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We are monitoring Air Quality: essentially concentrations. We collect time series and we periodically calibrate analysers in order to control the process. Knowing the specifications or assessing it experimentally we have, for each analyser, some low threshold on the dynamic range (measurement extent) where data become meaningless because of the method variability, imperfections & limitations.

Concentrations are physical real positive quantities. Therefore real concentration cannot be negative: whatever distribution it has, it should be positive. Analyser are real, and when measuring very low concentration, it can sometimes produce negative or null concentration due to variability of the process (we assume there is no bias in the method).

In analytical chemistry question of Detection Limit often arises. It is a common practice to truncate part of the data series bellow some 'arbitrary' threshold (modelled or experimentally assessed). Of course this truncature bias data, which leads to higher expectation. But using meaningless data bias too. We know that dealing with DL are not the perfect solution, it is a filtering technique with advantages and drawbacks.

My question is the following:

When (1) null is an hard boundary that cannot be crossed (real concentration distribution is positive); (2) process variability close to the lower boundary of dynamic range of the instrument cannot be assumed: symmetric or concentration independent, is it reasonable to set a negative or null Detection Limit to truncate our data? If so, what hypothesis must hold?

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    $\begingroup$ It may help to know that the correct technical term for this process is censoring (not truncation) and that suitable statistical techniques exist to avoid bias. See censoring for more. The issue of how to set a detection limit is complicated (I know of over 20 different definitions of a detection limit!) and entire books have been devoted to addressing the questions you ask. $\endgroup$
    – whuber
    Commented Feb 12, 2014 at 17:33
  • $\begingroup$ I am unclear what you mean by hypothesis? In signal processing it is not considered unreasonable at all to remove, ignore or (depending on the problem) set to zero, data points that are considered 'artifact'. Setting data below this threshold to zero seems reasonable in this context. $\endgroup$
    – BGreene
    Commented Feb 13, 2014 at 9:39
  • $\begingroup$ @whuber Could provide us some references to those dedicated books. Thank you. $\endgroup$
    – jlandercy
    Commented Feb 13, 2014 at 19:52
  • $\begingroup$ Lloyd A. Currie, ed., Detection in Analytical Chemistry: Importance, Theory, and Practice. American Chemical Society, Washington, DC, 1986. It's supposed to be available on the NIST website but currently the site is down. $\endgroup$
    – whuber
    Commented Feb 13, 2014 at 19:57
  • $\begingroup$ @whuber: unfortunately while NIST is online again, the link is broken. (and at ACS it's behind a paywall, of course) $\endgroup$ Commented Jul 20, 2019 at 15:41

2 Answers 2

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I strongly believe that back-calculated "analyte contents" (i.e., concentrations, quantities or amounts) should not be censored, truncated, replaced or altered: noisy estimates of the analyte content are not the same as physically constrained true analyte contents. Below is my short list of detection limit publications, etc. Reference 6 is the single most important paper on the topic, in my judgement, and I have not yet obtained references 25 and 26: I look forward to reading them as soon as I can get to my university's actual library.

  1. H. Kaiser, “Die Berechnung der Nachweisempfindlichkeit”, Spectrochim. Acta 3 (1947) 40-67.
  2. B. Altshuler, B. Pasternak, “Statistical Measures of the Lower Limit of Detection of a Radioactivity Counter”, Health Phys. 9 (1963) 293-298.
  3. H. Kaiser, “Zum Problem der Nachweisgrenze”, Z. Anal. Chem. 209 (1965) 1-18.
  4. H. Kaiser, “Zur Definition der Nachweisgrenze, der Garantiegrenze und der dabei benutzen Begriffe: Fragen und Ergebnisse der Diskussion”, Z. Anal. Chem. 216 (1966) 80-94.
  5. P.B. Adams, W.O. Passmore, D.E. Campbell, “Symposium on Trace Characterization - Chemical and Physical”, National Bureau of Standards, Paper 14 (1966).
  6. L.A. Currie, “Limits for Qualitative and Quantitative Determination - Application to Radiochemistry”, Anal. Chem. 40 (1968) 586-593.
  7. H. Kaiser, “Quantitation in Elemental Analysis”, Anal. Chem. 42 (1970) 24A-41A.
  8. H. Kaiser, “Part II Quantitation in Elemental Analysis”, Anal. Chem. 42 (1970) 26A-59A.
  9. A. Hubaux, G. Vos, “Decision and detection limits for linear calibration curves”, Anal. Chem. 42 (1970) 849-855.
  10. P.W.J.M. Boumans, “A tutorial review of some elementary concepts in the statistical evaluation of trace element measurements”, Spectrochim. Acta Part B 33B (1978) 625-634.
  11. J.A. Glaser, D.L. Foerst, G.D. McKee, S.A. Quave, W.L. Budde, “Trace analyses for wastewaters”, Environ. Sci. Tech. 15 (1981) 1426-1435.
  12. C.A. Clayton, J.W. Hines, T.D. Hartwell, P.M. Burrows, “Demonstration of a Technique for Estimating Detection Limits with Specified Assurance Probabilities”, Research Triangle Institute Technical Report 2757/05-01F, EPA Contract No. 68-01-6826; Research Triangle Institute, Research Triangle Park, NC, 1986.
  13. C.A. Clayton, J.W. Hines, P.D. Elkins, “Detection limits with specified assurance probabilities”, Anal. Chem. 59 (1987) 2506-2514.
  14. L.A. Currie, “Detection: Overview of Historical, Societal, and Technical Issues”, Chapter 1 in Detection in Analytical Chemistry : importance, theory and practice, ACS Symposium Series, ACS, Washington, DC, 1987 (published early 1988), 1- 62. (Currie was editor for this volume.)
  15. P.W.J.M. Boumans, “Detection limits and spectral interferences in atomic emission spectrometry”, Anal. Chem. 66 (1994) 459A-467A.
  16. R. Ferrús, M.R. Egea, “Limit of discrimination, limit of detection and sensitivity in analytical systems”, Anal. Chim. Acta 287 (1994) 119-145.
  17. D.M. Rocke, S. Lorenzato, “A Two-Component Model for Measurement Error in Analytical Chemistry”, Technometrics 37 (1995) 176-184.
  18. L.A. Currie, “Detection: International update, and some emerging di-lemmas involving calibration, the blank, and multiple detection decisions”, Chemom. Intell. Lab. Syst. 37 (1997) 151-181.
  19. D. Coleman, J. Auses, N. Grams, “Regulation – From an industry perspective or Relationships between detection limits, quantitation limits, and significant digits”, Chemom. Intell. Lab. Syst. 37 (1997) 71-80.
  20. J. Mocak, A. M. Bond, S. Mitchell, G. Scollary, for IUPAC, “A Statistical Overview of Standard (IUPAC and ACS) and New Procedures for Determining the Limits of Detection and Quantification: Application to Voltammetric and Stripping Techniques”, Pure Appl. Chem. 69 (1997) 297-328. IUPAC ©1997.
  21. J. Vogelgesang, J. Hädrich, “Limits of detection, identification and determination: a statistical approach for practitioners”, Accred Qual Assur 3 (1998) 242-255.
  22. M. Thompson, “Do we really need detection limits?”, Analyst 123 (1998) 405-407.
  23. L.A. Currie, “Detection and quantification limits: origins and historical overview”, Anal. Chim. Acta 391 (1999) 127-134.
  24. L.A. Currie, “Detection and quantification limits: origins and historical overview”, Journal of Radioanalytical and Nuclear Chemistry 245 (2000) 145-156.
  25. W. Huber, “Do we need a limit of determination?”, Accred Qual Assur 7 (2002) 256-257.
  26. W. Huber, “Basic calculations about the limit of detection and its optimal determination”, Accred Qual Assur 8 (2003) 213-217.
  27. M.C. Ortiz, L.A. Sarabia, A. Herrero, M. S. Sánchez, M.B. Sanz, M.E. Rueda, D. Giménez, M.E. Meléndez, “Capability of detection of an analytical method evaluating false positive and false negative (ISO 11843) with partial least squares”, Chemom. Intell. Lab. Syst. 69 (2003) 21-33.
  28. K. Linnet, M. Kondratovich, “Partly Nonparametric Approach for Determining the Limit of Detection”, Clinical Chemistry 50 (2004) 732-740.
  29. I. Lavagnini, F. Magno, “A statistical overview on univariate calibration, inverse regression, and detection limits: application to gas chromatography/mass spectrometry technique”, Mass Spectrometry Reviews 26 (2007) 1-18.
  30. C.G. Fraga, A.M. Melville, B.W. Wright, “ROC-curve approach for determining the detection limit of a field chemical sensor”, Analyst 132 (2007) 230-236.
  31. D. MacDougall, W.B. Crummett, ACS Committee on Environmental Improvement, “Guidelines for Data Acquisition and Data Quality Evaluation in Environmental Chemistry”, Anal. Chem. 52 (1980) 2242-2249.
  32. G.L. Long, J.D. Winefordner, “Limit of detection: a closer look at the IUPAC definition” Anal. Chem. 55 (1983) 712A-724A.
  33. J. Foley, J. Dorsey, “Clarification of the Limit of Detection in Chromatography”, Chromatographia 18 (1984) 503-511.
  34. M.D. Wilson, D.M. Rocke, B. Durbin, H.D. Kahn, “Detection limits and goodness-of-fit measures for the two-component model of chemical analytical error”, Anal. Chim. Acta 509 (2004) 197-208.
  35. E. Voigtman, “Limits of detection and decision. Part 1”, Spectrochim. Acta Part B 63 (2008) 115-128.
  36. E. Voigtman, “Limits of detection and decision. Part 2”, Spectrochim. Acta Part B 63 (2008) 129-141.
  37. E. Voigtman, “Limits of detection and decision. Part 3”, Spectrochim. Acta Part B 63 (2008) 142-153.
  38. E. Voigtman, “Limits of detection and decision. Part 4”, Spectrochim. Acta Part B 63 (2008) 154-165.
  39. E. Desimoni, B. Brunetti, “About estimating the limit of detection of heteroscedastic analytical systems”, Anal. Chim. Acta 655 (2009) 30-37.
  40. L. Brüggemann, P. Morgenstern, R. Wennrich, “Comparison of international standards concerning the capability of detection for analytical methods”, Accred Qual Assur 15 (2010) 99-104.
  41. M.C. Ortiz, L.A. Sarabia, M.S. Sánchez, “Tutorial on evaluation of type I and type II errors in chemical analyses: From the analytical detection to authentication of products and process control”, Anal. Chim. Acta 674 (2010) 123-142.
  42. E. Voigtman, K.T. Abraham, “Statistical behavior of ten million experimental detection limits”, Spectrochim. Acta Part B 66 (2011) 105-113.
  43. E. Voigtman, K.T. Abraham, “True detection limits in an experimental linearly heteroscedastic system. Part 1”, Spectrochim. Acta Part B 66 (2011) 822-827.
  44. E. Voigtman, K.T. Abraham, “True detection limits in an experimental linearly heteroscedastic system. Part 2”, Spectrochim. Acta Part B 66 (2011) 828-833.
  45. L.V. Rajaković, D.D. Marković, V.N. Rajaković-Ognjanović, D.Z. Antanasijević, “Review: The approaches for estimation of limit of detection for ICP-MS analysis of arsenic”, Talanta 102 (2012) 79-87.
  46. M. Belter, A. Sajnog, D. Barałkiewicz, “Over a century of detection and quantification capabilities in analytical chemistry – Historical overview and trends”, Talanta 129 (2014) 606-616.
  47. A. Wysoczanski, E. Voigtman, “Receiver operating characteristic - curve limits of detection”, Spectrochim. Acta Part B 100 (2014) 70-77.
  48. D. Badocco, I. Lavagnini, A. Mondin, A. Tapparo, P. Pastore, “Limit of detection in the presence of instrumental and non-instrumental errors: study of the possible sources of error and application to the analysis of 41 elements at trace levels by ICP-MS technique”, Spectrochim. Acta Part B 107 (2015) 178-184.
  49. D. Coleman, L. Vanatta, “Detection limits: editorial comments and introduction”, Am Lab 39(12) 2007 24-25.
  50. D. Coleman, L. Vanatta, “Part 27 – Receiver Operating Characteristic (ROC) Curves”, Am Lab 39(16) 2007 28-29.
  51. D. Coleman, L. Vanatta, “Part 28 – Statistically Derived Detection Limits”, Am Lab 39(20) 11-12/2007, 24-27.
  52. D. Coleman, L. Vanatta, “Part 29 – Statistically Derived Detection Limits (continued)”, Am Lab 40(3) 2/2008, 44-46.
  53. D. Coleman, L. Vanatta, “Part 30 – Statistically Derived Detection Limits (concluded)”, Am Lab 40(12) 6-7/2008, 34-37.
  54. D. Coleman, L. Vanatta, “Part 34 – Detection-Limit Summary”, Am Lab 41(6) 5/2009, 50-52.
  55. E. Bernal, “Limit of Detection and Limit of Quantification Determination in Gas Chromatography”, Advances in Gas Chromatography, 2014. http://dx.doi.org/10.5772/57341
  56. R.J.N. Bettencourt da Salva, “Spreadsheet for designing valid least-squares calibrations: A tutorial”, Talanta 148 (2016) 177-190.
  57. H. Evard, A. Kruve, I. Leito, “Tutorial on estimating the limit of detection using LC-MS analysis, Part I: theoretical review”, Anal. Chim. Acta 942 (2016) 23-39.
  58. H. Evard, A. Kruve, I. Leito, “Tutorial on estimating the limit of detection using LC-MS analysis, Part II: practical aspects”, Anal. Chim. Acta 942 (2016) 40-49.
  59. J. Jiménez-Chacón, M. Alvarez-Prieto, “Modelling uncertainty in a concentration range”, Accred Qual Assur 14 (2009) 15-27.
  60. J. Jiménez-Chacón, M. Alvarez-Prieto, “An approach to detection capabilities estimation of analytical procedures based on measurement uncertainty”, Accred Qual Assur 15 (2010) 19-28.
  61. M. Thompson, B.J. Coles, “Use of the ‘characteristic function’ for modelling repeatability precision”, Accred Qual Assur 16 (2011) 13-19.
  62. M. Thompson, “Uncertainty functions, a compact way of summarising or specifying the behavior of analytical systems”, Trends in Analytical Chemistry 30 (2011) 1168-1175.
  63. M. Thompson, S.L.R. Ellison, “Towards an uncertainty paradigm of detection capability”, Anal. Methods 5 (2013) 5857-5861.
  64. J. Fonollosa, A. Vergara, R. Huerta, S. Marco, “Estimation of the limit of detection using information theory measures”, Anal. Chim. Acta 810 (2014) 1-9.
  65. M.H. Ramsey, S.L.R. Ellison, “Uncertainty factor: an alternative way to express measurement uncertainty in chemical measurement”, Accred Qual Assur 20 (2015) 153-155.
  66. C.A. Holstein, M. Griffin, J. Hong, P.D. Sampson, “Statistical Method for Determining and Comparing Limits of Detection of Bioassays”, Anal. Chem. 87 (2015) 9795-9801.
  67. I. Janiga, J. Mocak, I. Garaj, “Comparison of Minimum Detectable Concentration with the IUPAC Detection Limit”, Measurement Science Review 8(5) (2008) 108-110.
  68. J. Tellinghuisen,“Calibration: Detection, Quantification, and Confidence Limits Are (Almost) Exact When the Data Variance Function is Known”, Anal. Chem., in press.

Books

  1. C. Liteanu, I. Rica, Statistical Theory and Methodology of Trace Analysis, Ellis Horwood, Chichester, UK ©1980.

  2. R.D. Gibbons, D.E. Coleman, Statistical Methods for Detection and Quantification of Environmental Contamination, John Wiley & Sons, NY ©2001.

  3. K. Danzer, Analytical Chemistry: Theoretical and Metrological Fundamentals, Springer-Verlag Berlin Heidelberg, Germany ©2007.

  4. E. Voigtman, Limits of Detection in Chemical Analysis, John Wiley & Sons, NY ©2017.

Book chapters

  1. L.A. Currie, “Detection: Overview of Historical, Societal, and Technical Issues”, Chap. 1 in Detection in Analytical Chemistry : importance, theory and practice, ACS Symposium Series, ACS, Washington, DC, 1987 (published 1988), 1-62.

  2. H. van der Voet, “Detection Limits. Encyclopedia of Environmetrics”, Vol. 1, pp. 504-515. A.H. El-Shaarawi, W.W. Piegorsch (eds.), Wiley, Chichester, UK, ©2002.

Official guidance publications

  1. IUPAC, “Nomenclature, symbols, units and their usage in spectrochemical analysis – II. Data interpretation, Analytical chemistry division”, Spectrochim. Acta Part B 33 (1978) 241-245.

  2. ISO 3534-1, “Statistics-Vocabulary and symbols-Part 1: Probability and general statistical terms”, ISO, Genève, 1993.

  3. ISO, “Guide to the Expression of Uncertainty in Measurement” ISO, Geneva,1993. (ISBN 92-67-10188-9) (Reprinted 1995) Reissued as ISO Guide 98–3, 2008, also available as JCGM 100:2008 from http://www.bipm.org

  4. L.A. Currie, W. Horwitz, for IUPAC, “IUPAC recommendations for defining and measuring detection and quantification limits”, Analusis Magazine 22 (1994) M24-M26.

  5. L.A. Currie, G. Svehla, for IUPAC, “Nomenclature for the presentation of results of chemical analysis”, Pure Appl. Chem. 66 (1994) 595-608. IUPAC © 1994.

  6. L.A. Currie, for IUPAC, “Nomenclature in evaluation of analytical methods including detection and quantification capabilities”, Pure Appl. Chem. 67 (1995) 1699-1723. IUPAC © 1995.

  7. M. Thompson, R. Wood, for IUPAC, “Harmonized Guidelines for Internal Quality Control in Analytical Chemistry Laboratories”, Pure Appl. Chem. 67 (1995) 649-666. IUPAC ©1995.

  8. J. Mocak, A. M. Bond, S. Mitchell, G. Scollary, for IUPAC, “A Statistical Overview of Standard (IUPAC and ACS) and New Procedures for Determining the Limits of Detection and Quantification: Application to Voltammetric and Stripping Techniques”, Pure Appl. Chem. 69 (1997) 297-328. IUPAC ©1997.

  9. ISO 11843-1, “Capability of Detection - Part 1: Terms and definitions” ISO, Genève,1997.

  10. K. Danzer, L.A. Currie, for IUPAC, “Guidelines for Calibration in Analytical Chemistry”, Pure Appl. Chem. 70 (1998) 993- 1014. IUPAC ©1998.

  11. ISO 11843-2, “Capability of Detection - Part 2: Methodology in the linear calibration case” ISO, Genève, 2000.

  12. Commission Decision 2002/657/EC, implementing Council Directive 96/23/EC concerning the performance of analytical methods and the interpretation of results.

  13. M. Thompson, S.L.R. Ellison, R. Wood, Harmonized Guidelines for Single Laboratory Validation of Methods of Analysis (IUPAC Technical Report), Pure Appl. Chem. 74 (2002) 835–855.

  14. NCCLS, EP17 P 1st Ed., “Protocols for Determination of Limits of Detection and Limits of Quantitation”, (2004). Also known as CLSI EP17 A 1st Ed. (2004). Superceded by CLSI EP17 A2 2nd Ed. (2012).

  15. Revised Assessment of Detection and Quantitation Approaches, EPA, Washington, DC 20460, EPA-821-B-04-005, 2004.

  16. Multi-Agency Radiological Laboratory Analytical Protocols (MARLAP) Manual Volume III, Section 20A.3.1, 20-54 - 20-58, 2004 (United States agencies in MARLAP are EPA, DoD, DoE, DHS, NRC, FDA, USGS and NIST. This lengthy document is available as a PDF online.)

  17. ISO/IEC 3534, “Statistics – Vocabulary and Symbols – Part 1: General Statistical Terms and Terms Used in Probability” Genève, 2006.

  18. ISO 11843-5, “Capability of Detection - Part 5: Methodology in the linear and non-linear calibration cases” ISO, Genève, 2008.

  19. BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, OIML, “International vocabulary of metrology – Basic and general concepts and associated terms (VIM), 3rd Ed., 2008. With minor corrections also available as JCGM 200: 2012 from http://www.bipm.org/vim.

  20. DIN 32645:2008-11, Chemische Analytik-Nachweis-, Erfassungsund Bestimmungsgrenze, Normenausschuss Materialprüfung im DIN.

  21. EURACHEM/CITAC Guide, “Quantifying Uncertainty in Analytical Measurement”, 3rd Ed., 2012.

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  • $\begingroup$ That's a short list, thank you for sharing. $\endgroup$
    – jlandercy
    Commented May 28, 2019 at 17:07
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    $\begingroup$ Great list! As you happily mix German language references with English language references: the Danzer book (3) is available in German as well as in English. $\endgroup$ Commented Jul 20, 2019 at 15:36
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I will offer a pragmatic solution that I believe has minimal potential for introducing bias in situations where the numbers of such measurements are modest relative to the overall numbers. I will assume that the instrument has a capacity for measurement which is understood and that there is some lower detection limit (LDL) below which accuracy is low. In some of my work with medical lab values of enzyme activity that value might be 2 or 3. What I do is take all the values below that limit and assign them to the midpoint between 0 and the LDL. I think using the word "null" may be misleading, because in some statistical packages it has a different meaning than "0.0". Admittedly, this crude practice is not endorsed by authorities (Helsel and Curie) and you should seek out more complete analyses if attempting to calibrate a method or have a substantial number of values in teh out-of-range regions.

You can get a more carefully reasoned (and illustrated) answer by @cbeleites here: It may be helpful to separate the problems where calibration is not the underlying task from those where that is the scientific concern. I also see that @whuber cites Helsel as the authority and that Helsel maintains a very helpful website.

I've been told by a statistician of good repute that a similar practice is sometimes needed when the upper values are very sparse if one is encountering numerical stability problems, and in some cases he has chosen to lump values above the 99.9th percentile at that value.

One can also regress on order statistics which is one of the methods offered in the NADA package in R.

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  • $\begingroup$ Thanks for answering, this might have lead to a Necromancer Badge isn't it? $\endgroup$
    – jlandercy
    Commented Nov 13, 2014 at 17:50
  • $\begingroup$ I'm in the process of improving my knowledge in the area and was surprised your question had no answer. After doing some more searching I've discovered that there is quite a bit in CV that already addresses the question, some of it citing the authors of the NADA R-package as authorities. $\endgroup$
    – DWin
    Commented Nov 13, 2014 at 17:52
  • $\begingroup$ Assigning fixed values between 0 and the LDL is specifically what Helsel says not to do--and he is very firm about that. I'm a little less dogmatic than he appears to be, because I have seen that this simple procedure can be ok for (a) some graphs, (b) some quick-and-dirty analyses, and (c) when sensitivity analyses show there will be little impact. Regardless, none of this seems to address the original question about how to set an appropriate level for censoring the data. $\endgroup$
    – whuber
    Commented Nov 13, 2014 at 18:21
  • $\begingroup$ Agreed, in part at least. I didn't think the question was how to set a limit but rather what to do with values below the limit. I see that my practice has been haphazard. I'm studying the material in your note and the citations to Curie's work. Have not been able to find his book on the NIST website, although inexpensive versions are available in Amaxon. $\endgroup$
    – DWin
    Commented Nov 13, 2014 at 18:40

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