This is related to noise (instrumental and/or of the method), gaussian distribution and confidence interval around the zero. I prefer LOD=1.96*SD to obtain a 95% confidence interval for LOD.

3*SD is related to a confidence interval at 99.7%.

This means that the experiment was performed thrice and data beyond 3 sigma limit can not be the part of confidence limit. 

Dear Mohammed 

I totally agree with Guiseppe, it's about C.I 

99.7% of the observations in a normal distribution (bell shaped) lie with +/-3 standard deviations of the mean just repeating the same thing as Guiseppe.

Mohammed,

The two-sigma or three-sigma confidence intervals are used by people who believe that their data follow - more or less - the normal distribution.

What to do, if one has doubts about the normality of his/her data? Use Tukey  fences instead. They are based on the 1st quartile (Q1) and 3rd quartile (Q2) of the distribution of the analysed data  : 

       [Q1 - 1.5*IQR,    Q3 + 1.5*IQR]

where IQR = Q3 - Q1,  is called the Inter-Quartile range.

Observations beyond the Tukey fences are suspected to be outliers, that is coming from  a different population  as the analysed one.

Tukey's fences are used in graphs called box-and-whisker plots, (or simply box-plots) proposed also by J, Tukey and popular in exploratory data analysis. They show how the distribution of the data looks like, in particular, looking at a box-plot one may judge if the sample distribution is likely to be a normal (i.e. Gaussian) distribution.

Both classical two-sigma confidence intervals and Tukey;s fences designate only some limits beyond which  data points  are very unlikely to appear. 

One may have much fun (and information about the shape of the data) drawing the boxplots with the whiskers and Tukey's fences.

Have fun with visualisation of your data

Anna

One simply wants his/her model to detect at best a  + or - 3-sigma deviation from the mean when the observations are normal.

LOD pretty well covered above, but it seems people have missed the bit about limit of quantification. Before I cover that I just want to check that the SD you mention is the SD for repetitions for blank samples? The standard  approach is to work from the blanks for determining LOD. Other approaches exist to estimate it from standard errors within a calibration, but if blanks are possible (in many real world calibrations they aren't, but in most controlled experiments they are) then use blanks.

I've not encountered defining LOQ by multiplying by SD of blanks by 10 before, but it is always set more stringent. The usual way of quantifying LOQ is to test a range of samples from the concentration that matches the LOD (minimum detectable quantity) up to e.g. 10x this concentration. You then identify what concentration at which the result is significantly different to your LOD (using the variation in the reps about each set of data). LOQ is the lowest concentration at which you can definitively be sure that the value measured is significantly above the LOD.

LOD indicates the baseline for qualitative confirmation (i.e. analyte was detectable/not detectable).

LOQ indicates the baseline for quantitative judgement on the sample (i.e. we can be confident that the sample was approximately this concentrated).

What "limits" to use depends on two things; one is the probability that the parameter does not lie within the limits, and the other is on the value of that large a spread.  In addition, there is the problem  of the posterior distribution of the parameter,  which is not in general normal.    The procedures you suggest, and most of what is suggested by the other responders, are for probabilities before the action is taken, and assuming normality.  If the sample is sufficiently large, and the problem sufficiently regular, normality can be a good approximation, but this is not always so.

Without knowing the information asked for in the first sentence, I cannot provide what I  consider to be the appropriate procedure.   Classical statistics fails to consider the importance of close limits.

According to (Pukelsheim,1994), the "Three Sigma Rule" has been proved for random variables with a Lebesgue (continuous) unimodal density with finite variance, as a special case derived from the Visochanskij-Petunin inequality (1980).

(Visochanskij and Petunin,1980) establishes the following upper bound of (4/81)

This is a fine argument for the probabilistic result for a unimodal distribution.

But the rule is used to get intervals for the parameter after after seeing the observations.  This is a totally different situation.  It could in some cases be used in Bayesian situations.  And their result only gives a .05 probability, which is much              more than the usual assumed .0026 probability from the normal distribution.               In any case, the decision arguments are missing from a general rule of this type.

This is related to noise , So (3*SD) is related to a confidence interval at 99%.

Dear Mohammed, 

Plese see if this is usefull to you.

Limit of Blank, Limit of Detection and Limit of Quantitation

David A Armbruster1,* and Terry Pry2

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This article has been cited by other articles in PMC.

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Summary

Limit of Blank (LoB), Limit of Detection (LoD), and Limit of Quantitation (LoQ) are terms used to describe the smallest concentration of a measurand that can be reliably measured by an analytical procedure.

LoB is the highest apparent analyte concentration expected to be found when replicates of a blank sample containing no analyte are tested.

LoB = meanblank + 1.645(SDblank)

LoD is the lowest analyte concentration likely to be reliably distinguished from the LoB and at which detection is feasible. LoD is determined by utilising both the measured LoB and test replicates of a sample known to contain a low concentration of analyte.

LoD = LoB + 1.645(SD low concentration sample)

LoQ is the lowest concentration at which the analyte can not only be reliably detected but at which some predefined goals for bias and imprecision are met. The LoQ may be equivalent to the LoD or it could be at a much higher concentration.

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Introduction

Sensitivity, Analytical Sensitivity, Functional Sensitivity, Lower Limit of Detection, LoB, LoD, and LoQ are terms used to describe the smallest concentration of a measurand that can be reliably measured by an analytical procedure. There has often been a lack of agreement within the clinical laboratory field as to the terminology best suited to describe this parameter. Likewise, there have been various methods for estimating it. Clinical laboratorians have perhaps been lax in dealing with this analytical issue because, in many cases, the ability of a laboratory test to detect a very small amount of measurand is not clinically significant. For example, the medical decision levels for glucose and cholesterol are so far above the lower analytical limits of these tests that it is highly unlikely that clinical action will depend on measurements of these analytes at such low concentration. Nevertheless, it is important to fully characterise the analytical performance of every clinical laboratory test in order to understand its capability and limitations, and to ensure that it is “fit for purpose.” Moreover, defining the limits of an assay at low concentration is directly related to its dynamic range, or analytical measurement range.

To provide a standard method for determining LoB, LoD and LoQ, Clinical and Laboratory Standards Institute (CLSI) has published the guideline EP17, Protocols for Determination of Limits of Detection and Limits of Quantitation.1 The Figure taken from their document illustrates the distinction of LoB, LoD and LoQ values. Typically, LoQ will be found at a higher concentration than LoD, but how much higher depends on the specifications for bias and imprecision used to define it. ‘Analytical sensitivity’ defined as the slope of the calibration curve is sometimes used as a synonym for LoD. However, because LoD may well reside at some concentration below the linear range of an assay, where the calibration curve is no longer valid, this usage should be avoided.

Figure

Relationship between LoB, LoD and LoQ. The solid line defines the LoB and represents the distribution of results for a blank specimen. As modern analysers seldom report results of less than zero, the frequency of “zero” results is artificially ...

All of these parameters are related but have distinct definitions and should not be confused. The intent is to define the smallest concentration of analyte that can be detected with no guarantee about the bias or imprecision of the result by an assay, the concentration at which quantitation as defined by bias and precision goals is feasible, and finally the concentration at which the analyte can be quantitated with a linear response.

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Limit of Blank

EP17 defines LoB as the highest apparent analyte concentration expected to be found when replicates of a sample containing no analyte are tested.1 Note that although the samples tested to define LoB are devoid of analyte, a blank (zero) sample can produce an analytical signal that might otherwise be consistent with a low concentration of analyte.

LoB is estimated by measuring replicates of a blank sample and calculating the mean result and the standard deviation (SD).

LoB=meanblank+1.645(SDblank)

Assuming a Gaussian distribution of the raw analytical signals from blank samples, the LoB represents 95% of the observed values. (Note: Typical modern clinical analysers don’t routinely display the actual analytical signal but automatically convert it to a concentration value. The raw analytical signal is preferable for establishing LoB as analysers may report all signal values below a certain fixed limit as “zero concentration”).

The remaining 5% of blank values represent a response that could actually be produced by a sample containing a very low concentration of analyte. Statistically, this false positivity is known as a Type I (or α) error (Figure). Conversely, while a sample that actually contains analyte is expected to exceed the LoB, it must also be recognised that a proportion of very low concentration samples will produce responses less than the LoB, representing Type II (or β) error (Figure). Thus, EP17 acknowledges that the overlap of the analytical responses of blank and low concentration is a statistical reality, and uses LoB as a reasonable starting point for estimating the LoD.

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Limit of Detection

Although reagent package inserts may state that an assay has a dynamic range that extends from zero concentration to some upper limit, typically an assay is simply not capable of accurately measuring analyte concentrations down to zero. Sufficient analyte concentration must be present to produce an analytical signal that can reliably be distinguished from “analytical noise,” the signal produced in the absence of analyte.

LoD is the lowest analyte concentration likely to be reliably distinguished from the LoB and at which detection is feasible. It is therefore greater than LoB (Figure). A traditional and typical approach to estimate LoD consists of measuring replicates, usually n=20, of a zero calibrator or blank sample, determining the mean value and SD, and calculating LoD as the mean +2 SD. Variations of this approach use the mean plus 3, 4, or even 10 SDs to provide a more conservative LoD. The assumption is that if analyte is present, it will produce a signal greater than the analytical noise in the absence of analyte. This is a simple and quick method. The weakness is that there is no objective evidence to prove that a low concentration of analyte will indeed produce a signal distinguishable from a blank (zero concentration) sample. As Needleman and Romberg noted, “It defines only the ability to measure nothing”.2

An alternative approach utilises analysis of samples containing small but known concentrations of the substance of interest (be it a drug, hormone or other analyte).3,4 The advantage of this empirical approach is that objective data is used to compare the analytical response of blank and low concentration samples to determine conclusively what concentration of analyte is necessary to distinguish its presence from its absence. Various analytical specifications (e.g. a minimum signal-to-noise ratio for a chromatographic method or a minimum absorbance requirement for a spectrophotometric procedure) can be applied to ensure that the LoD is meaningful and clearly distinguishable from a negative or blank sample.

As defined in EP17, LoD is determined by utilising both the measured LoB and test replicates of a sample known to contain a low concentration of analyte.1 The mean and SD of the low concentration sample is then calculated according to:

LoD=LoB+1.645(SDlow concentration sample)

Again assuming a Gaussian distribution of the low concentration samples, 95% of values will exceed the previously defined LoB, and only 5% of low concentration samples will produce values below the LoB and erroneously appear to contain no analyte. Manufacturers are expected to establish the LoB and LoD using two or more instruments and reagent lots to capture the expected performance of the typical population of analysers and reagents. A recommended practical number of LoB and LoD samples to be used by a manufacturer to establish these parameters is 60, while a laboratory verifying a manufacturer’s LoD (and possibly the LoB) is 20.

Once a provisional LoD is established, it can be confirmed by examining the observed values for samples containing the LoD concentration. Some LoD sample values are expected to be less than the estimated LoD (Figure), but when using 1.645 SD, no more than 5% of the values should be less than the LoB. If the observed LoD sample values meet this criterion, the LoD is considered established or verified. If more than 5% (roughly 1 out of 20 observations) of the LoD sample values fall below LoB, the LoD is too low and must be re-estimated (i.e. by testing a sample of higher concentration that will generate a higher mean and SD and thus a higher LoD).

This is an abbreviated and simplified description of the EP17 protocol. The guideline contains considerably more statistical detail and guidance, including the use of non-parametric (non-Gaussian) techniques if necessary. Readers are encouraged to consult EP17 for a complete explanation of this method for establishing and verifying LoD.1

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Limit of Quantitation

LoQ is the lowest concentration at which the analyte can not only be reliably detected but at which some predefined goals for bias and imprecision are met. “Functional sensitivity” is defined as the concentration that results in a CV=20% (or some other predetermined % CV), and is thus a measure of an assay’s precision at low analyte levels (without addressing bias).5 It was originally developed as a clinical diagnostic tool to characterise thyroid stimulating hormone (TSH) assay performance in distinguishing euthyroid from hyperthyroid patients at low TSH concentrations. It can be expected that the LoD lies somewhere below an assay’s functional sensitivity.

The LoQ may be equivalent to the LoD or it could be at a much higher concentration (Figure); it cannot be lower than the LoD. A LoD provides an estimate of bias and imprecision at very low analyte concentration. If the observed bias and imprecision at the LoD meet the requirements for total error for the analyte (i.e. the assay is “fit for purpose”) then: LoQ=LoD. If the analytical goals are not met at the LoD, a slightly higher analyte concentration must be tested to determine the LoQ.

The Table provides a brief summary of the features of the LoB, LoD, and LoQ.

Table

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Conclusions

It is important to fully characterise the analytical performance of clinical laboratory tests in order to understand their capability and limitations, and to ensure that they are “fit for purpose.” The terms LoB, LoD, and LoQ describe the smallest concentration of a measurand that can be reliably measured by an analytical procedure.

To establish these parameters a manufacturer would test a large number of sample replicates to increase the robustness and the statistical confidence of the estimate (Table 1). In addition, a manufacturer establishing the LoB, LoD, or LoQ should perform studies using more than one analyser and one lot of reagents to encompass the variability that users can expect to encounter in the field. Clinical laboratories can validatethese parameters using a smaller number of samples and likely will use only one analyser and one lot of reagents.

LoB and LoD are important for tests used to discriminate between the presence or absence of an analyte (e.g. drugs, troponin, human chorionic gonadotrophin) and LoQ, to reliably measure low levels of hormones (e.g. TSH) for clinical diagnosis and management and should be incorporated as part of any method evaluation.

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Footnotes

Competing Interests: None declared.

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References

1. Clinical and Laboratory Standards Institute. Protocols for Determination of Limits of Detection and Limits of Quantitation, Approved Guideline. Wayne, PA USA: CLSI; CLSI document EP17. 2004.

2. Needleman SB, Romberg RW. Limits of linearity and detection of some drugs of abuse. J Anal Toxicol.1990;14:34–8. [PubMed]

3. Lawson GM. Defining limit of detection and limit of quantitation as applied to drug of abuse testing: striving for a consensus. Clin Chem. 1994;40:1218–9. [PubMed]

4. Armbruster DA, Tillman MD, Hubbs LM. Limit of detection (LOD)/limit of quantitation (LOQ): comparison of the empirical and the statistical methods exemplified with GC-MS assays of abused drugs.Clin Chem. 1994;40:1233–38. [PubMed]

5. Hay ID, Bayer MF, Kaplan MM, Klee GG, Larsen PR, Spencer CA. American Thyroid Association assessment of current free thyroid hormone and thyrotropin measurements and guidelines for future clinical assays. The Committee on Nomenclature of the American Thyroid Association. Clin Chem. 1991;37:2002–8. [PubMed]

Have a nice time

Helena

helena

Mohammed

There may be some confusion here. In scientific circles (especially Medical Laboratory work) there are terms and measures quoted.

LoD - Limit of detection

is the lowest analyte concentration likely to be reliably distinguished from the highest apparent analyte concentration (LoB) and at which detection is feasible. LoD is determined by utilising both the measured LoB and test replicates of a sample known to contain a low concentration of analyte

LoQ (limit of quantification (or quantitation)

LoQ is the lowest concentration at which an analyte can not only be reliably detected but at which some predefined goals for bias and imprecision are met. The LoQ may be equivalent to the LoD or it could be at a much higher concentration

In a normal distribution 99.73% of the data should be within +/- 3 times your standard deviation around the mean and the distribution extends asymptotically so it’s impossible to state where the limits exist for all the variation. It is not a practical discussion when solving real problems.

Lowest expected variation. If you use +/- 1.96 times your standard deviation around the mean, this will give you where you would expect 95% of the data to occur.

To provide a standard method for determining LoB, LoD and LoQ, Clinical and Laboratory Standards Institute (CLSI) has published the guideline EP17, Protocols for Determination of Limits of Detection and Limits of Quantitation:

Clinical and Laboratory Standards Institute. Protocols for Determination of Limits of Detection and Limits of Quantitation, Approved Guideline. Wayne, PA USA: CLSI; CLSI document EP17. 2004

A traditional and typical approach to estimate LoD consists of measuring replicates, of a zero calibrator or blank sample, determining the mean value and SD, and calculating LoD as the mean +2 SD. Variations of this approach use the mean plus 3, 4, or even 10 SDs to provide a more conservative LoD

A bit "dirty" sounding approach for some practical uses is to remember that for a normally distributed population, the standard deviation remains constant for all samples. Therefore, for an analytical analysis, you can prepare several dilutions with at least 3 replicates of each and check the standard deviation. Make dilutions until you find that the standard deviation of the replicates has increased to such an extent that the sample no longer "belongs" to this population. What is "increased too much"? Check the distribution curve (chi-square) for standard deviation and select an appropriate cutoff.