usfda_emea 20070306 14:39 Posting: # 559 Views: 16,578 

Dear, I am looking for help on estimation of 90% confidence interval for log transformed data of AUC (0inf) for a bioequivalence study. This study involved twoway crossover, twoperiod, twosequence design with a total of 33 subjects completing the study. Each sequence had Period I TR=16 and RT=17, and in Period II TR=17 and RT=16; thus, total of 33 completed the study. In order to estimate the 90% confidence interval, can some one help me with the formula and a worked out example for the ease of understanding. OR Guide me to a site where this can be found. I believe leastsquares mean estimate for each formulation needs to be used to form the difference, together with the standard error. However, in lack of exact formula i am seeking the help of experts out there. Thanks in advance for the help. USFDA_EMEA Edit: Category changed. [Helmut] 
Helmut Hero Vienna, Austria, 20070306 16:46 @ usfda_emea Posting: # 560 Views: 14,590 

Dear USFDA_EMEA! I guess that’s not your real name » I am looking for help on estimation of 90% confidence interval for log transformed data of AUC (0inf) for a bioequivalence study. This study involved twoway crossover, twoperiod, twosequence design with a total of 33 subjects completing the study. Each sequence had Period I TR=16 and RT=17, and in Period II TR=17 and RT=16; thus, total of 33 completed the study. OK, the number of subjects in each sequence is n_{1} and n_{2} , respectively, where n = n_{1} + n_{2} .In your case: n_{1} = 17 The assignment of sequence 1 to RT follows the literature, but is only a convention (if you want you may call TR sequence 1 as long as you keep this convention through all calculations…) » In order to estimate the 90% confidence interval, can some one help me with the formula and a worked out example for the ease of understanding. OR Guide me to a site where this can be found. Let’s see. Which formula are you using right now? If you have something like 1/n in it, for unbalanced data you have to replace it by 1/n1 + 1/n2 .» I believe leastsquares mean estimate for each formulation needs to be used to form the difference, together with the standard error. However, in lack of exact formula i am seeking the help of experts out there. Yes, you are on the right track! Let's call Xt the LSM of the test, and Xr the LSM of the reference (logscale). sigmaw is the within (or intra) subject standard deviation (sqrt(MSE) from your ANOVA), t(1alpha,n1+n22) is the 0.95 quantile of the central tdistribution with n1+n22 degrees of freedom. Then the upper/lower 90% confidence limits (log scale) are given by
Xt  Xr ± t(1alpha,n1+n22) × sqrt(MSE) × sqrt[ 1/2 × (1/n1 + (1/n2) ] Just a reminder: don’t try evaluations in M$Excel; the tquantile (TINV) is implemented in a rather queer way: in order to get the correct value for t(1alpha,n1+n22) , you would have to use TINV(2 × alpha, n1+n22)!See http://www.practicalstats.com/xlsstats/excelstats.html http://www.mis.coventry.ac.uk/~nhunt/pottel.pdf I would heartly recommend two textbooks:
— Regards, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
drshiv Regular India, 20070306 18:49 @ Helmut Posting: # 561 Views: 14,414 

Dear USFDA_EMEA, For unbalanced designs the solution provided by HS is a good. Other method is to use GLM procedure in SAS (if you are using SAS) to calculate MSE before calculation of CI. Then you can use the formula givn by HS to calculate CI, maybe using Excel (off course after validation of the formula). Dr. Shiv Edit: Full quote removed (see this post)! [Helmut] 
usfda_emea 20070307 06:21 @ Helmut Posting: # 564 Views: 14,243 

Dear HS, » I guess that’s not your real name » Let’s call Xt the LSM of the test, and Xr the LSM of the reference (logscale).Can you help me with the formula for LSM (I wish to do a manual calculation, for verification purposes) and a brief explaination about why this method is used for unbalanced study? Thanks a million for the other formula and the references. Truly, USFDA_EMEA 
Helmut Hero Vienna, Austria, 20070307 12:04 @ usfda_emea Posting: # 565 Views: 14,434 

Dear USFDA_EMEA! » » Let's call Xt the LSM of the test, and Xr the LSM of the reference (logscale).» » Can you help me with the formula for LSM (I wish to do a manual calculation, for verification purposes) and a brief explaination about why this method is used for unbalanced study? The calculation is not different from a balanced study. I’ll give you an example for the reference: Calculate the (arithmetic) mean of logtransformed values in sequence 1 (if seq 1 = RT from period 1) as Xr1 = sum ( Xr1 … n1 ) / n1 Calculate the (arithmetic) mean of logtransformed values in sequence 2 (if seq 2 = TR from period 2) as Xr2 = sum ( Xr1 … n2 ) / n2 LSM for the reference is the arithmetic mean of Xr1 and Xr2 The difference Xt – Xr is also called MLE (the Maximum Likelihood Estimator) of the true difference Mu(t) – Mu(r) , which is slightly biased according to the degree of ‘unbalancedness’.A better estimate of the true difference is MVUE (the Minimum Variance Unbiased Estimator), however, its calculation is a little bit tricky (involves the Γdistribution). For details see Chow & Liu’s book. — Regards, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
usfda_emea 20070307 12:34 @ Helmut Posting: # 566 Views: 14,215 

Dear HS, 
Ohlbe Hero France, 20140730 15:04 @ Helmut Posting: # 13319 Views: 8,220 

Dear Helmut, » » Can you help me with the formula for LSM (I wish to do a manual calculation, for verification purposes) and a brief explaination about why this method is used for unbalanced study? » » The calculation is not different from a balanced study. » » I’ll give you an example for the reference [...] How do you calculate the LSM in a full replicate design study (RTRT and TRTR, with some subjects missing here and there) ? — Regards Ohlbe 
Helmut Hero Vienna, Austria, 20140731 02:54 @ Ohlbe Posting: # 13322 Views: 8,238 

Dear Ohlbe, » How do you calculate the LSM in a full replicate design study (RTRT and TRTR, with some subjects missing here and there) ? Manually? Never done that. If you are thinking about RSABE (in their FDA and EMA flavors), that’s a strange pot of tea. To quote Patterson & Jones^{1}: […] simulation findings support that if a subject is missing both reference observations or the test observation, no bias is introduced. It’s a complete mess. Subjects with incomplete data are excluded from the estimation of σ²_{WR}, but kept in the calculation of the upper 95% boundary (FDA) or excluded for the calculation of the 90% CI (EMA, all fixed effects. A mixedeffects model is not acceptable for EMA). To be honest, currently there are so many flaws in these methods that I would not bother to calculate anything by hand. Accept what you get from the FDA’s and EMA’s SAScode (or alternative software) and set this question aside until somebody finds a method to deal with the more important question: How to prevent inflation^{1,2} of the patient’s risk (according to my latest simulations: EMA up to 9.6% and FDA up to 22.5%). This is a nightmare.
— Regards, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
Ohlbe Hero France, 20140731 09:47 @ Helmut Posting: # 13323 Views: 8,157 

Dear Helmut, » Manually? Never done that. » It’s a complete mess. Subjects with incomplete data are excluded from the estimation of σ²_{WR}, but kept in the calculation of the upper 95% boundary (FDA) or excluded for the calculation of the 90% CI (EMA, all fixed effects. A mixedeffects model is not acceptable for EMA). I would not think for even a split second of trying to recalculate manually the 90 % CI or σ²_{WR}... But I was hoping there would be a simple way to at least recalculate LSMs (or to understand how they are calculated). Thanks anyway ! — Regards Ohlbe 
VStus Regular Poland, 20170301 14:07 @ Helmut Posting: # 17116 Views: 3,284 

Dear Helmut, Dear all, I have limited data about partial replicate trial (2x3x3) with RSABE, which I want to use to assess the feasibility of same study under EMA regulation (assuming that EU has the same Reference product as US). Reported parameters: Number of subjects (observations per treatment arm) Swr estimated using PROC GLM Point Estimate (%) from RSABEapproach 95% CI upper confidence bound Is it feasible to try calculation of 90% CIs? Can I apply the formula below? 90% CI = log(Point_Estimate) +/ Swr * qt(0.05, df))*100, where: df=2*n3 for partial replicate (2x3x3); df=3*n4 for full replicate (2x2x4); Thank you very much in advance! Regards, VStus 
Helmut Hero Vienna, Austria, 20170303 13:34 @ VStus Posting: # 17126 Views: 3,251 

Hi VStus, » I have limited data about partial replicate trial (2x3x3) with RSABE, which I want to use to assess the feasibility of same study under EMA regulation (assuming that EU has the same Reference product as US). » Reported parameters: » Number of subjects (observations per treatment arm) » Swr estimated using PROC GLM » Point Estimate (%) from RSABEapproach » […] » Is it feasible to try calculation of 90% CIs? Yes. » Can I apply the formula below? » 90% CI = log(Point_Estimate) +/ Swr * qt(0.05, df))*100, No (see this post). Let’s compare the EMA’s Q&A dataset II (n = 24; n_{1} = n_{2} = n_{3} = 8), evaluated by the FDA’s RSABE (PE = 1.022644, s_{wR} = 0.11397298) with results of the EMA’s methods: PE 102.26%; 90% CI: Methods A/B 97.32–107.46%, Method C 97.05–107.76%.
Pretty close but in this case (s_{wR} <0.294) the FDA requires a mixedeffects model for ABE, where s²_{wR} = 0.013246498 (PE 102.26%, 90% CI 97.05–107.76%). Hence, we have to take that into account.
— Regards, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 
VStus Regular Poland, 20170303 15:31 @ Helmut Posting: # 17127 Views: 3,172 

Dear Helmut, Thank you very much! I've missed recent post of Detlew when using search function... PS: I like this update
n < c(8,8,8) Regards, VStus 
d_labes Hero Berlin, Germany, 20170304 14:49 @ VStus Posting: # 17132 Views: 3,119 

Dear Helmut, dear VStus, I assume that the s2wR you have (or swR) is the intrasubject variance estimated with R data. If this is the case the elaborate answer by Helmut is only valid if s2wR is equal to the MSE from the evaluation of all data, which is necessary for calculation of the point estimate and 90% CI of T versus R.But s2wR = MSE is not necessarily true in all cases, although it is in many cases a reasonable assumption.Moreover the FDA evaluation uses intrasubject constrasts to evaluate s2wR an µTµR including the 90%CI whereas the EMA recommended method is bases on common ANOVA (aka GLM in SAS speak. Both evaluation methods give different results. Usually slight differences, but ...Thus my answer to the question "Is it feasible to try calculation of 90% CIs?" for the EMA based on the results of a FDA evaluation is a clear No. What you get by Helmuts calculations is only an approximate solution which may be good enough but may also terrible fail. — Regards, Detlew 
ElMaestro Hero Denmark, 20170304 19:13 @ d_labes Posting: # 17133 Views: 3,092 

I like that post, d_labes , » But s2wR = MSE is not necessarily true in all cases, although it is in many cases a reasonable assumption....and accordingly when the optimizer fails, plugging in the GLM residual (expressed as MSE) as a new starter value for s2wR (and even S2wT where applicable) may be a very decent lastditch attempt. » Moreover the FDA evaluation uses intrasubject constrasts to evaluate s2wR What does this mean? Don't you usually get it from the REML covarance matrix? I think EMA's approach can be condensed into intraSubject contrasts to derive S2wR from completers. — I could be wrong, but… Best regards, ElMaestro A potentially biased estimator may be a relevant estimator. The case of REML speaks volumes. 
d_labes Hero Berlin, Germany, 20170305 12:18 @ ElMaestro Posting: # 17135 Views: 3,070 

Dear ElMaestro! » ... » » Moreover the FDA evaluation uses intrasubject constrasts to evaluate s2wR » What does this mean? Don't you usually get it from the REML covarance matrix? If you have a look at the SAS code in the progesterone guidance you will discover that s2wR is not obtained from the REML covariance matrix but rather from an ISC evaluation. Means you calculate RR within the subjects and use that difference in an ANOVA with the sequence as the soley effect. The REML estimates dont play a role, since a mixed model only comes into the play if CVwR is <=30% and conventional ABE has to be used. Here you may get s2wR from the REML covariance matrix but it is not used in ABE. That's one of the curiosities of the FDA approach, among others. » I think EMA's approach can be condensed into intraSubject contrasts to derive S2wR from completers. Here you err. The EMA approach calls for an evaluation via lm(), GLM or comparable using the R(eference) data only with the effects period and subject (sequence may be also included but doesnt change the estimate of s2wR). This gives some different results to an estimation via ISC as described above. Try it. BTW: Have a look into Implementation_scaledABE_sims.pdf in the doc subdirectory of R package PowerTOST . There you will find a cookbook manner description of the computations involved in the EMA or FDA approaches for scaled ABE.— Regards, Detlew 
ElMaestro Hero Denmark, 20170305 13:50 @ d_labes Posting: # 17136 Views: 3,045 

Thanks d_labes, » If you have a look at the SAS code in the progesterone guidance you will discover that s2wR is not obtained from the REML covariance matrix but rather from an ISC evaluation. Means you calculate RR within the subjects and use that difference in an ANOVA with the sequence as the soley effect. » The REML estimates dont play a role, since a mixed model only comes into the play if CVwR is <=30% and conventional ABE has to be used. Here you may get s2wR from the REML covariance matrix but it is not used in ABE. » That's one of the curiosities of the FDA approach, among others. Yes that is kind of strange. » » I think EMA's approach can be condensed into intraSubject contrasts to derive S2wR from completers. » » Here you err. The EMA approach calls for an evaluation via lm(), GLM or comparable using the R(eference) data only with the effects period and subject (sequence may be also included but doesnt change the estimate of s2wR). This gives some different results to an estimation via ISC as described above. Try it. I beg to differ. Chow and Liu showed how the problem expressed as a linear model, can be solved with equations all based on contrast of T and R, when we talk 222BE; this gave them the desired sw. Meaning you can get the quanitity you want by equations or by solving the linear model. That is why some regulators (none mentioned, none forgotten) don't even need to see an ANOVA. This situation being a linear model which has an analytical solution we can generalise it further and make equations where things are condensed into equations of a similar nature for your refreplicated design. I don't think it would be very difficult, actually, but I am not convinced that I myself could do it without further ado. The key here be that we rely on (start with) a linear model in which all the residual df's can be said to derive from the replication itself. — I could be wrong, but… Best regards, ElMaestro A potentially biased estimator may be a relevant estimator. The case of REML speaks volumes. 
VStus Regular Poland, 20170304 21:41 @ d_labes Posting: # 17134 Views: 3,072 

Dear Detlew, » But s2wR = MSE is not necessarily true in all cases, although it is in many cases a reasonable assumption.And s2wR != s2wT in most cases. Maybe that's the reason why 2x3x3 design is still so popular. It's not possible to assess s2wT... » What you get by Helmuts calculations is only an approximate solution which may be good enough but may also terrible fail. These calculations are intended only to get an than approximate assumption. Like a pilot study. Regards, VStus 
Helmut Hero Vienna, Austria, 20170306 13:40 @ VStus Posting: # 17138 Views: 2,999 

Hi VStus, » » But s2wR = MSE is not necessarily true in all cases, although it is in many cases a reasonable assumption.» And s2wR != s2wT in most cases. Maybe that's the reason why 2x3x3 design is still so popular. It's not possible to assess s2wT... No agency asks for s²_{wT} in the context of RSABE or ABEL (the FDA’s RSABE for NTIDs is another story but requires a 4period full replicate anyhow). Hence, why should one report it? I never understood why one would opt for the partial replicate. Does this equation hold? Partial replicate = lack of statistical knowledge + “we want only three periods” + given in the FDA’s guidance / the EMA’ Q&Adocument (as an example)? IMHO, at least in pilot studies full replicates are preferable (see there). I once struggled with a Romanian CRO. The sponsor decided to perform a 4period full replicate in order to possibly end up with a smaller sample size of the pivotal study if CV_{wT} < CV_{wR}. The CRO refused to calculate CV_{wT} (“… because not in accordance with the SAScode given in the Q&A.”)… Oh boy! Only after the sponsor threatened to go with another CRO, they gave in. — Regards, Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. ☼ Science Quotes 