M. 1. Parsons
Arizona State University Tempe, 85281
II
The Definition
About 250 years ago Robert Boyle determined the first limit of detection recorded in the literature (1). Hc determined from experimentation that 1 part of C1- in 4000 parts of water gave an observable precipitate with Ag+. Thus, he observed the qualitative limit for the reaction. Since Boyle's time the term detection limit has been used to a great extent, generally to the benefit of science; however, it has often been misused and misinterpreted. The only valid definition (in the author's opinion) for detection limits are those which are based on, or can be related to, a statistical approach. The value of repcating results from laboratory to laboratory cannot be overemphasized, and the only hope of universally accomplishing this for any method is by having either universal definitions or at least definitions which can be understood and reproducibly applied. For this reason and for the sake of simplicity and clarity, the author encourages the adoption of uniformly stated detection limit definitions in all areas of analytical chemistry. The first statistically valid definition for detection limits was probably given by Kaiser (2) in 1947 and developed by him (5, 4) with direct application to emission spectrographic analysis. This approach has since been adopted by numerous workers in many related fields, for example, Herrmann and Allzemade (5) and Ramirez-Mhoz (6) in flame photometry; Ramirez-Rf~noz(7) and Winefordner and coworkers in atomic absorption (8); and Winefordner and coworkers in atomic fluorescence flame spectrometry (9). Recently, Boumans and Rlaessen (10) have refined the emission spectrographic approach, and St. John, Ahcarthy, and Winefordner (11) have shown how this approach can be followed in essentially all analytical methods that use a recorded readout. It is hoped that this paper will correlate and help clarify the statistical detection limit definition, as well as present a new and more accurate approach for experimentally determining detection limits based on quantitative measurements. The Limit of Detection
The limit of detection can be most simply defined as that concentration of analysis species which can be detected at a specified confidence level. RIa~lyauthors have chosen a 99.7% confidence level (5); however, thc choice is arbitrary. This definition is completely compatable with any of the experimental definitions based on the measurement of an experimentally recorded signal-to-noise ratio (11) such as Winefordner's @, 9.12) and Slavin's (IS). It is generally agreed that the detection of any species is a function of the experimental conditions, 290
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01 Detection Limits
and that any number stated for a detection limit is meaningless without adequately stating these conditions. Some of these parameters are often overlooked, e.g., the time response characteristics of the electronic readout system and the number of measurements on which the statistical calculations are based. Any time that an experimental estimate of a standard deviation is obtained, whether it be from a series of signal minus background measuremcnts (5) or from noise measurements ( l l ) , the estimate is a definite function of the number of measurements actually performed, and this number should be stated. Furthermore, if the experimental measurement is of the noise associated with a given signal, the noise is a function of the time response of the electronic readout, and this response must be known in order to experimentally reproduce the measurements. Modification of any nature to a commercial instrument (however slight) should bc carefully stated in the experimental section of a journal article. It is felt that a short review is in order for the statistical approach as applied to the experimentally measured signal-to-noise ratio, S I N . The experimenter can measure the response and the pealz-to-peak noise for any signal, and in all cases he must contend with a backgronnd (or blank) signal as well as the signal due to the analysis species. This is true even in double beam instruments, except that the background and signal are measured simultaneously. Therefore, the actual measurement of the analysis signal, S , is by difference of the total response, ST,i.e., the backgronnd plus analysis signal, and the background signal, SB. Any definition based on the measurement of the S I N is statistically valid because the noise measurement is a direct measure of the standard deviation if the noise is white, i.e., a continuous function of frequency which is generally the case (11), and if the time interval during which the noise is measured is sufficiently long to include all of the high frequency noise. It should be pointed out that noise of very low frequency constitutes drift, and this must he compensated by means of background readings. White noise follows a normal error probability distribution. Therefore, the standard deviation, s, can be written as follows (11)
where N,,, is the root-mean-square noise signal, 8 , is an estimate of the standard deviation, and n is the number of noise observations. When the standard deviation is written in this form, the number of signal observations must equal the number of background
observations, and the noises on the signal and background are assumed to be equal. It must be remembered that the value of the estimate of the standard deviation, s, only approaches the true standard deviation, u,as the number of obsewations, n., approach infinity, i.e., s + u as n + . It has also been determined (11) that
-
N,,. 6Z 1/5 X N,,
(3)
where N,, is the average peak-to-peak noise signal displayed on the recorder chart. The measure of the confidence of any signal can be determined by Student's "t" which can now be defined in terms of the signal and roobmean-square noise as (11)
from which
or combining eqns. (2) and (3)
where t is Student's t value. Values for t can be found in any number of statistical tables for any specified confidence level (14). Thus, if a 99.5y0 confidence level is desired for a signal to be considered a "detected" signal by any given method and the method utilizes four sample and four blank measurements, t will equal 4.604 (14). Therefore, the limit of detection can be defined as that concentration which gives a signal-to-(rms) noise equal to 3.26, or a signal-to-(pp) noise equal to 0.651 by solving eqns. (5) or (6), respectively. What is the precision a t the limit of detection? Many workers apparently feel that it should be good enough for quantitative analysis. This i s not so! Because a signal can be detected with a 99.5% confidence level is no sign that the value of the signal can be precisely measured, and precision of measurement is the criteria for quantitative analysis. It happens that if the number of obsewations is known, the precision and, more to the point, the relative standard deviation, RSD (often called the coefficient of variation), can be estimated directly from the S I N ratio. The RSD is given by RSD
=
% X 100
(7)
and substituting for s from eqn. (2)
RSD
=
28.28 X .N\ m /i~~
Relating to our example detection limit of a SIN,,, of 3.26 which uses four measurements of S , eqn. (8) results in a RSD of 21.7%. This means that the average of four signal measurements (eqn. (1)) at the limit of detection is known to only about +23.1Q/, (*RSD X t/.\/ii, where t = 2.132 (14) a t the 95% confidence level). Thus, experimentally measured limits of detection are inaccuvate by definition.
The Limit of Determination
Herrmann and Alkemade (6) define the limit of determination as that concentration of analysis species which can be determined to a specified precision. Again, this precision limit is arbitrary. RamirezMinoz (6, 7) concurs with this definition. It is interesting to note, however, that these researchers conclude that the way to obtain the determination limit is by calculation from the measured detection limit. The author feels that this is unfortunate in that a properly defined determination limit should be capable of accurate measurement; therefore, the detection limit could be calculated from the determination limit more accurately than it could be experimentally measured. (Herrmann and Alkermade (5) assume that enough measurements are made to obtain a very good estimate of u, and that detection limits are based on the measurement of the signal only; however, the signal cannot be measured accurately at the detection limit, and this should be the most important consideration.) The S I N approach is the most practical for the exoerimental measurement of the determination limit. lnAfact,it is a simple matter to set the RSD (eqns. (8) or (9)) equal to the desired precision and solve for the appropriate S I N . For example, if a particular analysis demands a precision of +5y0 RSD, the solution of eqn. (9) gives a SIN,, of 2.83 a t the determination limit using four S observations. The sample concentration giving a S/N,, of 2.83 would thus be the determination limit concentration. Discussion
Winefordner and coworkers (8, 9, 12, 16-18) have shown that for many analytical methods the S/N,,, is a linear function of the analysis species concentration near the detection limits. This results in the interesting conclusion for these methods that the precision of measurement is an inverse function of the concentration of analysis species a t low levels. Whenever this is the case (and it should be to a good approximation for most methods), it should be far more accurate to set a determination limit at a specified value (say 5 or 10yo RSD), experimentally observe the concentration at this level, and extrapolate to the appropriately defined detection limit (say a confidence level of 99.5 or 99.9%). A simple example of this extrapolation is as follows: The limit of determination is first decided, e.g., that concentration which gives a SIN,, = 2.83 using four S measurements as in the example above, i.e., an RSD of +5%. This limit is experimentally obsewed by the usual measurements of signal and noise as a function of concentration. Next the limit of detection is defined, e.g., that concentration which gives a SIN,, = ,651 as in the example above, i.e., a 99.5% confidence level. For most analytical methods in the concentrat'1011 region between the limit of determination and the limit of detection the sample concentration will be a linear function of the S / N (8, 9, 12, 15-18); therefore the concentration a t the detection limit can be expressed as
where the S I N can be determined using either N,,, or Volume 46, Number 5, Moy 1969
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N,,, C is
t h e sample concentration, and t h e subscripts
detect. and determ. refer t o t h e detection limits and determination limits, respectively. quoted above will give t h e following
The
cxample
should be much more accurate This value of CDereC,. t h a n one which is experimentally obtained because t h e signal, S,is known t o 5% RSD instead of 21.7% RSD. There are many analytical techniques to which this method of statistically defined determination a n d detection limits can apply: spectrographic ( l o ) , flame emission ( I $ ) , atomic absorption ( 8 ) , atomic fluorescence (9), molecular absorption ( l 7 ) , a n d molecular fluorescence, as well as certain types of gas chromatography detectors ( I S ) ,and could undoubtedly be applied t o more met,hods. Valuable discussions are cont,ained in t h e references mentioned above concerning t h e variation of t h e detection limit, and thus t h e determination limit, with t,he various experimental parameters. It is pertinent t o note t h a t there h a s recently developed a theoretical approach t o t h e calculation of S I N from known or measureable parameters for many analytical techniques (8, 9, 12, 15-18). This development opens t h e door t o t h e calculation of limits of determination a s well as mathematically studying their variat,ion with experimental parameter variat,ion. It should be emphasized t h a t t h e S I N approach makes absolutely no statement about systematic errors. T h e detect,ion and subsequent handling of systematic errors must be dealt with individually; however, their importance must never be overlooked. This discussion has been restricted to techniques which follow t h e normal distribution curve; however, there are cases of methods which follow other distri-
I Normal distribution statistics e m be applied to quantit,at,ive analyseq utilizing emission spectrographic methods because the lognormal and normal distribution curves carrnat be distingoished when the RSD is small, cf. = 207, (20).
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bution curves. Ipor example, it has been shown t h a t t h e results of semi-quantitative cmission spectrography This follow a lognormal distribution curve (19, means t h a t t h e logarithms of t h e signals rather than the signals follow a normal distribution curve (21).
to).'
Literature Cited
(1) SZ.\HADV.~RY, FI.:RI~NC, "I~Iistoryof Analytical Chemistry," (Tmnslalor: SVI:E~I..\, GYULA) Pergamon Press, Oxford, 1966, p. 39.; BOYLI:,IIOBKRT, Philo~ophicalWorks, 3, 231 (1725). TI., Speetroehim., Ada, 3, 40 (1947). (2) KAISI:K, H., Optik, 21, 309 (1964). (3) K,AISI,:R, H., Z. Anal. Chem., 216, 80 (1966). (4) K.\ISISR, (5) III:RRMINN, I~OLIND, A N D ALKI~M.\DII, C. T. J., "Chemical Analysis by Flame Phot,omet,ry2'(2nd ed.), (T~anslator: 'GILI~I.:RT, P. T., JR.), Interscience (division of John W h y & Sons, Inc.), 1963, pp. 255-8. (6) Rn~nurz-Mir~oz, JUAN,Talanta, 13, 87 (1966). JUAN, SHIFRIN,N., .\ND I~KLL,A,, (7) ~
