Received: 20 November 2002Accepted: 11 March 2003Published online: 23 April 2003© Springer-Verlag 2003

Abstract The limit of detection(LOD) is based strictly on an accept-ed probability for a false positive de-cision (type 1 error). For the neededstandard deviation of the blank, a ba-sic value for the calculation, threeexperimental different methods aredescribed, which should be applica-ble in every case. Table values forsimple but exact calculation, not on-

ly for single, but also for mean val-ues, and with probabilities for thetype 1 error of 1% and 5% are pre-sented. Demands for an acceptableuncertainty of the LOD and for thevalidation of the result are specified.

Keywords Limit of detection · Table values · Uncertainty · Validation

Accred Qual Assur (2003) 8:213–217DOI 10.1007/s00769-003-0626-8 GENERAL PAPER

Walter Huber Basic calculations about the limit of detectionand its optimal determination

Principle

The limit of detection (LOD) is a concentration, whoseexceeding by an analytical result indicates that a samplecontains the analyte [1, 2]. The definition is based on acalculation as a part of a convention to limit a false posi-tive decision (type 1 error). The principle, which hasbeen introduced by Kaiser [1] for signals, has been trans-ferred to concentrations and was widely accepted bymany authors and institutions. Different definitions, likethe Detection limit [3] , the Capability of detection [4]and the Identification limit (Erfassungsgrenze) [2] areconsidered together with a false positive and a false neg-ative decision (type 2 error) and are therefore only rele-vant for samples with known negative detection (a type 2error cannot exist for samples with positive detection).Definitions for a Limit of determination [1, 2, 5] are alsonot discussed and this concept seems almost unnecessary[6].

Experimental details

The detection of an analyte is based on the statement ofan enhanced signal yS of the sample (which may containthe analyte) against a signal yB of the blank (which is as-

sumed to be devoid of the analyte and should contain thesample matrix). Two cases are possible:

1. The signals of the blank are all equal (often zero).Most macro methods belong to this group. The reso-lution of the analytical system is insufficient to dem-onstrate the real existing random errors of the signals.Although these methods are normally not suited fordetermining low contents, a LOD can be defined. Itcorresponds to the lowest signal that can be distin-guished clearly from the signal of the blank.If only a part of the values are equal, the method isnot suitable. However, the calculation of a standarddeviation with those values is also not allowed, be-cause a normal distribution does not exist. An approx-imate value for LOD can be obtained by Methods 2and 3 (see later).

2. The signals of the blank show normal distributed ran-dom errors. This is a simplified assumption, becauseat very low contents an asymmetric distribution, likethe logarithmic normal distribution, is more realistic(in most cases the signals cannot be negative). How-ever, for an easier calculation and remembering thelow precision of the resulting LOD (see later) the pos-tulate of a normal distribution can be accepted.

W. Huber (✉)Weimarerstrasse 69,67071 Ludwigshafen, Germanye-mail: walterhuber@tiscali.de

In case 2 a problem exists about positive detection: is thereason for a higher signal of the sample the existence ofan analyte, or does the sample act like a second blankand the reason is an extreme high random value? Posi-tive detection would produce a type 1 error in this case,detection of a non-existing analyte. The probability ofthat error, which is in reality a false positive decision,must be limited, and therefore a minimum distance be-tween the two signals yB and yS must be defined. It isbased on the null hypothesis yS−yB=0, against the alter-native hypothesis yS−yB>0 (one-sided questioning).Therefore a prediction interval ycrit (=yS−yB) for the sig-nal(s) of a sample is calculated, assuming the worst case,that it works like a second blank. Values above ycrit areconsidered as positive detection, choosing the alternativehypothesis (Fig. 1).

For the calculation a knowledge of the standard devi-ation of the blank signals is needed. There are three pos-sibilities for its determination:

1. Direct determination by repeated analysis of theblank.

2. Repeated analysis of a sample with a very low contentof the analyte. It can be assumed that the variances ofsample and blank are not different. At very low con-tents the variances are constant [7, 8].

3. Preparing a calibration function with very low con-tents of the analyte (together with the matrix) and cal-culating the standard deviation of the residuals. Ho-mogeneity of the variances is here more critical thanin Method 2, because the distance between blank andhighest calibration value is probably larger.

Method 1 is the only direct one, but it can be very diffi-cult to get an appropriate blank.

Method 2 is the most practical one, because it can be ap-plied without any problems in routine work, producing

realistic values. The ratio between the content of thesample and calculated LOD must be limited and has tobe ≤5 [5].

Method 3 is time consuming and suffers from the sameproblem as Method 1, but has the advantage that the cali-bration function quite near to the LOD is validated. Be-cause the exact value of the LOD is not known in ad-vance, it is possible that a second calibration with lowercontents is necessary. The ratio between the highest cali-bration value and the blank should be ≤10, lowering thedemands from Method 2 [9]. The homogeneity of thevariances is crucial, because the variance of the blank isthe decisive value.

Calculation

For Method 1 the prediction value for the critical signalycrit of the blank with a defined probability 1−α for thetype 1 error and f=n−1 degrees of freedom is:

(1)

where sB is the standard deviation of the blank signals, tthe percentiles of the t-distribution, m the number of de-terminations of the sample and n those of the blank.

The critical signal ycrit must be converted into a con-centration by division with the slope b of the calibrationline and will result in LOD [10]:

(2)

LOD can be calculated directly with Method 2, replacing

by sx, the standard deviation of the results of a sam-ple with a low content of an analyte (with n determina-tions, instead of a blank) [11, 12]:

(3)

For Eqs. (2) and (3) the standard deviation is the

only variable. Therefore it is possible, to combine thefactors, with the result of a simplified equation:

(4)

The coverage factor kn is defined as

(5)

Values for kn are specified in Table 1. The acceptedprobability 1−α for the type 1 error can be chosen to be1% or 5%, either for a single result from the sample

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Fig. 1 Frequency distribution of the blank signals and definition ofthe critical signal ycrit

(m=1), or for the mean of two results (m=2), which iscommon in quantitative analysis.

The calculation for Method 3 is quite different. LODis also a prediction interval, the intersection of the pre-diction limit of the calibration line with the ordinate,with following transformation to a concentration by divi-sion with b [13]:

(6)

where sy.x is the standard deviation of the residuals and x–

the mean of all concentrations of the calibration samples.The result for LOD is in essence very similar to the other

methods, because the differing term , which

describes the hyperbolic curvature of the prediction lim-it, makes only a small part of the total value. The degreesof freedom are f=n−2.

Quality of results

LOD should be as low as possible and its uncertaintyshould be small. For both demands (under given analyti-cal conditions) many data are desirable. However, ob-taining data is time consuming and the question arises,how many are really necessary. The known recommen-dations for LOD are not satisfying in this importantpoint, either unnecessarily high without giving any rea-sons: ten [2, 4], or undefined [5]. USEPA uses seven de-terminations as a minimum, without discussion of theuncertainty.

Table 1 and its graphical demonstration (Fig. 2), helpwith decision making.

For single values five or, better, six determinations ap-pear imperative. For 1−α=1% the factor kn is about 4.The classical factor 3 [14] needs nine or ten determina-

tions, much more effort with little effect. For 1−α=5% sixdeterminations seem necessary and will produce kn≈2.

For the mean of two values kn is lower by about fac-tor . Five determinations can be sufficient, resultingin kn-values of about 3 and <2 for the two probabilities.

The uncertainty of the LOD is defined by the uncer-tainty of the standard deviation. The confidence intervalof a standard deviation is asymmetric, because the upperlimit theoretically can be infinite and the lower one can-not become negative. For this reason an upper and a low-er confidence limit must be calculated with two factorsκ, derived from the χ2-distribution [2]:

– Upper confidence limit UCL=LOD×κU– Lower confidence limit LCL=LOD×κL

The values for κU and κL with a level of confidence of95% are presented in Table 2. They are demonstrated inFig. 3 and show, that about six determinations are neces-sary, until the imprecision of the LOD seems acceptable.This coincides well with the result about the coveragefactor kn.

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Table 1 Coverage factor kn for the calculation of LOD from stan-dard deviations by Eq. (<equationcite>4</equationcite>): LOD=sxkn

Number of Coverage factor kndeterminations n

Single result Mean of two results

1−α=5% 1−α=1% 1−α=5% 1−α=1%

2 7.73 38.97 6.31 31.823 3.37 8.04 2.67 6.364 2.63 5.08 2.04 3.935 2.34 4.10 1.78 3.136 2.18 3.63 1.65 2.757 2.08 3.36 1.56 2.528 2.01 3.18 1.50 2.379 1.96 3.05 1.45 2.26

10 1.92 2.96 1.42 2.19 Fig. 2 Coverage factors kn from Table 1. The values with n=2have been cancelled for better readability. For discussion see text

Table 2 Factors κL and κU for the calculation of lower and upperconfidence limits of standard deviations and derived data. The lev-el of confidence is 95%. For Method 3 the used value for n has tobe diminished by 1

n κL κU

2 0.446 31.913 0.521 6.2854 0.566 3.7295 0.599 2.8746 0.624 2.4537 0.644 2.2028 0.661 2.0359 0.675 1.916

10 0.688 1.826

The confidence limits are useful for comparison ofdifferent LODs. The results show, too, that an LOD isnormally not a very precise value. Differences of two re-sults must be rather large if they are to be considered assignificant. The precise and correct test for a significantdifference of two LODs would be an F-test with varianc-es.

A weak point of Methods 1 and 2 is the validation.Although errors with the value of the slope b are not crit-ical (due to the low precision of LOD), a source of grosserrors could be bad linearity of the calibration line nearthe LOD. Method 3 allows a validation, but the lowestdata point must be quite near to LOD.

Practical considerations

Standardisation means to make a choice between possi-bilities, but this paper does not intend to propose one.The parameters, which have to been chosen in advance,are:

1. Accepted probability of the type 1 error.2. Validity of the result either for only one or for the

mean of two results (more than two could also be cal-culated for special cases).

3. Number of determinations, in order to find a goodcompromise between acceptable precision and neces-sary effort.

An LOD for only a single sample is almost useless.Therefore it is important, that an LOD has a generalmeaning for a group of samples. The problem is the defi-nition of substance groups with equal LOD. This prob-lem cannot be solved theoretically. It is the responsibility

of the analyst how far the range of validity for the differ-ent kinds of samples reaches.

The determination of the signal of the blank is deci-sive for the application of the LOD. It can be extremelydifficult to get representative blanks. Fortunately it ispossible for many methods to get the signal of a blankwithout any problems. All methods which are based onthe evaluation of peaks (like chromatographic methods)show a baseline, which can be interpreted as a signal ofthe blank, equal to zero. The determination of the stan-dard deviation of that "blank" is difficult because onlythe noise can be measured. Therefore Method 2 is theideal method for getting first the standard deviation andthen the LOD. However, problems can arise if a valida-tion is necessary.

Example

Application of Method 2 for compound x in a sampleyielded four results:

Limits of detection by different definitions (LOD=sxk4):

All LODs are >mx/5, and therefore Method 2 is applica-ble in every case.

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Fig. 3 Confidence limits UCLand LCL for the standard devi-ation s=1, calculated from κUand κL. The confidence level is95%, the number of the degreesof freedom f=n−1. The valuesfor n=2 have been cancelled forbetter readability

Single result,

Mean of two results,

The uncertainty for LOD1, single=380 ng/l is

The true value is situated with a probability of 95% be-tween 215 and 1417 ng/l. The uncertainty is high, result-ing from only four determinations.

Conclusions

Calculation of the LOD by multiplying any standard de-viations with a constant factor is not an advisable meth-

od, because the influence of the degrees of freedom andperhaps the homogeneity of the variances is neglected.However, uncritical use of a statistical calculation canalso be unsatisfactory, if the uncertainty of the result isnot taken into account. The paper describes optimalstrategies to meet the demands for precise values as wellas minimal efforts for that purpose.

The three methods described should allow one to de-termine statistical correct standard deviations and LODswith all samples. Method 3 is also capable of validatingthe result if satisfying blanks are available.

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References

1. Kaiser HZ (1965) Fresenius Z AnalChem 209:1–18

2. DIN 326453. IUPAC. Orange book: Compendium of

analytical nomenclature, 2nd edn.Blackwell Scientific, Oxford

4. Funk W, Dammann V, Donnevert G(1992) Qualitätssicherung in derAnalytischen Chemie. VCH, Weinheim

5. Fleming J, Albus H, Neidhart B, Wegscheider W (1997) Accred QualAssur 2:51–52

6. Huber W (2002) Accred Qual Assur7:256–257

7. Huber W (1997) Accred Qual Assur2:376

8. US Environmental Protection Agency(USEPA) (1997) Open File Report, pp 99–193

9. Vogelgesang J, Hädrich J (1998) Accred Qual Assur 3:243–255

10. DIN 32 645, Leerwertmethode11. Huber W (2001) GIT 45:1308–130912. The definition of a Method detection

limit MLD=sx t (USEPA) is similar, butneglects that there are two uncertainties(blank and sample), and not just one

13. DIN 32 645, Kalibriergeradenmethode14. A very long list of papers with this fac-

tor begins with (1) and ends with (5)