# The New U.S. EPA Method TO-15A Blog Series - Part 6: Calibration Curve Fits

22 Nov 2020Our last several blogs covered the numerous ways to help clean up lab air to meet the new TO-15A guidelines for canister blanks. Absent from the previous blogs was how we were getting quantitative numbers, which is important given the new calibration guidelines in TO-15A. The original TO-15 method only mentioned using an average response factor calibration, with the acceptance requirement being the %RSD between points <30%, with at most 2 exceptions up to 40% (TO-15, section 10.5.5.1). TO-15A adds in guidance for linear and quadratic fits, as well as alternate weighting models (TO-15A, section 15.2.3). The acceptance requirement for calibrations is either <30% RSD for average response factor or r^{2} ≥0.995, as well as each calculated concentration of each calibration point being within ±30% of the true value. This allows for more accuracy in your results, provided that the best curve fit can be found.

I won’t go deep into the math behind it here to keep this short, but equal weighted curve fits, be they linear, quadratic, or average response factor (RF), tend to be biased towards the higher concentration levels. Likewise r^{2}, the coefficient of determination for linear and quadratic fits, also tends to be biased towards higher concentrations. This is because the calibration models are based on absolute concentrations (or in the case of r^{2}, absolute error), so the bigger the difference between your high and low points the less your low points matter. This can be countered by either clustering more calibration points at the lower end, or using 1/x or 1/x^{2} weighting. The table below shows how the different curve weighting types can change an example linear calibration.

Concentration | response | Equal wt. | absolute error | % error |

1 | 12 | 0.33 | 0.67 | 67% |

10 | 110 | 10.74 | 0.74 | 7% |

100 | 950 | 99.94 | 0.06 | 0% |

r^{2} |
0.999657 | |||

total error | 1.47 | 74% | ||

%RSE | 67% | |||

Concentration | response | 1/x wt. | absolute error | % error |

1 | 12 | 0.90 | 0.10 | 10% |

10 | 110 | 11.14 | 1.14 | 11% |

100 | 950 | 98.96 | 1.04 | 1% |

r^{2} |
0.998526 | |||

total error | 2.28 | 23% | ||

%RSE | 15% | |||

Concentration | response | 1/x^{2} wt. |
absolute error | % error |

1 | 12 | 0.99 | 0.01 | 1% |

10 | 110 | 10.69 | 0.69 | 7% |

100 | 950 | 93.77 | 6.23 | 6% |

r^{2} |
0.995172 | |||

total error | 6.92 | 14% | ||

%RSE | 9% | |||

Concentration | response | Avg. RF | absolute error | % error |

1 | 12 | 1.11 | 0.11 | 11% |

10 | 110 | 10.16 | 0.16 | 2% |

100 | 950 | 87.72 | 12.28 | 12% |

%RSD | 11.62% | |||

total error | 12.55 | 25% | ||

%RSE | 17% |

Table 1: Example comparison of curve fit types

As you can see, the equal weighted curve fit gives the best r^{2} and lowest absolute error, but has the highest total % error, off by 67% on the lowest calibration point. Due to the large error it would actually fail the ±30% criteria for calibration accuracy. As the weighting changes to 1/x or 1/x^{2} the total absolute error and r^{2} get worse, but the total % error improves. For comparison the average RF calibration has the worst absolute error and 2^{nd} worst total % error.

What does this mean for your TO-15A analysis? Since the blank guidelines have dropped an order of magnitude from 200 pptv to 20 pptv, your error at the low end of your calibration could increase dramatically. In the above example the 1/x, 1/x^{2}, and average RF calibrations all meet the TO-15A calibration criteria, so which would be the best choice? Accepting average RF as the default unless it fails the %RSD criteria can lead to rather large errors, as seen above. The same is true if r^{2} is used as a judgment of best fit. Calibration models should not be based solely on passing any specific QC, including blank values, so looking at the error on any one calibration point isn’t appropriate. Minimizing the total % error would be a good measure of the best calibration, in which case the 1/x^{2} weighted calibration above would be the most appropriate despite having the worst r^{2} value. Percent relative standard error (%RSE) is another good measure of overall calibration accuracy, and the NELAC Institute has a brief document on how to calculate %RSE at http://nelac-institute.org/docs/comm/emmec/Calculating%20RSE.pdf. The calculation is shown below, where n= number of cal points, x_{i }is the true value of the cal point, x'_{i }is the calculated value for the cal point, and p=1 for average RF curves, 2 for linear curves, 3 for quadratic curves. In Table 1 you can see that the curve fits with lower total % error also have a lower %RSE.

Fig 1: Relative Standard error calculation

So, was I able to solve all of my blank issues through selection of the best calibration? As seen in the table below illustrating my problem compounds, I was not.

Compound | pptv | Curve fit |

n-Pentane | 22 | Linear, equal weight |

Ethanol | ND | Linear, 1/x |

Acetonitrile | 15 | Linear, 1/x |

Carbon disulfide | ND | Average RF |

Isopropyl alcohol | ND | Average RF |

Methylene chloride | 21 | Linear, equal weight |

Acetone | 12 | Linear, 1/x |

Hexane | 162 | Linear, 1/x |

Tertiary butanol | ND | Average RF |

Tetrahydrofuran (THF) | 28 | Linear, equal weight |

2-Butanone (MEK) | 30 | Linear, 1/x |

Toluene | 12 | Average RF |

4-Methyl-2-2pentanone (MIBK) | ND | Average RF |

Table 2: Selected TO-15A blank results humidified to 50% RH with boiled water. Calibration with lowest total % error used.

The 20 pptv cleanliness requirement is a challenging target to achieve and I imagine many labs are going to struggle with it, especially if your air lab shares space with heavy solvent users. Does this mean that running TO-15A is out of reach? Not necessarily. Remember that the TO methods are compendium methods meant to be used as a basis for laboratories to develop their own procedures, not as a strict list of requirements that must be followed. As long as regulatory limits and customer requirements are met, documenting some compounds as having blank or canister cleanliness limits above 20 pptv should be an acceptable deviance. Of course, labs will have to be prepared to defend this to their auditing agencies, and work to improve their blank results to be prepared for changing regulatory limits, customer requirements and potential competition from other labs, but a few outliers in your quest for <20 pptv cleanliness shouldn’t stop you from adopting TO-15A.