acfalcao 20070916 22:34 Posting: # 1082 Views: 11,822 

Dear All My name is Amilcar Falcão and I am professor at the University of Coimbra, Portugal. I am looking for help regarding the execution and interpretation of threeway crossover bioequivalence trials (for academic purposes). I would like to have an example (explicit) where a threeway crossover study is appropriately analysed. Namely, I would like to know what kind of ANOVA should be performed and how the 90%CI should be calculated for the different combinations of study formulations (i.e. A vs B and A vs C). Perhaps this is a naive question, but in fact I am trying to look for it in literature and I have seen different approaches always without the adopted statistical pathway. Therefore, I would like to be sure regarding the correct way to deal with. I strongly appreciate if you could help me about this matter. Thank you for your attention. Best regards; Amílcar Falcão Laboratory of Pharmacology Faculty of Pharmacy University of Coimbra Coimbra  Portugal 
Helmut Hero Vienna, Austria, 20070917 20:47 @ acfalcao Posting: # 1088 Views: 11,147 

Dear Amilcar! » I would like to have an example (explicit) where a threeway crossover » study is appropriately analysed. You have to use a Williams' design (three period, six sequences); the topic was covered in previous threads. A 6×3 design is needed in order to 'extract' two 2×2 tables, which are also balanced. Although the full 6×3 table will be used in the analysis of AUC and C_{max}, you will need these 2×2s for the nonparametric analysis of T_{max} (unfortunatelly there's no confidence interval based nonparametric method available for more than two formulations/periods). The asterisks * denote pseudosequences and pseudoperiods, e.g. P1* means only that the treatment was given in a period prior to P2*  irrespective of the true study period:+++ > +++ and +++ A common mistake is to design the study as a set of 3×3 latin squares, which will lead (especially if the sample size is small and in the case of drop outs) to extremely inbalanced data sets: +++ > +++ and +++ » Namely, I would like to know what kind of » ANOVA should be performed and how the 90%CI should be calculated for the » different combinations of study formulations (i.e. A vs B and A vs C). For the design see Chapter 10 of SC Chow and Jp Liu Design and Analysis of Bioavailability and Bioequivalence Studies Marcel Dekker, New York (2nd ed. 2000), pp 302332 For a detailed discussion of variance balanced designs see Chapter 4 of B Jones and MG Kenward Design and Analysis of Crossover Trials Chapman & Hall/CRC, Boca Raton (2nd ed. 2003), pp 151204 For an example data set see Chapter 4 of S Patterson and B Jones Bioequivalence and Statistics in Clinical Pharmacology Chapman & Hall/CRC, Boca Raton (2006), pp 79132 Example 4.5 matches your design (one test formulation is compared to two reference formulations) You may download zipped datasets and programs (SAS/S+) from the CRC's website. If you don't have access to commercial software, S+ code will run with opensource R with minor modifications. Patterson/Jones give results in Table 4.12 (p 105) with: T/R (% Test vs. Reference 1) ++++ T/S (% Test vs. Reference 2) ++++ WinNonlin 5.2 comes up with: T/R (% Test vs. Reference 1) ++++ T/S (% Test vs. Reference 2) ++++ Results are slightly different (although both WinNonlin and SAS use GLM  not ANOVA; implementation, rounding, etc. is different  and of course 'proprietary information' and not documented). I assume Patterson/Jones' results were produced by SAS; I will check the results from their S+ code the next days. » Perhaps this is a naive question,... Not at all; little is published  and no worked examples at all. There's another point which is a little bit tricky: multiplicity. If you are testing one test (A) against two references (B, C), any impression of 'data dredging' must be avoided, e.g., calculation 90% CIs of A/B and A/C  and only picking out the best getting an approval. Since in the EU the Innovator's product from any European country may be used as the reference, you may run into problems (A vs B is BE, whereas A vs C is not BE). It may be wise to use 95% CIs instead (overall Bonferronicorrected patient's risk: alpha = 1(10.05/k)^{k}, where k is the number of simultaneous comparisons). 95% CI should also be applied in testing for dose proportionality of three dose levels (or 96.67% for four levels). IMHO the only case where 90% CIs should be used is the comparison of two test formulations against one reference, and only one of them will be further used in the approval process. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
acfalcao 20070917 23:02 @ Helmut Posting: # 1089 Views: 10,370 

Dear Helmut Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future. Best regards; Amílcar  Edit: Full quote removed. Please see this post! [HS] 
Helmut Hero Vienna, Austria, 20070918 12:47 @ acfalcao Posting: # 1092 Views: 10,512 

Dear Amilcar! » Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future. Please try it again (it worked yesterday, and today as well): CRC's website Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find: Example4.5.sas (the SAS code) Example45.SSC (the S+ code) exam45.dat (the raw data for S+) As I get form the SAScode a mixed model (degrees of freedom according to Kenward) was used, whereas WinNonlin uses Satterthwaite's method. The S+ code does not calculate BE, but is used for the example's figures only.  Edit: Wrong link corrected in both posts. [HS] — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Nirali Regular India, 20070921 06:49 @ Helmut Posting: # 1106 Views: 10,359 

Dear Helmut, Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)? Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"? In threeway crossover study, does we try to compare Equivalence of all three product together? Thanks & Regards, Nirali 
Helmut Hero Vienna, Austria, 20070921 13:04 @ Nirali Posting: # 1109 Views: 10,381 

Dear Nirali, » Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)? » Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"? » In threeway crossover study, does we try to compare Equivalence of all three product together? Comparisons are based on paired differences; IMHO a simultaneous test of T_{1}=T_{2}=R is not possible. WinNonlin always uses one reference in a run, i.e., T_{1}=R (one test), and T_{2}=R (the second test). If you are interested in T_{1}=T_{2} you have to start a second run (defining T_{2} as the reference), and will obtain two tests (T_{1}=T_{2} and R=T_{2}) within. BTW, why are you interested in T_{1}=T_{2}? — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Nirali Regular India, 20070925 08:05 @ Helmut Posting: # 1127 Views: 10,324 

Dear Helmut, Actully, I was under impression that WinNonlin is using the hypo: "MEAN(Test1)=MEAN(Test2)=MEAN(Ref)". As I understood from your reply (A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo: (1) R1=T and (2) R1=R2. same way, (B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2 pls do correnct me if i am wrong to understand. Thanks & Regards, Nirali 
Helmut Hero Vienna, Austria, 20070925 13:15 @ Nirali Posting: # 1132 Views: 10,435 

Dear Nirali! » Actully, I was under impression that WinNonlin is using the hypo: » "MEAN(Test1)=MEAN(Test2)=MEAN(Ref)". Again, we obtain only pairwise comparisons, not simultaneous ones. » (A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo: » (1) R1=T and (2) R1=R2. Not quite, but very close. If you run the BE wizard (R_{1}=reference), you will get (1a) T vs. R_{1}, and (1b) R_{2} vs. R_{1}. You have to run the BE wizard again (this time with R_{2} as reference), getting (2a) T vs. R_{2}, and (2b) R_{1} vs. R_{2}. In order to get comparisons of T=R_{1} and T=R_{2}, you have to run the wizard twice (results 1a and 2a), omitting 1b/2b. Of course PE and CL of 2b=1/1b. » same way, » (B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2 Yes this is correct; in this case only one run of the BE wizard is necessary, getting: (1a) T_{1} vs. R, and (1b) T_{2} vs. R. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Irene_I Junior Indonesia, 20180607 11:09 @ Helmut Posting: # 18861 Views: 2,595 

Dear Helmut, I'm currently looking for statistics methods for William Design in Bioequivalence Study and I found this thread. I tried to download the SAS code from the link you've attached, but apparently the link doesn't work for now. » Please try it again (it worked yesterday, and today as well): CRC's website » Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find: » Example4.5.sas (the SAS code) » Example45.SSC (the S+ code) » exam45.dat (the raw data for S+) I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this. Thank you. Best Regards, Irene I 
Helmut Hero Vienna, Austria, 20180607 13:04 @ Irene_I Posting: # 18862 Views: 2,595 

Hi Irene, » I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this. The files of the 2005 edition are gone. However, here is the link for the 2016 edition. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Irene_I Junior Indonesia, 20180608 11:09 @ Helmut Posting: # 18869 Views: 2,571 

Dear Helmut, Thanks a lot! It is really a great help to me.. Regards, Irene I 
Irene_I Junior Indonesia, 20180612 09:21 @ Helmut Posting: # 18886 Views: 2,463 

Dear Helmut, Could I ask you some questions about this example? After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intrasubject CV from this value, right? So, My question is, Do we only get one Intrasubject CV from threecomparison of the treatment (TR, ST, SR) while we got three CI (TR, ST, SR)? I am looking forward for your help. Thank you very much. Best Regards, Irene I 
Helmut Hero Vienna, Austria, 20180612 12:42 @ Irene_I Posting: # 18888 Views: 2,444 

Hi Irene, » After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intrasubject CV from this value, right? So, My question is, Do we only get one Intrasubject CV from threecomparison of the treatment (TR, ST, SR) while we got three CI (TR, ST, SR)? I’m not equipped with … Hence, my understanding of the code is limited. You are right, in the model you get only one (pooled) estimate of the variance. That’s not a good idea, since it will “work” only if intrasubject variances would be identical and the treatment differences at least very similar. Otherwise, the treatment estimates will be biased and the type I error is not controlled. See also this post. At the 2^{nd} Conference of the Global Bioequivalence Harmonisation Initiative (Rockville, September 2016) Pina D’Angelo gave a presentation “Testing for Bioequivalence in Higher‐Order Crossover Designs: Two‐at‐a‐Time Principle Versus Pooled ANOVA” showing exactly that. Here some of her slides: Purpose
Introduction Statistical Concerns:
Simulated Data: Summary
Closing Remarks
If you are interested in all pairwise comparisons, generate three data sets (values or T&R, S&R, T&S, i.e., exclude all values of the respective other treatment S, T, R) whilst keeping the codes for sequence and period. This will give you data sets which represent an IBD (incomplete block design). Run the usual model on them. This approach is also recommended in the EMA’s BEGL: In studies with more than two treatment arms (e.g. a three period study including two references, one from EU and another from USA […], the analysis for each comparison should be conducted excluding the data from the treatments that are not relevant for the comparison in question. (my emphasis)Example 4.5 (Phoenix/WinNonlin 8, mixed effects, Satterthwaite’s degrees of freedom): PE (%) 90% CI (%) s²_{w} CV_{w} (%) It’s clear that the variances are not identical. The ones of T/R are smaller than the ones of S/R. If we apply the pooled model the CIs of T/R will be wider than in the IBD model and the CIs of S/R narrower. BTW, I don’t understand what the purpose of the carry variable in Byron’s code is.— Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Irene_I Junior Indonesia, 20180613 05:02 @ Helmut Posting: # 18896 Views: 2,365 

Dear Helmut, Thank you for your explanation. I am sure this will be helpful not only for me, but also others who looking for information about this topic too. Best Regards, Irene I 
Helmut Hero Vienna, Austria, 20180613 15:02 @ Irene_I Posting: # 18898 Views: 2,316 

Hi Irene, » Thank you for your explanation. You are welcome. A picture tells more than a thousand words (AUC only). The PE and its CI of T/R are similar. The small bias is caused by the fact that the average PE of S/R and T/S (√141.09×83.12=108.29) is close to the T/R with 116.05. The other comparisons show negative biases. The CI of S/R is narrower in the pooled model and the CI of T/S is wider. We don’t want to go there. — Cheers, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
d_labes Hero Berlin, Germany, 20180613 15:31 @ Helmut Posting: # 18899 Views: 2,281 

Dear Helmut, » BTW, I don’t understand what the purpose of the carry variable in Byron’s code is.As far as I see it: It serves for nothing because it is not used except in the class statement of Proc Mixed .The class statement serves the purpose of declaring factors even if a variable is numeric. Nothing more. The carry variable seems to me a left over from old times where carryover was an issue.— Regards, Detlew 