personally consider it interesting) it strikes me as better suited for crossvalidated SE. Sep 16 '20 at 18:14
• I agree with Buck that this appears to be more about statistical analysis and less about chemistry. Another alternative, if you want an answer specifically from the perspective of quantitative experimental science (rather than from pure mathematicians), is mattermodeling.stackexchange.com. Statistics questions are fine there, but the answer would still come from a chemist or physicist. Sep 16 '20 at 18:20
• Thank you both for your recommendations. I'll take this on over to crossvalidated then.
– MRJ
Sep 16 '20 at 20:42