1、Designation: E2935 13E2935 14 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
2、 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 methodology for conducting equivalence testing on numeric
3、al data from two sources todetermine if their true means are similar within predetermined limits.1.2 Applications include (1) equivalence testing for bias against an accepted reference value, (2) determining equivalence of twotest methods, test apparatus, instruments, reagent sources, or operators w
4、ithin a laboratory, and (3) equivalence of two laboratoriesin a method transfer.1.3 The current guidance in this standard applies only to experiments conducted on a single material. Guidance is given fordetermining the amount of data required for an equivalence trial.1.4 The statistical methodology
5、for determining equivalence used is the “Two one-sided t-test” (TOST). The control of risksassociated with the equivalence decision is discussed.1.5 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.1.6 This standard does not pu
6、rport to address all of the safety concerns, if any, associated with its use. It is the responsibilityof the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatorylimitations prior to use.2. Referenced Documents2.1 ASTM Standards:2E17
7、7 Practice for Use of the Terms Precision and Bias in ASTM Test MethodsE456 Terminology Relating to Quality and StatisticsE2282 Guide for Defining the Test Result of a Test MethodE2586 Practice for Calculating and Using Basic Statistics3. Terminology3.1 DefinitionsSee Terminology E456 for a more ext
8、ensive listing of statistical terms.3.1.1 accepted reference value, na value that serves as an agreed-upon reference for comparison, and which is derived as: (1)a theoretical or established value, based on scientific principles, (2) an assigned or certified value, based on experimental work ofsome n
9、ational or international organization, or (3) a consensus or certified value, based on collaborative experimental work underthe auspices of a scientific or engineering group. E1773.1.2 bias, nthe difference between the expectation of the test results and an accepted reference value. E1773.1.3 confid
10、ence interval, nan interval estimate L, U with the statistics L and U as limits for the parameter and withconfidence level 1 , where Pr(L U) 1 . E25863.1.3.1 DiscussionThe confidence level, 1 , reflects the proportion of cases that the confidence interval L, U would contain or cover the trueparamete
11、r value in a series of repeated random samples under identical conditions. Once L and U are given values, the resulting1 This test method is under the jurisdiction of ASTM Committee E11 on Quality and Statistics and is the direct responsibility of Subcommittee E11.20 on Test MethodEvaluation and Qua
12、lity Control.Current edition approved Aug. 1, 2013Oct. 1, 2014. Published August 2013. Originally approved in 2013. Last previous edition approved in 2013 as E2935 13. DOI:10.1520/E2935-13.10.1520/E2935-14.2 For referencedASTM standards, visit theASTM website, www.astm.org, or contactASTM Customer S
13、ervice at serviceastm.org. For Annual Book of ASTM Standardsvolume information, refer to the standards Document Summary page on the ASTM website.This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the prev
14、ious version. Becauseit may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current versionof the standard as published by ASTM is to be considered the official document.Copyright ASTM In
15、ternational, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1confidence interval either does or does not contain it. In this sense “confidence” applies not to the particular interval but only tothe long run proportion of cases when repeating the procedure many tim
16、es.3.1.4 confidence level, nthe value, 1 , of the probability associated with a confidence interval, often expressed as apercentage. E25863.1.4.1 Discussion is generally a small number. Confidence level is often 95 % or 99 %.3.1.5 confidence limit, neach of the limits, L and U, of a confidence inter
17、val, or the limit of a one-sided confidence interval.E25863.1.6 degrees of freedom, nthe number of independent data points minus the number of parameters that have to be estimatedbefore calculating the variance. E25863.1.7 equivalence, nsimilarity between two population parameters within predetermin
18、ed limits.3.1.8 intermediate precision conditions, nconditions under which test results are obtained with the same test method using testunits or test specimens taken at random from a single quantity of material that is as nearly homogeneous as possible, and withchanging conditions such as operator,
19、 measuring equipment, location within the laboratory, and time. E1773.1.9 mean, nof a population, , average or expected value of a characteristic in a population of a sample,X sum of theobserved values in the sample divided by the sample size. E25863.1.10 population, nthe totality of items or units
20、of material under consideration. E25863.1.11 population parameter, nsummary measure of the values of some characteristic of a population. E25863.1.12 precision, nthe closeness of agreement between independent test results obtained under stipulated conditions. E1773.1.13 repeatability, nprecision und
21、er repeatability conditions. E1773.1.14 repeatability conditions, nconditions where independent test results are obtained with the same method on identical testitems in the same laboratory by the same operator using the same equipment within short intervals of time. E1773.1.15 repeatability standard
22、 deviation (sr), nthe standard deviation of test results obtained under repeatability conditions.E1773.1.16 sample, na group of observations or test results, taken from a larger collection of observations or test results, whichserves to provide information that may be used as a basis for making a de
23、cision concerning the larger collection. E25863.1.17 sample size, n, nnumber of observed values in the sample. E25863.1.18 sample statistic, nsummary measure of the observed values of a sample. E25863.1.19 test result, nthe value of a characteristic obtained by carrying out a specified test method.
24、E22823.1.20 test unit, nthe total quantity of material (containing one or more test specimens) needed to obtain a test result asspecified in the test method. See test result. E22823.2 Definitions of Terms Specific to This Standard:3.2.1 bias equivalence, nequivalence of a population mean with an acc
25、epted reference value.3.2.2 equivalence limit, E, nin equivalence testing, a limit on the difference between two population parameters.3.2.2.1 DiscussionIn certain applications, this may be termed practical limit or practical difference.3.2.3 equivalence test, na statistical test conducted within pr
26、edetermined risks to confirm equivalence of two populationparameters.3.2.4 means equivalence, nequivalence of two population means.3.2.5 paired samples design, nin means equivalence testing, single samples are taken from the two populations at a numberof sampling points.3.2.5.1 DiscussionThis design
27、 is termed a randomized block design for a general number of populations sampled, and each group of data within asampling point is termed a block.3.2.6 power, nin equivalence testing, the probability of accepting equivalence, given the true difference between twopopulation means.E2935 1423.2.6.1 Dis
28、cussionIn the case of testing for bias equivalence the power is the probability of accepting equivalence, given the true difference betweena population mean and an accepted reference value.3.2.7 two independent samples design, nin means equivalence testing, replicate test results are determined inde
29、pendently fromtwo populations at a single sampling time for each population.3.2.7.1 DiscussionThis design is termed a completely randomized design for a general number of populations sampled.3.3 Symbols:B = bias (7.1.1)dj = difference between a pair of test results at sampling point j (7.1.1)d = ave
30、rage difference (7.1.1)D = difference in sample means (6.1.2) (X1.1.2)E = equivalence limit (5.2.1)E1 = lower equivalence limit (5.2.1.1)E2 = upper equivalence limit (5.2.1.1)H0: = null hypothesis (X1.1.1)HA: = alternate hypothesis (X1.1.1)f = degrees of freedom for s (8.1.1) (X1.1.2)fi = degrees of
31、 freedom for si (6.1.1)fp = degrees of freedom for sp (6.1.2)n = sample size (number of test results) from a population (5.3) (6.1.3) (7.1.1) (8.1.1)ni = sample size from ith population (6.1.1)n1 = sample size from population 1 (6.1.2)n2 = sample size from population 2 (6.1.2)s = sample standard dev
32、iation (8.1.1)sB = sample standard deviation for bias (8.1.2)sd = standard deviation of the difference between two test results (7.1.1)sD = sample standard deviation for mean difference (6.1.3) (X1.1.2)si = sample standard deviation for ith population (6.1.1)si2 = sample variance for ith population
33、(6.1.1)s12 = sample variance for population 1 (6.1.2)s22 = sample variance for population 2 (6.1.2)sp = pooled sample standard deviation (6.1.2)sr = repeatability sample standard deviation (6.2)t = Students t statistic (6.1.4) (7.1.3) (8.1.3)t12,f = (1-)th percentile of the Students t distribution w
34、ith f degrees of freedom (X1.1.2)Xij = jth test result from the ith population (6.1)X = test result average (8.1.1)Xi = test result average for the ith population (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-)th percentile of the s
35、tandard normal distribution (X1.5.1) = consumers risk (5.2.2) (6.2) (7.2) = producers risk (5.3) = true mean difference between populations (5.3) = population mean (X1.4.1)i = ith population mean (X1.1.1) = approximate degrees of freedom for sD (X1.1.4) = standard deviation of the test method (5.2.3
36、)d = standard deviation of the true difference between two populations (7.2)() = standard normal cumulative distribution function (X1.5.1)3.4 Acronyms:3.4.1 ARV, naccepted reference value (5.1.2) (8.1) (X1.4)3.4.2 CRM, ncertified reference material (5.1.2) (8.1)3.4.3 ILS, ninterlaboratory study (6.2
37、)E2935 1433.4.4 LCL, nlower confidence limit (6.2.5) (7.2.3)3.4.5 TOST, ntwo one-sided t test (4.3) (Section 6) (Section 7) (Section 8) (Appendix X1)3.4.6 UCL, nupper confidence limit (6.2.5) (7.2.3)4. Significance and Use4.1 Laboratories conducting routine testing have a continuing need to evaluate
38、 test result bias, to evaluate changes for improvingthe test process performance, or to validate the transfer of a test method to a new location or apparatus. In all situations it mustbe demonstrated that any bias or innovation will have negligible effect on test results for a characteristic of a ma
39、terial.This standardprovides statistical methods to confirm that the mean test results from a testing process are equivalent to those from a referencestandard or another testing process, where equivalence is defined as agreement within prescribed limits, termed equivalence limits.4.1.1 The intra-lab
40、oratory applications in this practice include, but are not limited to, the following:(1) Evaluating the bias of a test method with respect to a certified reference material,(2) Evaluating bias due to a minor change in a test method procedure,(3) Qualifying new instruments, apparatus, or operators in
41、 a laboratory, and(4) Qualifying new sources of reagents or other materials used in the test procedure.4.1.2 This practice also supports evaluating bias systematic differences in a method transfer from a developing laboratory to areceiving laboratory.4.2 This practice currently deals only with the e
42、quivalence of population means. In this standard, a population refers to ahypothetical set of test results arising from a stable testing process that measures a characteristic of a single material.NOTE 1The equivalence concept can also apply to population parameters other than means, such as precisi
43、on, stated as variances, standard deviations,or relative standard deviations (coefficients of variation), linearity, sensitivity, specificity, etc.4.3 The data analysis for equivalence testing of population means in this practice uses a statistical methodology termed the “Twoone-sided t-test” (TOST)
44、 procedure which shall be described in detail in this standard (see X1.1). The TOST procedure will beadapted to the type of objective and experiment design selected.4.3.1 Historically, this procedure originated in the pharmaceutical industry for use in bioequivalence trials (1, 2),3 denoted asthe Tw
45、o One-Sided Test, and has since been adopted for other applications, particularly in testing and measurement applications(3, 4).4.3.2 The conventional Students t test used for detecting differences is not recommended for equivalence testing as it does notproperly control the consumers and producers
46、risks for this application (see X1.3).4.4 Risk ManagementThis practice provides recommendations for the design of an equivalence experiment, and two basicdesigns are discussed. Guidance is provided for determining the amount of data required to control the risks of making the wrongdecision in accept
47、ing or rejecting equivalence (see X1.2).4.4.1 The consumers risk is the probability of accepting equivalence when the actual bias or difference in means is equal tothe equivalence limit. This probability is controlled to a low level so that accepting equivalence gives a high degree of assurancethat
48、differences in question are less than the equivalence limit.4.4.2 The producers risk is the risk of falsely rejecting equivalence. If improvements are rejected this can lead to opportunitylosses to the company and its laboratories (the producers) or cause additional unnecessary effort in improving t
49、he testing process.5. Planning the Equivalence Study5.1 Objectives and Design SelectionThis practice supports two equivalence study objectives: (1) determining the biasmeansequivalence of a test method test results from two testing processes or (2) determining the meansbias equivalence of test resultsfrom two testing processes. In both objectivesa test method. In both objectives, two population means are compared forequivalence.5.1.1 Means EquivalenceThis study compares the aver
