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| corresponding to hypotheses of general form | corresponding to null and alternative statistical hypotheses of general form |
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| HA : -t $$ \leq \theta \leq t$$ | HA : -t $$ \leq \theta \leq$$ t |
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| which switches around the 'usual' hypotheses of form | where $$\theta$$ is a function of parameters of interest (e.g. a difference between two group means) and t is the effect size of minimal interest (e.g. minimum difference in a pair of group means which is of clinical interest). |
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| H0: $$\theta$$ = t and HA: $$\theta \neq$$ t | Equivalence tests are also known as reverse tests because they switch around the 'usual' hypotheses of form H0: $$\theta$$ = t and HA: $$\theta \ne$$ t |
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| Lew MJ (2006) illustrates equivalence tests for two-sample (un)paired t-tests. The critical p-values ('''some''' of which are presented in the tables at the back of this paper) may be computed for any sample sizes using the R codes below: | SAS and FORTRAN programs with help guides are available [http://zima04.zi-mannheim.de/wktsheq/ for free download] which run equivalence analyses for other statistical tests using methodology described in Wellek (2003). It is easier to run the SAS programs. After downloading change the file name from *.sas to *.sss before clicking on the icon. '''CBSUERS: If SAS is not on your PC it can be added on by one of our CBSU IT people.''' |
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| If the p-value from the student's t-test on the raw data is greater than bout2 there is no difference between the observed group means in detecting effect size, d, type II error, beta, for equal group sizes, n. | * [:FAQ/mageq: Suggested effect sizes, t] |
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| [TYPE INTO R THE DESIRED INPUTS D, N, AND BETA USING VALUES IN FORM BELOW]. | * [:FAQ/tequi: R code for equivalence test for (un)paired t-tests] |
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| {{{ d <- 0.5 n <- 11 beta <- 0.05 }}} [THEN COPY AND PASTE THE BELOW INTO R] {{{ cv <- sqrt(qf(p=beta,df1=1,df2=(2*n)-2,,ncp=((n*n*d*d)/(2*n)))) bout <- 2*pt(q=cv,df=(2*n)-2)-1 bout2 <- 1- bout print(bout2) }}} |
* [:FAQ/zequiv: R code for equivalence test for one sample z-test] |
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| As above but allowing different group sizes, n1 and n2. [TYPE INTO R THE DESIRED INPUTS D, N1, N2 AND BETA USING VALUES IN FORM BELOW]. {{{ d <- 1.5 n1 <- 10 n2 <- 10 beta <- 0.02 }}} [THEN COPY AND PASTE THE BELOW INTO R] {{{ cv <- sqrt(qf(p=beta,df1=1,df2=n1+n2-2,,ncp=((n1*n2*d*d)/(n1+n2)))) bout <- 2*pt(q=cv,df=n1+n2-2)-1 bout2 <- 1- bout print(bout2) }}} |
* [:FAQ/bin2: R code for equivalence test for two unrelated proportions] |
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| SAS and FORTRAN programs with help guides are available [http://zima04.zi-mannheim.de/wktsheq/ for free download] which run equivalence analyses for other statistical tests using methodology described in Wellek(2003). | * [:FAQ/mcnequiv:R code for equivalence test using McNemar's test] * [:FAQ/oneweq: R code for equivalence test for one-way ANOVA] |
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| Lew MJ (2006) Principles: When there should be no difference - how to fail to reject the null hypothesis ''Trends in Pharmacological Sciences'' '''27(5)''' 274-278. [http://www.sciencedirect.com/science Available to CBSU users on ScienceDirect] | Lew MJ (2006) Principles: When there should be no difference - how to fail to reject the null hypothesis. ''Trends in Pharmacological Sciences'' '''27(5)''' 274-278. [http://www.sciencedirect.com/science Available to CBSU users on ScienceDirect.] |
Statistical tests of equivalence
Wellek (2003) illustrates the application of a series of familiar statistical tests corresponding to null and alternative statistical hypotheses of general form
H0: $$\theta \leq $$-t or $$\theta \geq$$ t and HA : -t $$ \leq \theta \leq$$ t
where $$\theta$$ is a function of parameters of interest (e.g. a difference between two group means) and t is the effect size of minimal interest (e.g. minimum difference in a pair of group means which is of clinical interest).
Equivalence tests are also known as reverse tests because they switch around the 'usual' hypotheses of form
H0: $$\theta$$ = t and HA: $$\theta \ne$$ t
and so the emphasis is on verifying rather than rejecting hypotheses such as equality of group means or zero correlations. Failing to reject a null hypothesis is not the same as showing it to be valid.
SAS and FORTRAN programs with help guides are available [http://zima04.zi-mannheim.de/wktsheq/ for free download] which run equivalence analyses for other statistical tests using methodology described in Wellek (2003). It is easier to run the SAS programs. After downloading change the file name from *.sas to *.sss before clicking on the icon. CBSUERS: If SAS is not on your PC it can be added on by one of our CBSU IT people.
- [:FAQ/mageq: Suggested effect sizes, t]
- [:FAQ/tequi: R code for equivalence test for (un)paired t-tests]
- [:FAQ/zequiv: R code for equivalence test for one sample z-test]
- [:FAQ/bin2: R code for equivalence test for two unrelated proportions]
[:FAQ/mcnequiv:R code for equivalence test using McNemar's test]
- [:FAQ/oneweq: R code for equivalence test for one-way ANOVA]
References
Lew MJ (2006) Principles: When there should be no difference - how to fail to reject the null hypothesis. Trends in Pharmacological Sciences 27(5) 274-278. [http://www.sciencedirect.com/science Available to CBSU users on ScienceDirect.]
Wellek S (2003) Testing of statistical hypotheses of equivalence. Chapman and Hall/CRC Press.