1、Designation: D 6250 98 (Reapproved 2003)Standard Practice forDerivation of Decision Point and Confidence Limit forStatistical Testing of Mean Concentration in WasteManagement Decisions1This standard is issued under the fixed designation D 6250; the number immediately following the designation indica
2、tes the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (e) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers a logical basis for th
3、e derivation ofa decision point and confidence limit when mean concentrationis used for making environmental waste management deci-sions. The determination of a decision point or confidence limitshould be made in the context of the defined problem. Themain focus of this practice is on the determinat
4、ion of a decisionpoint.1.2 In environmental management decisions, the derivationof a decision point allows a direct comparison of a samplemean against this decision point, where similar decisions canbe made by comparing a confidence limit against a concentra-tion limit (for example, a regulatory lim
5、it, which will be usedas a surrogate term for any concentration limit throughout thispractice). This practice focuses on making environmentaldecisions using this kind of statistical comparison. Otherfactors, such as any qualitative information that may beimportant to decision-making, are not conside
6、red here.1.3 A decision point is a concentration level statisticallyderived based on a specified decision error and is used in adecision rule for the purpose of choosing between alternativeactions.1.4 This practice derives the decision point and confidencelimit in the framework of a statistical test
7、 of hypothesis underthree different presumptions. The relationship between deci-sion point and confidence limit is also described.1.5 Determination of decision points and confidence limitsfor statistics other than mean concentration is not covered inthis practice. This practice also assumes that the
8、 data arenormally distributed. When this assumption does not apply, atransformation to normalize the data may be needed. If otherstatistical tests such as nonparametric methods are used in thedecision rule, this practice may not apply. When there are manydata points below the detection limit, the me
9、thods in thispractice may not apply.2. Referenced Documents2.1 ASTM Standards:2D 4687 Guide for General Planning of Waste SamplingD 5792 Practice for Generation of Environmental DataRelated to Waste Management Activities: Development ofData Quality ObjectivesD 4790 Terminology of Aromatic Hydrocarbo
10、ns and Re-lated ChemicalsE 456 Terminology Relating to Quality and StatisticsE 1138 Terminology of Technical Aspects of Products Li-ability Litigation32.2 Other Documents:USEPA(1989a) StatisticalAnalysis of Ground-Water Moni-toring Data at RCRA Facilities. Interim Final Guidance.Office of Solid Wast
11、e Management Division, Washington,D.C. (PB89-15-1047)4USEPA (1989b) Methods for Evaluating the Attainment ofCleanup Standards. Vol. 1: Soils and Solid Media. Statis-tical Policy Branch (PM-223)4USEPA(1992) Statistical Methods for Evaluating the attain-ment of Superfund Cleanup Standards. Vol. 2: Gro
12、undwa-ter. DRAFT, Statistical Policy Branch, Washington, D.C4USEPA (1994) Guidance for the Data Quality ObjectivesProcess. EPA QA/G4, Quality Assurance ManagementStaff, USEPA, September, 199443. Terminology3.1 Definitions:3.1.1 decision point, nthe numerical value which causesthe decision maker to c
13、hoose one of the alternative actions (forexample, conclusion of compliance or noncompliance).3.1.1.1 DiscussionIn the context of this practice, thenumerical value is calculated in the planning stage and prior to1This practice is under the jurisdiction of ASTM Committee D34 on WasteManagement and is
14、the direct responsibility of Subcommittee D34.01.01 onPlanning for Sampling.Current edition approved April 10, 1998. Published December 1998.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volu
15、me information, refer to the standards Document Summary page onthe ASTM website.3Withdrawn.4Available from the Superintendent of Documents, U.S. Government PrintingOffice, Washington, DC 20402.1Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United
16、 States.the collection of the sample data, using a specified hypothesis,decision error, an estimated standard deviation, and number ofsamples. In environmental decisions, a concentration limit suchas a regulatory limit usually serves as a standard for judgingattainment of cleanup, remediation, or co
17、mpliance objectives.Because of uncertainty in the sample data and other factors,actual cleanup or remediation, for example, may have to go toa level lower or higher than this standard. This new level ofconcentration serves as a point for decision-making and is,therefore, termed the decision point.3.
18、1.2 confidence limits, nthe limits on either side of themean value of a group of observations which will, in a statedfraction or percent of the cases, include the expected value.Thus the 95 % confidence limits are the values between whichthe population mean will be situated in 95 out of 100 cases.D
19、47903.1.2.1 DiscussionA one-sided upper or lower confidencelimit can also be used when appropriate. An upper confidencelimit is a value below which the population mean is expectedto be with the specified confidence. Similarly, a lower confi-dence limit is a value above which the population mean isex
20、pected to be with the specified confidence. It is to be notedthat confidence limits are calculated after the collection ofsample data.3.1.3 decision rule, na set of directions in the form of aconditional statement that specify the following: (1) how thesample data will be compared to the decision po
21、int, (2) whichdecision will be made as a result of that comparison, and (3)what subsequent action will be taken based on the decisions.D 57923.1.3.1 DiscussionFor this practice, the comparison in (1)in 3.1.3 can be made in two equivalent ways: (1) a comparisonbetween the sample mean (calculated from
22、 the sample data)and a decision point (calculated during the planning stage), or(2) a comparison between a confidence limit(s) (calculatedfrom the sample data) and a regulatory limit.3.1.4 false negative error, noccurs when environmentaldata mislead decision maker(s) into not taking action specified
23、by a decision rule when action should be taken. D 57923.1.4.1 DiscussionFor this practice, this is an error de-fined in the context of a regulatory decision in waste manage-ment. In this context, it is an error in concluding that the truevalue is smaller than the regulatory limit when in fact it is
24、not.The calculation of the false negative error will depend on howthe hypotheses are framed (see Appendix X1).3.1.5 false positive error, noccurs when environmentaldata mislead decision maker(s) into taking action specified bya decision rule when action should not be taken. D 57923.1.5.1 DiscussionF
25、or this practice, this is an error de-fined in the context of a regulatory decision in waste manage-ment. In this context, it is an error in concluding that the truevalue is equal to or greater than the regulatory limit when infact it is not. The calculation of the false positive error willdepend on
26、 how the hypotheses are framed (see Appendix X1).3.1.6 hypothesis, na supposition or conjecture put for-ward to account for certain facts and used as a basis for furtherinvestigation by which it may be proved or disproved.E 11383.1.6.1 DiscussionFor this practice, a hypothesis is apostulation of wha
27、t the true value is, typically framed for thepurpose of making a statistical test of the hypothesis. In astatistical test, there are two competing hypotheses: the nullhypothesis and the alternative hypothesis. The null hypothesisis a hypothesis “put up” for consideration and is the presumedhypothesi
28、s of choice before the data are collected. The alter-native hypothesis is favored only when the data reject the nullhypothesis.3.1.7 statistic, na quantity calculated from a sample ofobservations, most often to form an estimate of some popula-tion parameter. E 4564. Significance and Use4.1 Environme
29、ntal decisions often require the comparisonof a statistic to a decision point or the comparison of aconfidence limit to a regulatory limit to determine which of twoalternate actions is the proper one to take.4.2 This practice provides a logical basis for statisticallyderiving a decision point, or a
30、confidence limit as an alterna-tive, for different underlying presumptions.4.3 This practice is useful to users of a planning processgenerally known as the data quality objectives (DQO) process(see Practice D 5792), in which calculation of a decision pointis needed for the decision rule.5. Overview
31、of Decision Point Determination5.1 The determination of a decision point is usually a part ofan overall planning process. For example, the decision rule inthe DQO planning process often includes the specification of adecision point. A brief summary of the steps needed todetermine a decision point is
32、 given below.5.1.1 State the problem and the decision rule (see Section6),5.1.2 Consider the alternative presumptions in the hypoth-eses based on the relative consequences of false positive andfalse negative errors (see 7.6),5.1.3 Choose the form of the hypotheses to be used in thedecision rule base
33、d on the chosen presumption (see 7.5 through7.6 and Fig. 1),5.1.4 Obtain an estimated standard deviation and the num-ber of samples used in that estimation,5.1.5 Specify acceptable decision errors (see Section 8), and5.1.6 Calculate the decision point (see Section 8).5.2 The following sections discu
34、ss in practical terms thetopics of decision rule, presumptions and test of hypothesis,calculation of a decision point for specified decision errors,ways to control decision errors, and the use of a confidencelimit as an alternative approach in decision-making.6. Decision Rule in Waste Management Dec
35、isions6.1 A decision rule is constructed according to a problemstatement defined and agreed to by all the parties concerned,through a planning process. The decision rule can be carriedout in two similar ways.6.1.1 When Using A Decision Point:6.1.1.1 The general construct of the decision rule in this
36、case is:D 6250 98 (2003)2If sample mean! $ decision point!, then one action!. Otherwise, alternate action!.6.1.1.2 Because a decision point is needed in the abovedecision rule, this practice provides a logical basis for devel-oping such a decision point. Because the above decision rulecan also be ca
37、rried out similarly using confidence limits, it isalso presented that way in 6.2.6.1.1.3 Note that when data can be measured with certainty,the regulatory limit defines the decision point. For example,sample data taken from a totally homogeneous population, inthe absence of measurement error, have n
38、o variability. Thismeans that the standard deviation of the data is zero and thedecision point is reduced to the regulatory limit (see 8.6.3).6.1.1.4 When data cannot be measured precisely or thepopulation is not totally homogeneous, this variability needs tobe incorporated to obtain a decision poin
39、t. The decision pointthen includes both the original regulatory limit and a margin ofuncertainty that is reflected in the standard deviation, which isa component in the calculation of the decision point (see 8.6.3).The way to incorporate this uncertainty depends on how ahypothesis is formulated and
40、which presumption is adopted.This is discussed in Section 7.6.1.1.5 An example of carrying out the decision rule using adecision point is:If average concentration of cadmium in a truck load! $ decisionpoint!, then dispose of the waste fly ash in an RCRA landfill!.Otherwise, dispose the waste fly ash
41、 in a sanitary landfill!.6.1.1.6 The inputs needed for the calculation of the decisionpoint in 6.1.1.5 are: form of the hypotheses to be tested,acceptable maximum decision error, number of samples, andestimated standard deviation. The standard deviation shouldinclude all the sources of variation in
42、the sampling andmeasurement processes. Decision errors include the falsepositive error and false negative error. Details are given inSection 8.6.2 When Using Confidence Limit:6.2.1 The general construct of the decision rule in this caseis:FIG. 1 Decision Point Determination for Mean ConcentrationD 6
43、250 98 (2003)3If confidence limit!.or and (regulatory limit), then (dispose of the waste flyash in a RCRA landfill). Otherwise, (dispose of the waste flyash in a sanitary landfill).6.2.2.2 If lower confidence limit of mean concentration ofcadmium) 0, the false positive error in Eq X1.3 becomes:Prob
44、x $La! 5 Prob$x Lrd1!# / s/=n! $La Lr d1!# / s/=n!%(X1.4)0), the false negative error becomes smaller.X1.3.4 It is to be noted that the false negative error is theexact complement of the chosen false positive error when thetrue value is at Lr; there the two errors total 100 %.X1.3.5 When the populat
45、ion standard deviation, s,isre-placed by sample standard deviation, s, in Eq X1.8, thez-statistic is replaced by the t-statistic. The calculation of thefalse negative error now involves a non-centrality parameterand a statistician needs to be consulted.X1.4 Equivalency Between Decision Point Approac
46、h andTest of Hypothesis Approach:X1.4.1 The decision rule using the decision point given inEq X1.7 can be made to be equivalent to one using the lower100 (1-p) % confidence limit in a statistical test of hypothesis.X1.4.2 Recall that the decision rule using a decision point isto conclude exceedance
47、of the regulatory limit if x $ La.X1.4.3 But La=Lr+t1-p,n-1s/=n, from Eq X1.7. Thus x$Labecomes x $ Lr+t1-p,n-1s/=n.X1.4.3.1 Rearranging:x t12p,n21s/=n$Lr(X1.9)TABLE X1.1 False Negative Error (q) Under PresumptionNumber 1NOTE 1For Some Values of False Positive Error (p), and True Valuesd2/(s=n).AFal
48、sePositiveError,pIf True Value As Numberof Standard ErrorsAway from RegulatoryLimit, or d2/(s/=n), IsFalseNegativeError,q0.05 0 0.951 0.742 0.363 0.090.10 0 0.901 0.612 0.243 0.040.20 0 0.801 0.442 0.123 0.02AWhere the true value = regulatory limit when d2/(s=n)=0.D 6250 98 (2003)11X1.4.3.2 Note tha
49、t the left-hand side of Eq X1.9 is the lower100 (1-p) % confidence limit in the framework of a test ofhypothesis. Thus, the decision point approach from Eq X1.7and the hypothesis test approach from Eq X1.9 are equivalentin decision-making.X1.4.3.3 The decision rule using the lower confidence limitin Eq X1.9 is carried out as follows:(1) If (xt1-p,n-1s/=n) 0. Thus,False positive decision error:5 probability of sayingthat the true mean concentration is not lowerthan the regulatory limit when it is so, (X1.17)5 probability that the data observe
