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. However, the exact threshold (level of decision) used to decide when a signal
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
analytical chemistry
Analytical chemistry studies and uses instruments and methods to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separati ...
, the detection limit, lower limit of detection, or LOD (limit of detection), often mistakenly confused with the analytical sensitivity, is the lowest quantity of a substance that can be distinguished from the absence of that substance (a ''
blank value'') with a stated
confidence level
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
(generally 99%). The detection limit is estimated from the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
of the blank, the
standard deviation of the blank, the slope (
analytical sensitivity) of the
calibration plot and a defined
confidence factor (e.g. 3.2 being the most accepted value for this arbitrary value).
Another consideration that affects the detection limit is the adequacy and the
accuracy
Accuracy and precision are two measures of ''observational error''.
''Accuracy'' is how close a given set of measurements ( observations or readings) are to their ''true value'', while ''precision'' is how close the measurements are to each oth ...
of the
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
used to predict concentration from the raw analytical signal.
As a typical example, from a
calibration plot following a
linear equation taken here as the simplest possible model:
:
where,
corresponds to the signal measured (e.g. voltage, luminescence, energy, etc.), "" the value in which the straight line cuts the ordinates axis, "" 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 "" the value of the quantity (e.g. temperature, concentration, pH, etc.) to be determined from the signal
, the LOD for "" is calculated as the "" value in which
equals to the average value of blanks "" plus "" times its standard deviation "" (or, if zero, the standard deviation corresponding to the lowest value measured) where "" is the chosen confidence value (e.g. for a confidence of 95% it can be considered = 3.2, determined from the limit of blank).
Thus, in this didactic example:
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.
The figure below illustrates the relationship between the blank, the ''limit of detection'' (LOD), and the ''limit of quantitation'' (LOQ) by showing the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
for
normally distributed measurements at the blank, at the LOD defined as 3 ×
standard deviation of the blank, and at the LOQ defined as 10 × standard deviation of the blank. For a signal at the LOD, the
alpha error (probability of
false positive
A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resul ...
) is small (1%). However, the
beta error (probability of a
false negative
A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resul ...
) 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.
Instrument detection limit
Most
analytical instruments produce a signal even when a blank (
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
without
analyte
An analyte, component (in clinical chemistry), or chemical species is a substance or chemical constituent that is of interest in an analytical procedure. The purest substances are referred to as analytes, such as 24 karat gold, NaCl, water, etc. ...
) is analyzed. This signal is referred to as the noise level. The IDL is the analyte concentration that is required to produce a signal greater than three times the
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 from the measured concentrations of those standards.
The detection limit (according to
IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
) is the smallest
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', '' number concentration'', ...
, or the smallest absolute
amount
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
, 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 (
) is given by:
:
where,
is the
mean value
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the ''arithm ...
of the signal for a reagent blank measured multiple times, and
is the known
standard deviation for the reagent blank's signal.
Other approaches for defining the detection limit have also been developed. In
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
absorbance
Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". Alternatively, for samples which scatter light, absorbance may be defined as "the negative lo ...
at a given
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, t ...
. 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
interfering
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extra ...
substances, instrument... .
Method detection limit
Often there is more to the analytical method than just performing a
reaction
Reaction may refer to a process or to a response to an action, event, or exposure:
Physics and chemistry
*Chemical reaction
*Nuclear reaction
* Reaction (physics), as defined by Newton's third law
*Chain reaction (disambiguation).
Biology and m ...
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
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 is then determined. The one-sided
Student's t-distribution is determined and multiplied versus the determined
standard deviation. For seven samples (with six degrees of freedom) the t value for a 99%
confidence level
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
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
geochemistry
Geochemistry is the science that uses the tools and principles of chemistry to explain the mechanisms behind major geological systems such as the Earth's crust and its oceans. The realm of geochemistry extends beyond the Earth, encompassing the ...
,
seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
,
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
,
dendrochronology,
climatology
Climatology (from Greek , ''klima'', "place, zone"; and , ''-logia'') or climate science is the scientific study of Earth's climate, typically defined as weather conditions averaged over a period of at least 30 years. This modern field of study ...
,
life sciences in general, and in many other fields impossible to enumerate extensively, the problem is wider and deals with
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
extraction out of a
background of noise. It involves complex
statistical analysis procedures and therefore it also depends on the
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
s used,
the hypotheses and the simplifications or approximations to be made to handle and manage
uncertainties. When the data resolution is poor and different signals overlap, different
deconvolution
In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deco ...
procedures are applied to extract parameters. The use of different
phenomenological, mathematical and
statistical models 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
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References
Further reading
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External links
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Downloads of articles (a.o. harmonization of concepts by ISO and IUPAC) and an extensive list of references
{{DEFAULTSORT:Detection Limit
Analytical chemistry
Measurement
Background radiation