| ObjectiveNot only usually applied in clinical trials of new drugs,equivalence evaluations are extensively used in other medical territories,such as whether two test methods can replace each other,whether two preventive measures have the same effects,etc. To solve these issues,researchers will be facing ethical problems if the control group is set to be blank.On the contrary,it is fight without question if the control group is set to be positive and equivalence tests are applied.However,because of some objective reasons,it is difficult to put it into effect in various practical research fields despite the equivalence assessment is right.For example,many researchers make the conclusions of 'equivalence' or 'no difference' when P value is larger thanαwith traditional significance test,which regard equivalence test and traditional test as the same methods.Accordingly,some so called new methods and new measures are extended,while those existing methods and measures that have absolute effects are replaced.As a result,the great loss or harm to health would probably be caused.Some domestic researches have written articles to emphasize the differences between traditional test and equivalence test,and introduce the software about equivalence test.However,both inside and outside the country,there have been no reports both on the relation laws between them and on the attempt to replace equivalence test with traditional significance test.So how to avoid the existing misuses between the two test methods is urgently to be resolved.This study will explore the relation laws under different designs and data,and try to probe into the probability to make an equivalence conclusion when the P value is larger thanαwith traditional significance test based on the equivalence boundary and this study results.At last,this study results will be expressed in forms as briefly as possible for consulting.And then,unnecessary to learn theories about the equivalence test,researchers in different fields will be able to make decisions about equivalence or not just by means of traditional significance test and results of this study.This may avoid the great loss or harm to health caused by misuses of the two test methods.Besides,when reading literatures,researchers can make a right judgment by utilizing information from literatures and this study results.This study can also enrich contents of equivalence test.To state conveniently,the traditional significance test is called 'difference test' for short in this paper,which is to compare whether the statistic difference exists between groups.MethodThis study concerns three designs,which are two independent-sample(means), paired-sample(means)and two independent- sample(proportions).Through simulating with programs I had made,the research was completed on MATLAB software packages.All the programs concerned in difference test were verified by SPSS 13.0,and those concerned in equivalence test were verified by EquivTestTM2.0. According to the formulae from statistics teaching materials and relevant literatures of clinical pharmacology trials,we calculated sample sizes of the two kinds of tests.All the sample sizes agree with the results of nQuery Advisor.1.The decision of critical P values to make equivalence conclusions by difference test(1)Set up parametersFor measurement data:α(0.05),β(0.10,0.20),δ(0.10μs,0.15μs,0.20μs), CV(0.10,0.20,0.30,0.40.SDs of the two compared populations were set to be equal and CV=σ/μs.Equivalence bounds were set to be equal to the difference of population means,which were｜μT—μS｜=δ.For enumeration data:α(0.05,),β(0.10,0.20),δ(0.05,0.10,0.15),Ï€S(0.60～0.90),Ï€T(0.50～0.95).Equivalence bounds were set to be equal to the difference of population means,which were｜πT—πS｜=δ.Sample sizes were calculated according to the formulae of difference test.(2)Compute the P values of difference testsWe sampled units from the two populations repeatedly,and made difference tests for those samples whose sampling results were｜(?)T—(?)S｜＜δor｜PT—PS｜＜δand then acquired the P values.(3)Make equivalence testequivalence tests were made for the samples whose P values were larger thanα. And the P value was recorded as PE if they were equivalent,otherwise,it was recorded as PNE.(4)Define P(cut)which would be applied to make equivalence conclusions base on difference tests.The 99thpercentile of PNEwas defined to be Pcut.When the P value of difference test was equal to or larger than Pcut,the conclusion of equivalence would be made. (5)Check PcutAccording to the level of Pcut,the rate of erroneous judgment were calculated from samples which were｜(?)T—(?)S｜＜δor｜PT—PS｜＜δ.(6)Define P(cut)under different parametersP(cut)under various parameters were listed in the form for inquiring.(7)The rates of erroneous judgment with different P value levels to make equivalence conclusionFixing the sample sizes calculated according to the formulae of difference test, and changing the levels of making equivalence judgment from small to large,and in the meanwhile,the rates of erroneous judgment were computed and listed in the forms for inquiring.A user would learn the risk from the forms to make equivalence conclusion with the P value of difference tests.The P values here to make equivalence judgment were recorded as Pvarto distinguish from Pcut.2.The sample sizes to make equivalence conclusions with difference test(1)Set parametersFor measurement data:α(0.05),β(0.10,0.20),δ(0.10μs,0.15μs,0.20μs), CV(0.10,0.20,0.30,0.40.SDs of the two populations were set to be equal and CV =σ/μs.Equivalence bounds were set to be less than the difference of population means,which were｜μT—μS｜＜δ.For enumeration data:α(0.05,),β(0.10,0.20),δ(0.05,0.10,0.15),Ï€S (0.60～0.90),Ï€T(0.50～0.95).Equivalence bounds were set to be less than the difference of population means,which were｜πT—πS｜＜δ.Sample sizes were calculated according to the formulae of equivalence test.(2)Compute the P value of difference testWe sampled units from the two populations repeatedly,made difference tests for those samples whose sampling results were｜(?)T—(?)S｜＜δor｜PT—PS｜＜δand then acquired the P values.(3)Make equivalence testWhile increasing sample sizes,equivalence tests were made for the samples whose P values were larger thanαuntil the test results of equivalence tests were all equivalent(the rate was set to be 99.6%in the program).Sample sizes at this moment were recorded as nCOand listed in the forms for inquiring.Results1.The critical P value to make an equivalence conclusion by difference test(1)Make equivalence conclusions according to PcutFor measurement data:except for some situations that sample sizes are very small,Pcutwill get smaller if variation get larger or sample size get larger,and Pcut will get larger if equivalence bounds get larger.So the corresponding Pcutwill lower down if the practical sample size is bigger than this study.The probability of making equivalence conclusions according to Pcutis less thanα+2β×0.01(e.g.＜0.05+0.002=0.052).The rates of erroneous judgment vary according to the level of variation.Fixing the variation,the rate of erroneous judgment will get smaller if the sample size gets larger.For example,under the circumstances of two independent-sample design,the results of checking shows that the rate of erroneous judgment are less than 0.052 when parameters are set as follows:α=0.05,β=0.10,δ=0.10μSFor enumeration data:There was no obvious changing trend among Pcut.Like measurement data,Pcutwould get smaller if sample size got larger.The probability of making equivalence conclusions according to Pcutis less thanα+2β×0.01.The rates of erroneous judgment vary according to the level of populations proportions.And it will get smaller if sample size gets larger.For example,the results of checking shows that the rates of erroneous judgment are 0.024～0.038;0.036～0.039;0.048～0.051;0.037～0.047 respectively with the increasing of population proportions,which are less than 0.052 when parameters are set as follows:α=0.05,β=0.10,δ=0.10μS.(2)Relations between Pvarand the rate of erroneous judgmentWith fixed sample size,the rate of erroneous judgment will lower down if the Pvargets large gradually which comes from difference tests.The rate of erroneous judgment is relatively high when Pvaris barely higher thanαwhich indicates the compared may not be equivalent when the difference is not significant.Except for few situations,the rates of erroneous judgment are close to each other for all the three designs.So a brief form can be made for inquiring.The maximum value of the same situation is adopted to control typeâ… error.If the rate of erroneous judgment is set to be dependent variable,and the Pvaris set to be independent variable,we can get models which show cubic relations both whenβis equal to 0.10 and 0.20.The two model are Y=0.29ï¼2.22X+6.24X2ï¼5.87X3 and Y=0.56ï¼3.83X+10.18X2ï¼9.31X3 respectively,whose R2s are both higher than 0.98 with statistic significance(P=0.000).2.The sample size to make an equivalence conclusion with difference test(1)Two independent sample design(means)If nE is set to be basis,we can make equivalence conclusions when the sample size increases by 0.5-1.4 times with the condition that the difference of population means is less than equivalence bounds and P value is larger thanα.As the sample sizes to make equivalence conclusions(nco)is at least 1.5 times as large as no,so the power of difference test is nearly 100%.And if the test result is stillP＞α,we can only think the compared are equivalent. Under the condition thatαandβare fixed,nCOwill get larger if the variation or difference between population means get larger;and it will get smaller ifδgets larger.Under the condition thatαandβare fixed,the increased proportion based on nE will get smaller with the increasing of variations;and it will get larger with the increasing of difference between population means;and it will keep stable with the change ofδ.(2)Paired-sample designWe can not get the proper sample size of making equivalence conclusions with such a design.(3)Two independent-sample design(proportions)With fixedÏ€S andÏ€S-Ï€T,nCOhas relations with the relative magnitudes betweenÏ€S andÏ€T though the sample sizes needed to make equivalence test are equal.And nCO will be smaller whenÏ€T is far from 0.50.On the contrary,it will be larger.With fixed equivalence bound and difference betweenÏ€S andÏ€T,nCOwill lower down when both proportions get far from 0.50.Conelusions1.To make an equivalence conclusion according to PcutFor measurement data:According to the practical sample size and variation(SD/(?)s),a user can make an equivalence conclusion if the P value of difference test is equal to or larger than Pcutlisted in the forms.The probability to commit error is less thanα+2β×0.01.For enumeration data:According to the practical sample size and the relative magnitude between two sample proportions,a user can make an equivalence conclusion if the P value of difference test is equal to or larger than Pcutlisted in the forms.The probability to commit error is less thanα+2β×0.01. 2.To make an equivalence conclusion according to PvarIf sample sizes are calculated with equivalence bounds set to be differences of two means or two proportions,a user can make the decision of equivalence or not according to the P values of difference test,whose rates of erroneous judgment are listed in the forms.If necessary,the models can be applied to compute rates of erroneous judgment.3.To make an equivalence conclusion according to nCOTo get 100%coincidence,though sample sizes change with certain laws,it has relative many limitations and is inconvenient to be applied.For two independent-sample design(proportions),it is inconvenient to make equivalence conclusions by means of nCObecause of too many limitations.This study may enrich the contents of equivalence test,and in the meanwhile,a convenient choice is provided for researchers to make equivalence tests.This study is intended to solve the existing problems with large quantities at present,that is to make 'no difference' or 'equivalence' conclusions directly when the P value is larger thanα.This study just manages to find the laws or conditions to make equivalence conclusions when P value of difference test is larger thanα.So it should be emphasized that we do not deny that it may be equivalent when P value is less than or equal toα. |
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