D. Harvey - Modern Analytical Chemistry (794078), страница 30
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Aspart of the study, samples taken at different times from a fermentationproduction vat were analyzed for the concentration of monensin using boththe electrochemical and microbiological procedures. The results, in parts perthousand (ppt),* are reported in the following table.Sample1234567891011Microbiological129.589.676.652.2110.850.472.4141.475.034.160.3Electrochemical132.391.073.658.2104.249.982.1154.173.438.160.1Determine whether there is a significant difference between the methods atα = 0.05.*1 ppt is equivalent to 0.1%.1400-CH04 9/8/99 3:55 PM Page 93Chapter 4 Evaluating Analytical Data93SOLUTIONThis is an example of a paired data set since the acquisition of samples over anextended period introduces a substantial time-dependent change in theconcentration of monensin. The comparison of the two methods must be donewith the paired t-test, using the following null and two-tailed alternativehypotheses––H0: d = 0HA: d ≠ 0Defining the difference between the methods asd = Xelect – Xmicrowe can calculate the difference for each sampleSampled12.821.43–3.046.05–6.66–0.579.7812.79–1.6104.011–0.2The mean and standard deviation for the differences are 2.25 and 5.63,respectively.
The test statistic ist exp =d nsd=2.25 115.63= 1.33which is smaller than the critical value of 2.23 for t(0.05, 10). Thus, the nullhypothesis is retained, and there is no evidence that the two methods yielddifferent results at the stated significance level.A paired t-test can only be applied when the individual differences, di, belongto the same population. This will only be true if the determinate and indeterminateerrors affecting the results are independent of the concentration of analyte in thesamples.
If this is not the case, a single sample with a larger error could result in avalue of di that is substantially larger than that for the remaining samples. Including–this sample in the calculation of d and sd leads to a biased estimate of the true meanand standard deviation. For samples that span a limited range of analyte concentrations, such as that in Example 4.21, this is rarely a problem. When paired data spana wide range of concentrations, however, the magnitude of the determinate and indeterminate sources of error may not be independent of the analyte’s concentration.
In such cases the paired t-test may give misleading results since the paired data–with the largest absolute determinate and indeterminate errors will dominate d. Inthis situation a comparison is best made using a linear regression, details of whichare discussed in the next chapter.4F.5 OutliersOn occasion, a data set appears to be skewed by the presence of one or more datapoints that are not consistent with the remaining data points.
Such values are calledoutliers. The most commonly used significance test for identifying outliers is Dixon’sQ-test. The null hypothesis is that the apparent outlier is taken from the same population as the remaining data. The alternative hypothesis is that the outlier comes from adifferent population, and, therefore, should be excluded from consideration.The Q-test compares the difference between the suspected outlier and its nearest numerical neighbor to the range of the entire data set.
Data are ranked fromsmallest to largest so that the suspected outlier is either the first or the last dataoutlierData point whose value is much larger orsmaller than the remaining data.Dixon’s Q-testStatistical test for deciding if an outliercan be removed from a set of data.1400-CH04 9/8/99 3:55 PM Page 9494Modern Analytical Chemistrypoint. The test statistic, Qexp, is calculated using equation 4.23 if the suspected outlier is the smallest value (X1)Qexp =X 2 − X1X n − X14.23or using equation 4.24 if the suspected outlier is the largest value (Xn)Qexp =X n − Xn −1X n − X14.24where n is the number of members in the data set, including the suspected outlier.It is important to note that equations 4.23 and 4.24 are valid only for the detectionof a single outlier.
Other forms of Dixon’s Q-test allow its extension to the detectionof multiple outliers.10 The value of Qexp is compared with a critical value, Q(α, n), ata significance level of α. The Q-test is usually applied as the more conservative twotailed test, even though the outlier is the smallest or largest value in the data set.Values for Q(α, n) can be found in Appendix 1D.
If Qexp is greater than Q(α, n),then the null hypothesis is rejected and the outlier may be rejected. When Qexp isless than or equal to Q(α, n) the suspected outlier must be retained.EXAMPLE 4.22The following masses, in grams, were recorded in an experiment to determinethe average mass of a U.S. penny.3.0673.0493.0392.5143.0483.0793.0943.1093.102Determine if the value of 2.514 g is an outlier at α = 0.05.SOLUTIONTo begin with, place the masses in order from smallest to largest2.5143.0393.0483.0493.0673.0793.0943.1023.109and calculate QexpQexp =3.039 − 2.514X 2 − X1== 0.8823.109 − 2.514X 9 − X1The critical value for Q(0.05, 9) is 0.493.
Since Qexp > Q(0.05, 9) the value isassumed to be an outlier, and can be rejected.The Q-test should be applied with caution since there is a probability, equal toα, that an outlier identified by the Q-test actually is not an outlier. In addition, theQ-test should be avoided when rejecting an outlier leads to a precision that is unreasonably better than the expected precision determined by a propagation of uncertainty. Given these two concerns it is not surprising that some statisticians caution against the removal of outliers.11 On the other hand, testing for outliers canprovide useful information if we try to understand the source of the suspected outlier.
For example, the outlier identified in Example 4.22 represents a significantchange in the mass of a penny (an approximately 17% decrease in mass), due to achange in the composition of the U.S. penny. In 1982, the composition of a U.S.penny was changed from a brass alloy consisting of 95% w/w Cu and 5% w/w Zn, toa zinc core covered with copper.12 The pennies in Example 4.22 were thereforedrawn from different populations.1400-CH04 9/8/99 3:55 PM Page 95Chapter 4 Evaluating Analytical Data954G Detection LimitsThe focus of this chapter has been the evaluation of analytical data, including theuse of statistics.
In this final section we consider how statistics may be used to characterize a method’s ability to detect trace amounts of an analyte.A method’s detection limit is the smallest amount or concentration of analytethat can be detected with statistical confidence. The International Union of Pureand Applied Chemistry (IUPAC) defines the detection limit as the smallest concentration or absolute amount of analyte that has a signal significantly larger than thesignal arising from a reagent blank. Mathematically, the analyte’s signal at the detection limit, (SA)DL, is(SA)DL = Sreag + zσreag(n A )DL(S A )DLk(S )= A DLkThe value for z depends on the desired significance level for reporting the detectionlimit. Typically, z is set to 3, which, from Appendix 1A, corresponds to a significance level of α = 0.00135.
Consequently, only 0.135% of measurements made onthe blank will yield signals that fall outside this range (Figure 4.12a). When σreag isunknown, the term zσreag may be replaced with tsreag, where t is the appropriatevalue from a t-table for a one-tailed analysis.13In analyzing a sample to determine whether an analyte is present, the signalfor the sample is compared with the signal for the blank. The null hypothesis isthat the sample does not contain any analyte, in which case (SA)DL and Sreag areidentical.
The alternative hypothesis is that the analyte is present, and (SA)DL isgreater than Sreag. If (SA)DL exceeds Sreag by zσ(or ts), then the null hypothesis isrejected and there is evidence for the analyte’s presence in the sample. The probability that the null hypothesis will be falsely rejected, a type 1 error, is the same asthe significance level. Selecting z to be 3 minimizes the probability of a type 1error to 0.135%.Significance tests, however, also are subject to type 2 errors in which the nullhypothesis is falsely retained. Consider, for example, the situation shown in Figure4.12b, where SA is exactly equal to (SA)DL. In this case the probability of a type 2error is 50% since half of the signals arising from the sample’s population fall belowthe detection limit.
Thus, there is only a 50:50 probability that an analyte at theIUPAC detection limit will be detected. As defined, the IUPAC definition for thedetection limit only indicates the smallest signal for which we can say, at a significance level of α, that an analyte is present in the sample. Failing to detect the analyte, however, does not imply that it is not present.An alternative expression for the detection limit, which minimizes both type 1and type 2 errors, is the limit of identification, (SA)LOI, which is defined as 14(SA)LOI = Sreag + zσreag + zσsampProbabilitydistributionfor blank4.25where Sreag is the signal for a reagent blank, σreag is the known standard deviation for the reagent blank’s signal, and z is a factor accounting for the desiredconfidence level. The concentration, (C A ) DL , or absolute amount of analyte,(nA)DL, at the detection limit can be determined from the signal at the detectionlimit.(C A )DL =detection limitThe smallest concentration or absoluteamount of analyte that can be reliablydetected.(a)SreagProbability distributionfor blank(b)SreagSreagProbabilitydistributionfor sample(SA)DLProbability distributionfor blank(c)(SA)DLProbabilitydistributionfor sample(SA)LOIFigure 4.12Normal distribution curves showing thedefinition of detection limit and limit ofidentification (LOI).
The probability of a type1 error is indicated by the dark shading, andthe probability of a type 2 error is indicatedby light shading.limit of identificationThe smallest concentration or absoluteamount of analyte such that theprobability of type 1 and type 2 errorsare equal (LOI).1400-CH04 9/8/99 3:55 PM Page 9696Modern Analytical Chemistrylimit of quantitationThe smallest concentration or absoluteamount of analyte that can be reliablydetermined (LOQ).SreagNeverdetectedAs shown in Figure 4.12c, the limit of identification is selected such that there is anequal probability of type 1 and type 2 errors.
The American Chemical Society’sCommittee on Environmental Analytical Chemistry recommends the limit ofquantitation, (SA)LOQ, which is defined as15(SA)LOQ = Sreag + 10σreagSA???LowerconfidenceintervalAlwaysdetectedUpperconfidenceintervalFigure 4.13Establishment of areas where the signal isnever detected, always detected, and whereresults are ambiguous. The upper and lowerconfidence limits are defined by the probabilityof a type 1 error (dark shading), and theprobability of a type 2 error (light shading).Other approaches for defining the detection limit have also been developed.16The detection limit is often represented, particularly when used in debates overpublic policy issues, as a distinct line separating analytes that can be detected fromthose that cannot be detected.17 This use of a detection limit is incorrect.
Definingthe detection limit in terms of statistical confidence levels implies that there may bea gray area where the analyte is sometimes detected and sometimes not detected.This is shown in Figure 4.13 where the upper and lower confidence limits are defined by the acceptable probabilities for type 1 and type 2 errors. Analytes producing signals greater than that defined by the upper confidence limit are always detected, and analytes giving signals smaller than the lower confidence limit are neverdetected.
Signals falling between the upper and lower confidence limits, however,are ambiguous because they could belong to populations representing either thereagent blank or the analyte. Figure 4.12c represents the smallest value of SA forwhich no such ambiguity exists.4H KEY TERMSalternative hypothesis (p. 83)binomial distribution (p. 72)central limit theorem (p. 79)confidence interval (p. 75)constant determinate error (p.
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