1、Designation: E2935 17 An American National StandardStandard Practice forConducting Equivalence Testing in Laboratory Applications1This standard is issued under the fixed designation E2935; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revi
2、sion, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon () indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice provides statistical methodology for con-ducting equivalence testing on numerical data
3、 from twosources to determine if their true means or variances differ byno more than predetermined limits.1.2 Applications include (1) equivalence testing for biasagainst an accepted reference value, (2) determining meansequivalence of two test methods, test apparatus, instruments,reagent sources, o
4、r operators within a laboratory or equiva-lence of two laboratories in a method transfer, and (3)determining non-inferiority of a modified test procedure versusa current test procedure with respect to a performance charac-teristic.1.3 The guidance in this standard applies to experimentsconducted on
5、a single material at a given level of the test resultor on multiple materials covering a range of selected testresults.1.4 Guidance is given for determining the amount of datarequired for an equivalence trial. The control of risks associ-ated with the equivalence decision is discussed.1.5 The values
6、 stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.6 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safet
7、y, health, and environmental practices and deter-mine the applicability of regulatory limitations prior to use.1.7 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Decision on Principles for theDevelopment of I
8、nternational Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:2E177 Practice for Use of the Terms Precision and Bias inASTM Test MethodsE456 Terminology Relating to Quality and Statistic
9、sE2282 Guide for Defining the Test Result of a Test MethodE2586 Practice for Calculating and Using Basic StatisticsE3080 Practice for Regression Analysis2.2 USP Standard:3USP Validation of Alternative MicrobiologicalMethods3. Terminology3.1 DefinitionsSee Terminology E456 for a more exten-sive listi
10、ng of statistical terms.3.1.1 accepted reference value, na value that serves as anagreed-upon reference for comparison, and which is derivedas: (1) a theoretical or established value, based on scientificprinciples, (2) an assigned or certified value, based on experi-mental work of some national or i
11、nternational organization, or(3) a consensus or certified value, based on collaborativeexperimental work under the auspices of a scientific orengineering group. E1773.1.2 bias, nthe difference between the expectation of thetest results and an accepted reference value. E1773.1.3 confidence interval,
12、nan interval estimate L, Uwith the statistics L and U as limits for the parameter andwith confidence level 1 , where Pr(L U) 1.E25863.1.3.1 DiscussionThe confidence level, 1 , reflects theproportion of cases that the confidence interval L, U wouldcontain or cover the true parameter value in a series
13、 of repeatedrandom samples under identical conditions. Once L and U aregiven values, the resulting confidence interval either does ordoes not contain it. In this sense “confidence” applies not to theparticular interval but only to the long run proportion of caseswhen repeating the procedure many tim
14、es.1This test method is under the jurisdiction of ASTM Committee E11 on Qualityand Statistics and is the direct responsibility of Subcommittee E11.20 on TestMethod Evaluation and Quality Control.Current edition approved Oct. 1, 2017. Published November 2017. Originallyapproved in 2013. Last previous
15、 edition approved in 2016 as E2935 16. DOI:10.1520/E2935-17.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at serviceastm.org. For Annual Book of ASTMStandards volume information, refer to the standards Document Summary page onthe ASTM website.3
16、Available from U.S. Pharmacopeial Convention (USP), 12601 TwinbrookPkwy., Rockville, MD 20852-1790, http:/www.usp.org.Copyright ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United StatesThis international standard was developed in accordance with internat
17、ionally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.13.1.4 confidence level, nthe value, 1 , of the prob
18、abilityassociated with a confidence interval, often expressed as apercentage. E25863.1.4.1 Discussion is generally a small number. Confi-dence level is often 95 % or 99 %.3.1.5 confidence limit, neach of the limits, L and U, of aconfidence interval, or the limit of a one-sided confidenceinterval. E2
19、5863.1.6 degrees of freedom, nthe number of independentdata points minus the number of parameters that have to beestimated before calculating the variance. E25863.1.7 equivalence, ncondition that two population param-eters differ by no more than predetermined limits.3.1.8 intermediate precision cond
20、itions, nconditions un-der which test results are obtained with the same test methodusing test units or test specimens taken at random from a singlequantity of material that is as nearly homogeneous as possible,and with changing conditions such as operator, measuringequipment, location within the la
21、boratory, and time. E1773.1.9 mean, nof a population, , average or expectedvalue of a characteristic in a population; of a sample, Xsum ofthe observed values in the sample divided by the sample size.E25863.1.10 percentile, nquantile of a sample or a population,for which the fraction less than or equ
22、al to the value isexpressed as a percentage. E25863.1.11 population, nthe totality of items or units ofmaterial under consideration. E25863.1.12 population parameter, nsummary measure of thevalues of some characteristic of a population. E25863.1.13 precision, nthe closeness of agreement betweenindep
23、endent test results obtained under stipulated conditions.E1773.1.14 quantile, nvalue such that a fraction f of the sampleor population is less than or equal to that value. E25863.1.15 repeatability, nprecision under repeatabilityconditions. E1773.1.16 repeatability conditions, nconditions where inde
24、-pendent test results are obtained with the same method onidentical test items in the same laboratory by the same operatorusing the same equipment within short intervals of time. E1773.1.17 repeatability standard deviation (sr), nthe standarddeviation of test results obtained under repeatabilitycond
25、itions. E1773.1.18 sample, na group of observations or test results,taken from a larger collection of observations or test results,which serves to provide information that may be used as a basisfor making a decision concerning the larger collection. E25863.1.19 sample size, n, nnumber of observed va
26、lues in thesample. E25863.1.20 sample statistic, nsummary measure of the ob-served values of a sample. E25863.1.21 standard deviationof a population, , the squareroot of the average or expected value of the squared deviationof a variable from its mean; of a sample, s, the square root ofthe sum of th
27、e squared deviations of the observed values in thesample from their mean divided by the sample size minus 1.E25863.1.22 test result, nthe value of a characteristic obtainedby carrying out a specified test method. E22823.1.23 test unit, nthe total quantity of material (containingone or more test spec
28、imens) needed to obtain a test result asspecified in the test method. See test result. E22823.1.24 variance, 2,s2,nsquare of the standard deviationof the population or sample. E25863.2 Definitions of Terms Specific to This Standard:3.2.1 bias equivalence, nequivalence of a populationmean with an acc
29、epted reference value.3.2.2 equivalence limit, E, nin equivalence testing, a limiton the difference between two population parameters.3.2.2.1 DiscussionIn certain applications, this may betermed practical limit or practical difference.3.2.3 equivalence test, na statistical test conducted withinprede
30、termined risks to confirm equivalence of two populationparameters.3.2.4 means equivalence, nequivalence of two populationmeans.3.2.5 non-inferiority, ncondition that the difference inmeans or variances of test results between a modified testingprocess and a current testing process with respect to ap
31、erformance characteristic is no greater than a predeterminedlimit in the direction of inferiority of the modified process tothe current process.3.2.5.1 DiscussionOther terms used for non-inferior are“equivalent or better” or “at least equivalent as.”3.2.6 paired samples design, nin means equivalence
32、testing, single samples are taken from the two populations at anumber of sampling points.3.2.6.1 DiscussionThis design is termed a randomizedblock design for a general number of populations sampled, andeach group of data within a sampling point is termed a block.3.2.7 power, nin equivalence testing,
33、 the probability ofaccepting equivalence, given the true difference between twopopulation means.3.2.7.1 DiscussionIn the case of testing for bias equiva-lence the power is the probability of accepting equivalence,given the true difference between a population mean and anaccepted reference value.3.2.
34、8 range equivalence, nequivalence of two populationmeans over a range of test result values.3.2.9 slope equivalence, nequivalence of the slope of alinear statistical relationship with the value one (1).3.2.10 two independent samples design, nin meansequivalence testing, replicate test results are de
35、termined inde-pendently from two populations at a single sampling time foreach population.E2935 1723.2.10.1 DiscussionThis design is termed a completelyrandomized design for a general number of sampled popula-tions.3.2.11 two one-sided tests (TOST) procedure, na statisti-cal procedure used for testi
36、ng the equivalence of the param-eters from two distributions (see equivalence).3.3 Symbols:a = intercept estimate (8.1.3)B = bias (7.1.1)b = slope estimate (8.1.3)dj= difference between a pair of test results at samplingpoint j (7.1.1)d= average difference (7.1.1)D = difference in sample means (6.1.
37、2)(X1.1.2)E = equivalence limit (5.2)E1= lower equivalence limit (5.2.1)E2= upper equivalence limit (5.2.1)ei= residual estimate (8.3.3)f = degrees of freedom for s (9.1.1)(X1.1.2)F1-=(1)thpercentile of the F distribution (10.3.1)fi= degrees of freedom for si(6.1.1)fp= degrees of freedom for sp(6.1.
38、2)() = the cumulative F distribution function (X1.6.3)H0: = null hypothesis (X1.1.1)Ha: = alternate hypothesis (X1.1.1)n = sample size (number of test results) from a popu-lation (5.4)(6.1.3)(7.1.1)(9.1.1)ni= sample size from ithpopulation (6.1.1)n1= sample size from population 1 (6.1.2)n2= sample s
39、ize from population 2 (6.1.2)R = ratio of two sample variances (5.5.2.1)r = sample correlation coefficient (8.3.2)5 = ratio of two population variances (X1.6.3)SXX= sum of squared deviations of X from their mean(8.1.3.2)SXY= sum of products of deviations of X and Y from theirmeans (8.1.3.2)SYY= sum
40、of squared deviations of Y from their mean(8.1.3.2)s = sample standard deviation (9.1.1)sB= sample standard deviation for bias (9.1.2)sd= standard deviation of the difference between twotest results (7.1.1)sD= sample standard deviation for mean difference(6.1.3)(X1.1.2)si= sample standard deviation
41、for ithpopulation (6.1.1)si2= sample variance for ithpopulation (6.1.1)s12= sample variance for population 1 (6.1.2)s12= variance of test results from the current process(10.3.1)s22= sample variance for population 2 (6.1.2)s22= variance of test results from the modified process(10.3.1)sp= pooled sam
42、ple standard deviation (6.1.2)sr= repeatability sample standard deviation (6.2)t = Students t statistic (6.1.4)(7.1.3)(9.1.3)t12,f=(1)thpercentile of the Students t distributionwith f degrees of freedom (X1.1.2)Xij= jthtest result from the ithpopulation (6.1)UCLR= upper confidence limit for 5 (10.3.
43、1)X= test result average (9.1.1)Xi= test result average for the ithpopulation (6.1.1)X1= test result average for population 1 (6.1.3)X2= test result average for population 2 (6.1.3)Z12=(1)thpercentile of the standard normal distribu-tion (X1.6.1) = (alpha) intercept parameter (8.1.1) = consumers ris
44、k (5.2.2)(6.2)(7.2) = (beta) slope parameter (8.1.1) = producers risk (5.4.1) = true mean difference between populations (5.4.1) = (delta) measurement error of X (X3.1.1) = (epsilon) measurement error of Y (X3.1.1) = (eta) true mean of Y (X3.1.1) = (theta) angle of the straight line to the horizonta
45、laxis (8.1.4.1)= estimate of (8.1.4.1)2= (kappa squared) information size (X3.3) = (lambda) ratio of measurement error variances of Yover X (8.1.1.1) = population mean (X1.4.1)i= ithpopulation mean (X1.1.1) = (nu) probability associated with informative confi-dence interval (X3.3.2) = approximate de
46、grees of freedom for sD(X1.1.4) = (xi) true mean of X (X3.1.1) = standard deviation of the test method (5.2)d= standard deviation of the true difference betweentwo populations (7.2)2= measurement error variances of Y (8.1.1)2= measurement error variances of X (8.1.1) = (tau) perpendicular distance f
47、rom line to origin(X3.1.3)() = standard normal cumulative distribution function(X1.6.1) = (phi) half width of confidence interval for (8.1.4.2) = (omega) width of the equivalence interval for (X3.2)3.4 Acronyms:3.4.1 ARV, naccepted reference value (5.5.1.1)(9.1)(X1.4)3.4.2 CRM, ncertified reference
48、material (5.5.1.1)(9.1)3.4.3 ILS, ninterlaboratory study (6.2)3.4.4 LCL, nlower confidence limit (6.2.5)(7.2.3)3.4.5 TOST, ntwo one-sided tests (5.5.1) (Section 6)(Section 7) (Section 9)(Appendix X1)3.4.6 UCL, nupper confidence limit (6.2.5)(7.2.3)4. Significance and Use4.1 Laboratories conducting r
49、outine testing have a continu-ing need to make improvements in their testing processes. Inthese situations it must be demonstrated that any changes willneither cause an undesirable shift in the test results from thecurrent testing process nor substantially affect a performancecharacteristic of the test method. This standard providesguidance on experiments and statistical methods needed todemonstrate that the test results from a modified testing process
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