Two one-sided tests

Schuirmann, D. J. A Comparison of the Two One-Sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability. J. Pharmacol. Biopharm., 15, 1987, 657-680. 

Schuirmann DJ (1987) A comparison of two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability Journal of Pharmacokinetics and... 

Parametric methods, i.e. those based on normal distribution theory, are recommended for the analysis of log-transformed bioequivaience measures. The general approach is to construct a 90% confidence interval for the quantity pT-pR and to reach a conclusion of pharmacokinetic equivalence if this confidence interval is within the stated limits. The nature of parametric confidence intervals means that this is equivalent to carrying out two one-sided tests of the hypothesis at the 5% level of significance 10, 77). The antilogs of the confidence limits obtained constitute the 90% confidence interval for the ratio of the geometric means between the multisource and comparator products. 

Now, it is a fact that in drug development two-tailed tests are commonly performed, whatever the comparator. (An exception is tests of equivalence where two one-sided tests are commonly performed. This confusing state of affairs will be covered in Chapters 15... 

As an example to help understand the effect of equivalence on sample size, consider a case where we wish to show that the difference in FEVi between two treatments is not greater than 200 ml and where the standard deviation is 350 ml for conventional type I and type II errors rates of 0.05 and 0.2. If we assume that the drugs are in fact exactly identical, the sample size needed (using a Normal approximation) is 53. If we allow for a true difference of 50 ml this rises to 69. On the other hand, if we wished to demonstrate superiority of one treatment over another for a clinically relevant difference of 200 ml with the other values as before, a sample size of 48 would suffice. Thus, in the best possible case a rise of about 10% in the sample size is needed (from 48 to 53). This difference arises because we have two one-sided tests each of which must be significant in order to prove efficacy. To have 80% power each must (in the case of exact equality) have approximately 90% power(because 0.9 x 0.9 = 0.8). The relevant sum of z-values for the power calculation is thus 1.2816-1-1.6449 = 2.93 as opposed to for a conventional trial 0.8416 -I-1.9600 = 2.8. The ratio of the square of 2.93 to 2.8 is about 1.1 explaining the 10% Increase in sample size. 

Let 8,8 >0, be a limit of indifference . True absolute differences greater than this constitute a nonignorable distinction between drugs. We wish to show that —8 < Ij, — jj, < 8. An approximately adequate way to do this is to carry out two one-sided tests of significance. Let — ix = t, then we can test ... 

The current evaluation criteria are based on the two one-sided test approach, also commonly referred to as the Confidence Interval Approach or Average Bioequivalence, which determines whether the average values for the pharmacokinetic parameters measured after the administration of test and reference products are comparable. This approach involves the calculation of a 90% confidence interval about the ratio of the averages of T and R products for AUC and values. To establish bioequivalence, the AUC and of the T product should not be less than 0.80 (80%) or greater than 1.25 (125%) of the R product based on log-transformed data (i.e., a bioequivalence limit of 80 to 125%). For some time prior to the use of log-transformed data, the nontransformed data were used to assess bioequivalence. In 1989, it was realized that log transformation of the data enables a comparison based on the ratio of the two averages rather than the difference between the averages in an additive manner." Moreover, most biological data correspond to a log-normal distribution rather than to a normal distribution. 

Eor the average bioeqnivalence approach, two one-sided tests of hypothesis at the 5% level of significance are carried ont to constrnct the 99% confidence intervals. Eor the popnlation and individnal bioeqnivalence approach, an npper 95% confidence bonnd for the popnlation or individnal criterion is estimated, which shonld be less than or eqnal to the bioeqnivalence limit (i.e., 9/, 9 ). 

The assessment of bioequivalence is based on 90% confidence intervals for the ratio of the population geometric means (test/reference) for the parameters under consideration. This method is equivalent to two one-sided tests with the null hypothesis of bio-inequivalence at the 5% significance level. Two products are declared bioequivalent if upper and lower limits of the confidence interval of the mean (median) of log-transformed AUC and Cmax each fall within the a priori bioequivalence intervals 0.80-1.25. It is then assumed that both rate (represented by Cmax) and extent (represented by AUC) of absorption are essentially similar. Cmax is less robust than AUC, as it is a single-point estimate. Moreover, Cmax is determined by the elimination as well as the absorption rate (Table 2.1). Because the variability (inter- and intra-animal) of Cmax is commonly greater than that of AUC, some authorities have allowed wider confidence intervals (e.g., 0.70-1.43) for log-transformed Cmax, provided this is specified and justified in the study protocol. 

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