# ASTM E2935-17

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 revision, 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 ology for con-ducting equivalence testing on numerical data 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 s, test apparatus, instruments,reagent sources, or operators within a laboratory or equiva-lence of two laboratories in a transfer, and 3determining non-inferiority of a modified test procedure versusa current test procedure with respect to a perance charac-teristic.1.3 The guidance in this standard applies to experimentsconducted on 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 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 safety, 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 International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade TBT Committee.2. Referenced Documents2.1 ASTM Standards2E177 Practice for Use of the Terms Precision and Bias inASTM Test sE456 Terminology Relating to Quality and StatisticsE2282 Guide for Defining the Test Result of a Test E2586 Practice for Calculating and Using Basic StatisticsE3080 Practice for Regression Analysis2.2 USP Standard3USP Validation of Alternative Microbiologicals3. Terminology3.1 DefinitionsSee Terminology E456 for a more exten-sive listing 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 international organization, or3 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, nan interval estimate L, Uwith the statistics L and U as limits for the parameter andwith confidence level 1 , where PrL 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 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 times.1This test is under the jurisdiction of ASTM Committee E11 on Qualityand Statistics and is the direct responsibility of Subcommittee E11.20 on Test uation and Quality Control.Current edition approved Oct. 1, 2017. Published November 2017. Originallyapproved in 2013. Last previous edition approved in 2016 as E2935 16. DOI10.1520/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 ination, refer to the standards Document Summary page onthe ASTM website.3Available 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 internationally 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 probabilityassociated 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. E25863.1.6 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 conditions, nconditions un-der which test results are obtained with the same test using 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 laboratory, 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 equal 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 thues of some characteristic of a population. E25863.1.13 precision, nthe closeness of agreement betweenindependent 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-pendent test results are obtained with the same 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 repeatabilityconditions. 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 ination that may be used as a basisfor making a decision concerning the larger collection. E25863.1.19 sample size, n, nnumber of observed values 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 the 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 . E22823.1.23 test unit, nthe total quantity of material containingone or more test specimens needed to obtain a test result asspecified in the test . 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 Standard3.2.1 bias equivalence, nequivalence of a populationmean with an accepted 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 withinpredetermined 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 aperance 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 equivalencetesting, 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, 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.3.2.8 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 determined 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 testing the equivalence of the param-eters from two distributions see equivalence.3.3 Symbolsa intercept estimate 8.1.3B bias 7.1.1b slope estimate 8.1.3dj difference between a pair of test results at samplingpoint j 7.1.1d average difference 7.1.1D difference in sample means 6.1.2X1.1.2E equivalence limit 5.2E1 lower equivalence limit 5.2.1E2 upper equivalence limit 5.2.1ei residual estimate 8.3.3f degrees of freedom for s 9.1.1X1.1.2F1-1thpercentile of the F distribution 10.3.1fi degrees of freedom for si6.1.1fp degrees of freedom for sp6.1.2 the cumulative F distribution function X1.6.3H0 null hypothesis X1.1.1Ha alternate hypothesis X1.1.1n sample size number of test results from a popu-lation 5.46.1.37.1.19.1.1ni sample size from ithpopulation 6.1.1n1 sample size from population 1 6.1.2n2 sample size from population 2 6.1.2R ratio of two sample variances 5.5.2.1r sample correlation coefficient 8.3.25 ratio of two population variances X1.6.3SXX sum of squared deviations of X from their mean8.1.3.2SXY sum of products of deviations of X and Y from theirmeans 8.1.3.2SYY sum of squared deviations of Y from their mean8.1.3.2s sample standard deviation 9.1.1sB sample standard deviation for bias 9.1.2sd standard deviation of the difference between twotest results 7.1.1sD sample standard deviation for mean difference6.1.3X1.1.2si sample standard deviation for ithpopulation 6.1.1si2 sample variance for ithpopulation 6.1.1s12 sample variance for population 1 6.1.2s12 variance of test results from the current process10.3.1s22 sample variance for population 2 6.1.2s22 variance of test results from the modified process10.3.1sp pooled sample standard deviation 6.1.2sr repeatability sample standard deviation 6.2t Students t statistic 6.1.47.1.39.1.3t12,f1thpercentile of the Students t distributionwith f degrees of freedom X1.1.2Xij jthtest result from the ithpopulation 6.1UCLR upper confidence limit for 5 10.3.1X test result average 9.1.1Xi test result average for the ithpopulation 6.1.1X1 test result average for population 1 6.1.3X2 test result average for population 2 6.1.3Z121thpercentile of the standard normal distribu-tion X1.6.1 alpha intercept parameter 8.1.1 consumers risk 5.2.26.27.2 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 horizontalaxis 8.1.4.1 estimate of 8.1.4.12 kappa squared ination size X3.3 lambda ratio of measurement error variances of Yover X 8.1.1.1 population mean X1.4.1i ithpopulation mean X1.1.1 nu probability associated with inative confi-dence interval X3.3.2 approximate degrees of freedom for sDX1.1.4 xi true mean of X X3.1.1 standard deviation of the test 5.2d standard deviation of the true difference betweentwo populations 7.22 measurement error variances of Y 8.1.12 measurement error variances of X 8.1.1 tau perpendicular distance from line to originX3.1.3 standard normal cumulative distribution functionX1.6.1 phi half width of confidence interval for 8.1.4.2 omega width of the equivalence interval for X3.23.4 Acronyms3.4.1 ARV, naccepted reference value 5.5.1.19.1X1.43.4.2 CRM, ncertified reference material 5.5.1.19.13.4.3 ILS, ninterlaboratory study 6.23.4.4 LCL, nlower confidence limit 6.2.57.2.33.4.5 TOST, ntwo one-sided tests 5.5.1 Section 6Section 7 Section 9Appendix X13.4.6 UCL, nupper confidence limit 6.2.57.2.34. Significance and Use4.1 Laboratories conducting routine 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 perancecharacteristic of the test . This standard providesguidance on experiments and statistical s needed todemonstrate that the test results from a modified testing process