Detection limits statistical estimation

The limit of detection (LoD) has already been mentioned in Section 4.3.1. This is the minimum concentration of analyte that can be detected with statistical confidence, based on the concept of an adequately low risk of failure to detect a determinand. Only one value is indicated in Figure 4.9 but there are many ways of estimating the value of the LoD and the choice depends on how well the level needs to be defined. It is determined by repeat analysis of a blank test portion or a test portion containing a very small amount of analyte. A measured signal of three times the standard deviation of the blank signal (3sbi) is unlikely to happen by chance and is commonly taken as an approximate estimation of the LoD. This approach is usually adequate if all of the analytical results are well above this value. The value of Sbi used should be the standard deviation of the results obtained from a large number of batches of blank or low-level spike solutions. In addition, the approximation only applies to results that are normally distributed and are quoted with a level of confidence of 95%. 

J. Vial and A. Jardy, Experimental Comparison of the Different Approaches to Estimate LOD and LOQ of an HPLC Method, Anal. Chem. 1999, 71, 2672 G. L. Long and J. D. Winefordner, Limit of Detection, Anal. Chem. 1983,55, 713 A W. R. Porter, Proper Statistical Evaluation of Calibration Data, Anal. Chem. 1983,55, 1290A S. Geiss and J. W. Einmax, Comparison of Detection Limits... 

There are often data sets used to estimate distributions of model inputs for which a portion of data are missing because attempts at measurement were below the detection limit of the measurement instrument. These data sets are said to be censored. Commonly used methods for dealing with such data sets are statistically biased. An example includes replacing non-detected values with one half of the detection limit. Such methods cause biased estimates of the mean and do not provide insight regarding the population distribution from which the measured data are a sample. Statistical methods can be used to make inferences regarding both the observed and unobserved (censored) portions of an empirical data set. For example, maximum likelihood estimation can be used to fit parametric distributions to censored data sets, including the portion of the distribution that is below one or more detection limits. Asymptotically unbiased estimates of statistics, such as the mean, can be estimated based upon the fitted distribution. Bootstrap simulation can be used to estimate uncertainty in the statistics of the fitted distribution (e.g. Zhao Frey, 2004). Imputation methods, such as... 

Two measurable quantities relate to precision in EMP analysis of phosphates. The first quantity is the uncertainty of the analysis itself, i.e., standard deviation, which is the main measure of the uncertainty of either thermometry estimates or chemical age. The second quantity is the minimum detection limit, i.e., the smallest concentration of an element of interest generating characteristic X-ray counts that are statistically distinguishable from background X-rays. 

Difficulties arise when the concentration of an element is below the detection limit (DL). It is often standard practice to report these data simply as

generally depends on the amount of data below the detection hmit, the size of the data set, and the probability distribution of the measurements. When the number of < DL observations is small, replacing them with a constant is generally satisfactory (Clarke, 1998). The values that are commonly used to replace the < DL values are 0, DL, or DL/2. Distributional methods such as the marginal maximum likelihood estimation (Chung, 1993) or more robust techniques (Helsel, 1990) are often required when a large number of < DL observations are present. 

REM IT ALSO FINDS DETECTION LIMITS USING CONCEPTS FROM Hubaux and Vos, Analytical Chemistry, 1970, 42(8), 849-855 and from Koehn and Zimmerman, "Method Detection Limits, or How Low Can You Really Go Estimation of Analytical Method Reporting Limits by Statistical Procedures", paper presented at the EPA Conference on Analysis of Pollutants in the Environment Norfolk, VA May 13-14, 1987. 

Korn, L.R. Tyler, D.E. (2001) Robust estimation for chemical concentration data subject to detection limits. In Turrin Femholz, L., Morgenthaler, S. Stahel, W., eds. Statistics In genetics and In the environmental sciences. Basel, Switzerland, Birkhauser Verlag, pp. 41-64. 

Methods for determining the LOD that are based on the analysis of a field blank that does not contain the analyte of interest are problematic in many real-world applications because either such samples do not exist, or would be impossibly difficult to create. As such a circumstance is frequently encountered in environmental analysis, the USEPA adopted a detection limit procedure, termed the method detection limit (MDL), which focuses on an operational definition of detection limit. Specifically, the MDL is defined as the minimum concentration of a substance that can be measured and reported with 99% confidence that the analyte concentration is greater than zero. The MDL is determined from a replicate analysis of a sample of a specified matrix. Specifically, at least seven aliquots of sample, spiked to contain a concentration of from one to five times the method s estimated MDL, are analyzed. The MDL calculated from these results is statistically tested to determine its reasonableness. If the result fails the testing, this iterative process begins again with a new estimate of the MDL. 

Method detection limit (MDL) Involves measuring and LOD using blank matrix samples spiked with known amounts of the cahbration standard and taken through the entire extraction-clean-up-analysis procedure. MDL measures the ability of a specified measurement method to determine an analyte in a sample matrix, with data interpretation using a specified statistical method. The term MDL is often taken to mean the particular single point estimate determined using the prescription of the US EPA. 

What is the minimum number of counts I can be confident of detecting It is important to appreciate that the critical limit and upper limit are both a posteriori estimates based upon actual measured counts. They are statements of what has been achieved in the measurement. The detection limit answers the a priori question If you were to measure a sample, what would the count have to be for, say, 95 % certainty of detection it is, therefore, a statement of what might be achieved. Detection limit is often confused with the critical limit. However, if the sample activity did happen to be exactly Lq (distribution (b) in Figure 5.10), statistically we would only be able to be sure (or 95 % sure ) of detection in 50 % of cases because the counts would be distributed symmetrically about Lq. It is clear that Lq must be some way above Lq (see distribution (c) in Figure 5.10). 

And the equation of RSD (%) is RSD = (SD/3c) x 100. In general, these are standard statistical applications, in which five or six runs of FIA-ECD experiments are carried out for obtaining an exact estimate of the measurement SD and RSD. On the other hand, ISO 11843-7, which provides an SD and RSD of measurement to examine detection limits from stochastic aspects of the signal and noise in a chromatogram based on the function of mutual information (FUMI) theory [9], has been proposed in recent years. When an FIA signal has a peak height, H, a measurement RSD can be estimated based on the noise... 

For any analytical procedure, it is important to establish the smallest amount of an analyte that can be detected and/or measured quantitatively. In statistical terms, and for instrumental data, this is defined as the smallest amount of an analyte giving a detector response significantly different from a blank or background response (i.e. the response from standards containing the same reagents and having the same overall composition (matrix) as the samples, where this is known, but containing no analyte). Detection limits are usually based on estimates of the standard deviation of replicate measurements of prepared blanks. A detection hmit of two or three times the estimated standard deviation of the blanks above their mean, Xg, is often quoted, where as many blanks as possible (at least 5 to 10) have been prepared and measured. 

The above approaches work through the use of calibration plots (see Fig. B.3) where the measurement is performed enough times to be statistically relevant and where the error of the line of best fit is used with the relevant equation, such that, in the case of the LOD, LOD = >S],/q can be readily deduced as shown in Fig. B.3. Note that there are limitations where over- and under- estimation of the LOD and LOQ can become possible using this approach and interested readers are directed towards Ref. [4, 5]. Indeed, if the cahbration plot is of high quality, sj, will be very small, leading to an extremely low detection limit being reported. In practice, this will never be achieved due to the limitation in the experimental procedure of the instrumentation (such as noise and drift at low signal levels and so on) as well as chemical interferences, since typically the LOD and LOQ are reported in model systems (buffer solutions only). 

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