View source for Chemistry:Detection limit

{{Short description|The lowest signal or quantity observable with sufficient confidence above the background}}

The '''limit of detection''' ('''LOD''' or '''LoD''') is the lowest signal, or the lowest corresponding quantity to be determined (or extracted) from the signal, that can be observed with a sufficient degree of confidence or [[Statistical significance|statistical significance]]. However, the exact threshold (level of decision) used to decide when a signal [[Statistical significance|significantly emerges]] above the continuously fluctuating background noise remains arbitrary and is a matter of policy and often of debate among scientists, statisticians and regulators depending on the stakes in different fields.

==Significance in analytical chemistry==

In [[Chemistry:Analytical chemistry|analytical chemistry]], the '''detection limit''', '''lower limit of detection''', also termed '''LOD''' for '''limit of detection''' or '''analytical sensitivity''' (not to be confused with [[Sensitivity and specificity|statistical sensitivity]]), is the lowest quantity of a substance that can be distinguished from the absence of that substance (a ''[[Chemistry:Blank value|blank value]]'') with a stated [[Confidence interval|confidence level]] (generally 99%).<ref>{{GoldBookRef|title=detection limit|file=L03540}}</ref><ref>{{cite journal | doi = 10.1021/ac50064a004 | vauthors = MacDougall D, Crummett WB |title = Guidelines for Data Acquisition and Data Quality Evaluation in Environmental Chemistry | journal = Analytical Chemistry | volume = 52 | pages = 2242–49 | year =1980| issue = 14 |display-authors=0}}</ref><ref name=Saah1998>{{cite journal| author=Saah AJ, Hoover DR| title=[Sensitivity and specificity revisited: significance of the terms in analytic and diagnostic language]. | journal=Ann Dermatol Venereol | year= 1998 | volume= 125 | issue= 4 | pages= 291–4 | pmid=9747274 | doi= | pmc= | url=https://pubmed.ncbi.nlm.nih.gov/9747274  }}</ref> The detection limit is estimated from the [[Mean|mean]] of the blank, the [[Standard deviation|standard deviation]] of the blank, the slope ([[Sensitivity and specificity|analytical sensitivity]]) of the [[Probabilistic classification|calibration plot]] and a defined [[Confidence interval|confidence factor]] (e.g. 3.2 being the most accepted value for this arbitrary value).<ref name="cbr">{{cite journal | vauthors = Armbruster DA, Pry T | title = Limit of blank, limit of detection and limit of quantitation | journal = The Clinical Biochemist. Reviews | volume = 29 Suppl 1 | issue = 1 | pages = S49–S52 | date = August 2008 | pmid = 18852857 | pmc = 2556583 }}</ref>  Another consideration that affects the detection limit is the adequacy and the accuracy of the [[Model|model]] used to predict concentration from the raw analytical signal.<ref name="R_model" />

As a typical example, from a calibration plot following a [[Linear equation|linear equation]] taken here as the simplest possible model:

: <math>f(x) = ax + b </math>

where, <math>f(x)</math> corresponds to the signal measured (e.g. voltage, luminescence, energy, etc.), "{{mvar|b}}" the value in which the straight line cuts the ordinates axis, "{{mvar|a}}" the sensitivity of the system (i.e., the slope of the line, or the function relating the measured signal to the quantity to be determined) and "{{mvar|x}}" the value of the quantity (e.g. temperature, concentration, pH, etc.) to be determined from the signal <math>f(x)</math>,<ref>{{cite journal | vauthors = Quesada-González D, Stefani C, González I, de la Escosura-Muñiz A, Domingo N, Mutjé P, Merkoçi A | title = Signal enhancement on gold nanoparticle-based lateral flow tests using cellulose nanofibers | journal = Biosensors & Bioelectronics | volume = 141 | pages = 111407 | date = September 2019 | pmid = 31207571 | doi = 10.1016/j.bios.2019.111407 | hdl-access = free | hdl = 10261/201014 | s2cid = 190531742 | url = http://ddd.uab.cat/record/218082 }}</ref> the LOD for "{{mvar|x}}" is calculated as the "{{mvar|x}}" value in which <math>f(x)</math> equals to the average value of blanks "{{mvar|y}}" plus "{{mvar|t}}" times its standard deviation "{{mvar|s}}" (or, if zero, the standard deviation corresponding to the lowest value measured) where "{{mvar|t}}" is the chosen confidence value (e.g. for a confidence of 95% it can be considered {{mvar|t}} = 3.2, determined from the limit of blank).<ref name="cbr" />

Thus, in this didactic example:

<math display="block">\text{LOD for } x = \frac{\left(f(x)-b\right)} {a} = \frac{\left(y + 3.2s - b\right)} {a}</math>

There are a number of concepts derived from the detection limit that are commonly used.  These include the '''instrument detection limit''' ('''IDL'''), the '''method detection limit''' ('''MDL'''), the '''practical quantitation limit''' ('''PQL'''), and the '''limit of quantitation''' ('''LOQ''').  Even when the same terminology is used, there can be differences in the LOD according to nuances of what definition is used and what type of noise contributes to the measurement and calibration.<ref>{{citation | doi = 10.1021/ac00258a724 |author1=Long, Gary L. |author2=Winefordner, J. D. | title = Limit of detection: a closer look at the IUPAC definition | journal = Anal. Chem. | volume = 55 | issue =7 | pages = 712A–724A | year = 1983}}</ref>

The figure below illustrates the relationship between the blank, the ''limit of detection'' (LOD), and the ''limit of quantitation'' (LOQ) by showing the [[Physics:Probability density function|probability density function]] for [[Normal distribution|normally distributed]] measurements at the blank, at the LOD defined as 3 × [[Standard deviation|standard deviation]] of the blank, and at the LOQ defined as 10 × standard deviation of the blank. (The identical spread along Abscissa of these three functions is problematic.) For a signal at the LOD, the [[Type I and type II errors|alpha error]] (probability of [[False positives and false negatives|false positive]]) is small (1%).  However, the [[Type I and type II errors|beta error]] (probability of a [[False positives and false negatives|false negative]]) is 50% for a sample that has a concentration at the LOD (red line).  This means a sample could contain an impurity at the LOD, but there is a 50% chance that a measurement would give a result less than the LOD. At the LOQ (blue line), there is minimal chance of a false negative.

{{Wide image|LOD.png|625px|Illustration of the concept of detection limit and quantitation limit by showing the theoretical [[Normal distribution|normal distribution]]s associated with blank, detection limit (LOD), and quantitation limit (LOQ) level samples.}}

==Instrument detection limit==
Most [[Engineering:Scientific instrument|analytical instruments]] produce a signal even when a blank ([[Chemistry:Matrix (chemical analysis)|matrix]] without [[Chemistry:Analyte|analyte]]) is analyzed. This signal is referred to as the noise level. The instrument detection limit (IDL) is the analyte concentration that is required to produce a signal greater than three times the [[Standard deviation|standard deviation]] of the noise level. This may be practically measured by analyzing 8 or more standards at the estimated IDL then calculating the [[Standard deviation|standard deviation]] from the measured concentrations of those standards.

The detection limit (according to [[Organization:International Union of Pure and Applied Chemistry|IUPAC]]) is the smallest [[Chemistry:Concentration|concentration]], or the smallest absolute [[Physics:Amount of substance|amount]], of analyte that has a signal statistically significantly larger than the signal arising from the repeated measurements of a reagent blank.

Mathematically, the analyte's signal at the detection limit (<math>S_{dl}</math>) is given by:

: <math> S_{dl} = S_{reag} + 3 \ \sigma_{reag} </math>

where, <math>S_{reag}</math> is the mean value of the signal for a reagent blank measured multiple times, and <math>\sigma_{reag}</math> is the known [[Standard deviation|standard deviation]] for the reagent blank's signal.

Other approaches for defining the detection limit have also been developed. In [[Physics:Atomic absorption spectroscopy|atomic absorption spectrometry]] usually the detection limit is determined for a certain element by analyzing a diluted solution of this element and recording the corresponding [[Physics:Absorbance|absorbance]] at a given [[Wavelength|wavelength]]. The measurement is repeated 10 times. The 3σ of the recorded absorbance signal can be considered as the detection limit for the specific element under the experimental conditions: selected wavelength, type of flame or graphite oven, chemical matrix, presence of [[Physics:Interference (communication)|interfering]] substances, instrument... .

==Method detection limit==

Often there is more to the analytical method than just performing a reaction or submitting the analyte to direct analysis. Many analytical methods developed in the laboratory, especially these involving the use of a delicate scientific instrument, require a [[Chemistry:Sample preparation|sample preparation]], or a pretreatment of the samples prior to being analysed. For example, it might be necessary to heat a sample that is to be analyzed for a particular metal with the addition of acid first (digestion process). The sample may also be diluted or concentrated prior to analysis by means of a given instrument. Additional steps in an analysis method add additional opportunities for errors. Since detection limits are defined in terms of errors, this will naturally increase the measured detection limit. This "''global''" detection limit (including all the steps of the analysis method) is called the method detection limit (MDL). The practical way for determining the MDL is to analyze seven samples of concentration near the expected limit of detection. The [[Standard deviation|standard deviation]] is then determined. The one-sided [[Student's t-distribution]] is determined and multiplied versus the determined [[Standard deviation|standard deviation]]. For seven samples (with six degrees of freedom) the t value for a 99% [[Confidence interval|confidence level]] is 3.14. Rather than performing the complete analysis of seven identical samples, if the Instrument Detection Limit is known, the MDL may be estimated by multiplying the Instrument Detection Limit, or Lower Level of Detection, by the dilution prior to analyzing the sample solution with the instrument. This estimation, however, ignores any uncertainty that arises from performing the sample preparation and will therefore probably underestimate the true MDL.

==Limit of each model==

The problem of limit of detection, or limit of quantification, is encountered in all scientific disciplines. This explains the variety of definitions and the diversity of solutions developed to address the question. In the simplest cases as in nuclear and chemical measurements, definitions and approaches have probably received the clearer and the simplest solutions. In biochemical tests and in biological experiments depending on many more intricate factors, the situation involving false positive and false negative responses is more delicate to handle. In many other disciplines such as [[Chemistry:Geochemistry|geochemistry]], [[Earth:Seismology|seismology]], [[Astronomy:Astronomy|astronomy]], [[Physics:Dendrochronology|dendrochronology]], [[Earth:Climatology|climatology]], life sciences in general, and in many other fields impossible to enumerate extensively, the problem is wider and deals with [[Signal|signal]] extraction out of a [[Physics:Noise (spectral phenomenon)|background of noise]]. It involves complex statistical analysis procedures and therefore it also depends on the [[Model|model]]s used,<ref name="R_model">{{Cite web |title=R: "Detection" limit for each model |author= |work=search.r-project.org |date= |access-date=2022-01-04 |url= https://search.r-project.org/CRAN/refmans/bioOED/html/calculate_limit.html |language=English}}</ref> the hypotheses and the simplifications or approximations to be made to handle and manage [[Physics:Uncertainty|uncertainties]]. When the data resolution is poor and different signals overlap, different [[Deconvolution|deconvolution]] procedures are applied to extract parameters. The use of different [[Physics:Phenomenology|phenomenological]], mathematical and [[Statistical model|statistical model]]s may also complicate the exact mathematical definition of limit of detection and how it is calculated. This explains why it is difficult to find a general consensus about the precise mathematical definition of the notion of limit of detection. However, one thing is clear: it always requires a sufficient number of data (or accumulated data) and a rigorous statistical analysis to be statistically significant.

==Limit of quantification==

The limit of quantification (LoQ, or LOQ) is the lowest value of a signal (or concentration, activity, response...) that can be quantified with acceptable precision and accuracy.

The LoQ is the limit at which the difference between two distinct signals / values can be discerned with a reasonable certainty, ''i.e.'', when the signal is statistically different from the background. The LoQ may be drastically different between laboratories, so another detection limit is commonly used that is referred to as the '''Practical Quantification Limit''' (PQL).

== See also ==
* {{Annotated link|Background noise}}
* {{Annotated link|Background radiation}}
* {{Annotated link|Noise (electronics)|Electronic noise}}
* {{Annotated link|Noise (spectral phenomenon)}}
* {{Annotated link|Chemometrics}}
* {{Annotated link|Gamma spectroscopy#Calibration and background radiation}}
* {{Annotated link|Malmquist bias}}
* {{Annotated link|p-value}}
* {{Annotated link|Misuse of p-values}}
* {{Annotated link|Statistical significance}}

== References ==
{{Reflist}}

== Further reading ==
{{refbegin}}
* {{Cite journal| issn = 0017-9078| volume = 9| issue = 3| pages = 293–298| vauthors = Altshuler B, Pasternack B | title = Statistical measures of the lower limit of detection of a radioactivity counter| journal = Health Physics| access-date = 2022-01-03| date = 1963-03-01| doi = 10.1097/00004032-196303000-00005| pmid = 14040764| url = https://journals.lww.com/health-physics/Abstract/1963/03000/Statistical_Measures_of_the_Lower_Limit_of.5.aspx}}
* {{Cite journal| doi = 10.1021/ac60259a007| issn = 0003-2700| volume = 40| issue = 3| pages = 586–593| vauthors = Currie LA | title = Limits for qualitative detection and quantitative determination. Application to radiochemistry| journal = Analytical Chemistry| date = 1968}}
* {{Cite journal| doi = 10.1021/ac00258a001| issn = 0003-2700| volume = 55| issue = 7| pages = 712A–724A|  vauthors = Long GL, Winefordner JD | title = Limit of detection. A closer look at the IUPAC definition| journal = Analytical Chemistry| date = 1983}}
* {{cite journal | vauthors = Armbruster DA, Pry T | title = Limit of blank, limit of detection and limit of quantitation | journal = The Clinical Biochemist. Reviews | volume = 29 | issue = Suppl 1 | pages = S49–S52 | date = August 2008 | pmid = 18852857 | pmc = 2556583 }}
* {{Cite book| publisher = Publications Office| last = European Commission. Joint Research Centre.| title = Guidance document on the estimation of LOD and LOQ for measurements in the field of contaminants in feed and food.| location = Luxembourg| access-date = 2022-01-03| date = 2016| doi = 10.2787/8931| isbn = 9789279617683| url = https://data.europa.eu/doi/10.2787/8931}}
* {{Cite journal| publisher = Published by Beuth Verlag, a subsidiary of the DIN Group| title = DIN 32645 – Chemical analysis – Decision limit, detection limit and determination limit under repeatability conditions – Terms, methods, evaluation. Technical standard. Deutsches Institut für Normung, Berlin (DIN 32645:2008-11) {{!}} Via Engineering360| language = German| access-date = 2022-01-03| url = https://standards.globalspec.com/std/1114504/DIN%2032645| doi = 10.31030/1465413}}
{{refend}}

== External links ==
* <!-- <ref name="Geogebra_LoD"> -->{{Cite web |title=Limit of Detection – Interactive Java applet to illustrate some basic ideas of the limit of detection problem | vauthors = Evans WC |work=GeoGebra |date= 21 February 2019|access-date=2022-01-04 |url= https://www.geogebra.org/m/hugmwyvp}}
* {{Cite web |title=The R Language |author= |work=search.r-project.org |date= |access-date=2022-01-04 |url= https://search.r-project.org/CRAN/doc/html/index.html |language=English}}
* {{Cite journal| doi = 10.1144/geochem2011-106| issn = 1467-7873 | volume = 13| issue = 4| pages = 355–378| vauthors = Garrett RG | title = The 'rgr' package for the R Open Source statistical computing and graphics environment – a tool to support geochemical data interpretation| journal = Geochemistry: Exploration, Environment, Analysis| access-date = 2022-01-04| date = 2013-11-01| bibcode = 2013GEEA...13..355G | s2cid = 129059022 | url = https://geea.lyellcollection.org/content/13/4/355}}
* {{Cite web |title=R: "Detection" limit for each model |author= |work=search.r-project.org |date= |access-date=2022-01-04 |url= https://search.r-project.org/CRAN/refmans/bioOED/html/calculate_limit.html |language=English}}
* <!-- <ref name="R_Package_envalysis"> -->{{Cite web |title=R: Calibration data from DIN 32645 (Package envalysis version 0.5.1) |author=Deutsches Institut für Normung |work=search.r-project.org |date= |access-date=2022-01-04 |url= https://search.r-project.org/CRAN/refmans/envalysis/html/din32645.html |language=English}}
* [http://www.chemometry.com/Research/LOD.html Downloads of articles (a.o. harmonization of concepts by ISO and IUPAC) and an extensive list of references] {{Webarchive|url=https://web.archive.org/web/20090421153854/http://www.chemometry.com/Research/LOD.html |date=2009-04-21 }}

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