14 IV.2 Ratio of Cmaxmeans of test and reference drugs for each subject and. 15 IV.3 Probability profile for mean difference between test and reference drug for log AUC. 17 IV.4 Profile likelihood for the ratio of total standard deviations of test/reference drugs for.
18 IV.5 The profile probability for the ratio of within-subject standard deviations of test/reference. 19 IV.6 The profile probability for the mean difference of the test and reference means for log Cmax. 20 IV.7 The profile likelihood for the ratio of total standard deviations of test/reference means for.
21 IV.8 Profile likelihood for the ratio of standard deviations within the test/reference subject. Recently, the FDA task force examined several alternatives and proposed a scale of average reference bioequivalence (RSABE) approach, where the EU limits are scaled to the variability of the reference drug for HVD (ie, the limits are increasing proportionally with the variance) (Haidar et al., 2008a). The application of FDA's RSABE method to HVDs is summarized in Davit et al.
In the context of BE analysis, Choi et al. 2008) showed that the profile likelihood is a good approximation of the true likelihood.
Estimation of the mean difference, φ, and its standard error
In a full replicate 2×4 cross-over study, subjects are randomized to either sequence RTRT or TRTR. Each subject then receives both drug formulations twice in the order of RTRT or TRTR to which the subject is randomized. Between drug administrations there is a "washout" period to avoid the possible effect of drugs administered in the previous period(s), called "carry-over" effect.
Let be a random variable representing the log-transformed response (i.e. Y = log AUC or log Cmax) for the subjective period of sequencekwithi=1. The random errors, ei jk's, are independent, and ei jk is assumed to follow a normal distribution (N), N(0,στ(2jk)). Equations (III.6) and (III.9) to (III.11) are the simple moment estimators that can be obtained using a random effects model, also with an arbitrary intercept.
FDA and EMA’s RSABEs for HVDs
Likelihood function and likelihood approach
Using the optim() function in R, we obtained the profile likelihood for the mean differenceφ and the variance ratios for an example of 2×4 full replicate BE data presented in CHAPTER IV. We used the largest k, kmax, where 1/kLI is completely within the BE limit (for the mean or variance), as evidence for BE in the example. Ln(φ) to provide evidence for BE or BIE, where H1 and H0 are the interval hypotheses (I.1); the larger this GLR, the stronger the evidence for BE.
Simulations
For each simulated data, we used 4 approaches to estimate BE, including ABE (TOST), FDA RSABE, EMA RSABE, and a likelihood approach. According to the W R, we assessed whether these intervals lie within the relevant FDA or EMA RSABE limits (with or without the point score limit). Due to the relatively slow computational speed of the optim() function, we used an alternative method (lme() function, Appendix B) to obtain the profile likelihood for the mean difference and the corresponding profile LIs in the simulations.
We confirmed that the profile probabilities of the two computational methods using the example data were identical (Figure B.1). Sample data were taken from a full replicate 2×4 crossover design where sequence 1 is RTRT and sequence 2 is TRTR [modified from a sample data in Chapter 4 in Patterson and Jones (2005) with only complete data for 27 subjects for each sequence]. For the frequentist approach, the ABE was evaluated using TOST, FDA and EMA RSABE methods.
For the likelihood approach, in addition to ABE, PBE and IBE were also estimated using the total variances for the reference (σT R2 =σBR2 +σW R2 ) and test (σT T2 =σBT2 +σW T2 ) drugs, and the within- of the subject for reference drugs (σW R2 ) and test (σW T2 ).
TOST, FDA and EMA RSABEs
The estimates from the moment-based method are almost the same as those from the random-effects model. If the conventional BE test method is to be used, we would not conclude ABE for AUC and Cmax, as none of the 90% CIs fall completely within the limit. The individual Cmax ratios show that there are two extreme values (Figure IV.2) from sequence RTRT.
If the RSABE is used taking into account the within-subject variability, the conclusion may be different, as summarized in Table IV.1. The reference drug is indeed highly variable for both AUC and Cmax with W R equal to 0.337 and 0.546 respectively. The overall conclusion is that the test drug failed to demonstrate BE to its reference drug.
For EMA, RSABE is only allowed for Cmax, and the BE limit for AUC is always regardless of sW R. Neither the 90% CI nor the GMR point estimate for Cmaxis fully within their respective limits, making it impossible to make good inferences about BE for Cmaxas. Thus, BE was concluded for AUC according to FDA RSABE criteria, but we failed to conclude BE according to EMA RSABE criteria.
This is because the FDA RSABE is more permissive due to the larger scaling factor. Note that in the frequentist framework, if the data does not show BE, it does not imply BIE.
Likelihood approach
Profile likelihoods for log AUC
Within the probabilistic paradigm, it is easy to evaluate evidence for BE for the mean and variance together to address the PBE or IBE issues. For AUC, there is weak evidence for PBE when we combine the evidence for φ and σT T/σT R together.
Profile likelihoods for log C max
It is impossible to present evidence for EU or BIE using FDA or EMA's RSABE approaches. The red and blue dotted vertical lines in these figures represent the FDA and EMA RSABE limits for the respective %CV. We used the associated 95% criteria along with the point estimate cutoff to define the EU for FDA RSABE here.
2008b) demonstrated an improved power of RSABE as within-subject variation increases compared to the conventional ABE test method (ie, TOST) for 3×3 partial replicate design. 2012) compared the performance of the FDA RSABE with the EMA scaled approach in the partial replicate study design. Their results are consistent with our simulation results for the full replicate 2×4 cross-over design. Clearly, these frequentist approaches for BE may not guarantee any confidence level for GMR at the ABE limits of 0.8 and 1.25, even when RSABE is completed.
We used a more general numerical method [optim() function] to obtain the profile probabilities for the mean difference and variance ratios in the likelihood function (III.14). We have shown that, for the composite hypothesis such as the interval hypotheses in BE testing, the evidence can be represented by kmax (the largest so that 1/kLI is completely contained within the BE limit) or Zhang's GLR. Therefore, we may prefer to use GLR as the measure of strength of evidence for the interval hypotheses.
It is impossible to show evidence for BIE using regular approaches such as the FDA or EMA RSABE, as failure to reject the null does not imply that the null (BIE) is true. By considering the evidence for the mean and the variability together, our probabilistic approach also provides evidence for PBE or IBE, thus easily addressing the prescriptability or commutability issue. Rather, we advocate using the probabilistic approach to present evidence for BE in the way we have illustrated using the example data (CHAPTER IV).
It is true that large variability can widen the LI of the mean difference, which makes it more difficult to present evidence for EU. Sample size and the strength of evidence measured by GLR for interval hypotheses determine the probabilities of observing spurious evidence, weak evidence, and strong evidence (CHAPTER II) for BE. We suggest that evaluation of the evidence for EU should be decoupled from decision-making for the approval of generic drugs.
We recommend using the overall likelihood ratio (orkmax, if available) as evidence for interval hypotheses in bioequivalence testing. Similar to Figure II.1 for the simple hypotheses, we need to determine in the future how the probabilities of observing misleading and weak evidence would behave for the interval hypotheses (I.1) as a function of the ratio of the geometric mean, the sample size, and the generalized likelihood ratio cutoff.
Computer code
Comparison of profile likelihoods obtained using lme() and optim()
ABE, FDA RSABE and EMA RSABE
Moment estimators
Random-effects model: lme()
FDA RSABE
EMA RSABE for Cmax
Profile likelihoods using the optim() function
Profile likelihood for φ
Simulations
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Comparison of generic and innovator drugs: A review of 12 years of bioequivalence data from the United States Food and Drug Administration. Annals of Pharmacotherapy. A new statistical procedure for testing equivalence in two-arm comparative bioavailability trials. "Journal of Pharmacokinetics and Biopharmaceutics. Abuse of power: the widespread error of power calculations for data analysis." American statistician.
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