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Back to the forum  Query: 2018-02-24 01:18 CET (UTC+1h)
 
Louis52
Junior

2018-02-07 21:04

Posting: # 18375
Views: 1,805
 

 Power and type I adjustment for 222BE when several PK prams considered [General Sta­tis­tics]

Hello,

Can anyone direct me towards a thread where it is discussed/presented how to deal with alpha adjustment and power in case of multiplicity (AUC and Cmax considered at the same time)? Both CO and Parallel designs.

Thanks!
ElMaestro
Hero

Denmark,
2018-02-07 22:37

@ Louis52
Posting: # 18376
Views: 1,676
 

 Power and type I adjustment for 222BE when several PK prams considered

Hi Louis,

» Can anyone direct me towards a thread where it is discussed/presented how to deal with alpha adjustment and power in case of multiplicity (AUC and Cmax considered at the same time)? Both CO and Parallel designs.

There is no alpha adjustment.
You need to show BE for both, usually. Sometimes even AUC in several flavours.

AUC and Cmax are positively correlated. That's something you may wish to factor in for sample size. However, in practice this requires a model for the correlation itself (for which nothing generally can be said) and besides it actually works quite well if you just dimension it according to the worst-case metric which tends to be Cmax.

I could be wrong, but…


Best regards,
ElMaestro

- Bootstrapping for dissolution data is a relatively new hobby of mine.
d_labes
Hero

Berlin, Germany,
2018-02-08 14:12

@ Louis52
Posting: # 18381
Views: 1,640
 

 Power of two combined TOST

Dear Louis,

» Can anyone direct me towards a thread where it is discussed/presented how to deal with alpha adjustment and power in case of multiplicity (AUC and Cmax considered at the same time)? Both CO and Parallel designs.

have a loook at this post or this one.
As our ElMaestro already said: According to the intersection-union principle there is no need for adjusting alpha if you combine two TOST with AND. As long as the two combined test on their own have size alpha.

Power is another pair of shoes.
If the correlation is zero, the overall power is the product of the powers of the individual tests, i.e. may be considerable lower then the targeted power.
If the correlation is one, the overall power is the minimum of the powers of the individual tests. This is the foundation why we make a sample size calculation for both metrics and choose the resulting higher sample size.

But no one knows, at least not me, to what number the correlation should be set in practice.
The authors team of the R package PowerTOST is currently discussing that theme. May be we came out with some aid in the near future.

Be aware: It is not the correlation of the two PK metrics itself. To cite from the man page of function power.2TOST():
"rho: Correlation between the two PK metrics (e.g. AUC and Cmax) under consideration. This is defined as correlation between the estimator of the treatment difference of PK metric one and the estimator of the treatment difference of PK metric two."

Be further aware: The functions for power and sample size for the combination of 2 TOST are currently under revision since the implemented versions in V1.4-6 came out as statistical flawed. You need the development version of PowerTOST to get reliable values for the power of 2 TOST. See this thread how to get it.


Reference to look into:
Phillips KF.
Power for Testing Multiple Instances of the Two One-Sided Tests Procedure
Int J Biostat. 2009;5(1):Article 15. doi:10.2202/1557-4679.1169

Regards,

Detlew
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-08 21:40

@ d_labes
Posting: # 18382
Views: 1,620
 

 The mysterious ρ

Hi Louis,

» But no one knows, at least not me, to what number the correlation should be set in practice.
» The authors team of the R package PowerTOST is currently discussing that theme. May be we came out with some aid in the near future.

Current state of affairs about the mysterious ρ:

[image]


Runtime 0.78 seconds. 6176 subjects in 124 data sets of 98 2×2×2 studies (74 analytes).
Evaluated by: lm(log(PK) ~ sequence+subject+period+treatment, data=study)
              PE.AUC[study]  <- exp(summary(model.AUC)$coef["treatmentT", "Estimate"])
              PE.Cmax[study] <- exp(summary(model.Cmax)$coef["treatmentT", "Estimate"])
              model.rho <- lm(PE.Cmax ~ PE.AUC)
              rho       <- sqrt(summary(model.rho)$r.squared)

rho         : 0.6983



PS: No, I’m not underpowering my studies for Cmax. Some were so old that they were powered for an acceptance range of 75–133% or even 70–143%. ;-) Was too lazy to browse through the protocols…

[image]Regards,
Helmut Schütz 
[image]

The quality of responses received is directly proportional to the quality of the question asked. ☼
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ElMaestro
Hero

Denmark,
2018-02-09 09:58

@ Helmut
Posting: # 18385
Views: 1,555
 

 ρ ρ ρ your boat

Hi Helmut,

thank you for the info.
Out of curiosity, could you show a histogram of lnPEAUC-lnPECmax ?

Muchas gracias

I could be wrong, but…


Best regards,
ElMaestro

- Bootstrapping for dissolution data is a relatively new hobby of mine.
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-09 19:20

@ ElMaestro
Posting: # 18389
Views: 1,518
 

 ρ ρ ρ your boat

Hi ElMaestro,

» Out of curiosity, could you show a histogram of lnPEAUC-lnPECmax ?

Other way ’round:

[image]


[image]Regards,
Helmut Schütz 
[image]

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
ElMaestro
Hero

Denmark,
2018-02-09 11:32

@ Helmut
Posting: # 18387
Views: 1,545
 

 ...and a bonus question

Hi Helmut,

I have a bonus question, although not about the correlation itself.

If I look at the location of your PE's on the horisontal axis (the AUC axis) then it looks to me like there are more points on the higher side of 1 than on the lower side.

Perhaps it is only a graphical artifact? Or something with the selection of studies plotted?

If not, then I wonder why that wold be so? I imagine you don't really have the answer but I am curious about any speculation from your side or from others. My head is already abuzz with stabilities and overages and much other weird stuff from the odd sock drawer of speculation.

If this is not an artifact then I am almost thinking this would be worth a publication (if it had no already been shown here).

Same might apply for the vertical axis, not sure. Perhaps I am only hallewcinating?:-D

I could be wrong, but…


Best regards,
ElMaestro

- Bootstrapping for dissolution data is a relatively new hobby of mine.
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-09 19:43

@ ElMaestro
Posting: # 18390
Views: 1,527
 

 ...and a bonus question

Hi ElMaestro,

updated version:

[image]


Runtime 0.92 seconds. 7446 subjects in 141 data sets of 113 2×2×2 studies (86 analytes).
Evaluated by: lm(log(PK) ~ sequence+subject+period+treatment, data=study)
              PE.AUC[study]  <- exp(summary(model.AUC)$coef["treatmentT", "Estimate"])
              PE.Cmax[study] <- exp(summary(model.Cmax)$coef["treatmentT", "Estimate"])
              pearson <- cor.test(PE.Cmax, PE.AUC)
              rho     <- pearson$estimate
              CI      <- as.numeric(pearson$conf.int)

rho         : 0.6534 (95% CI: 0.5473 … 0.7389)


» If I look at the location of your PE's on the horisontal axis (the AUC axis) then it looks to me like there are more points on the higher side of 1 than on the lower side.

cat(sprintf("%.1f%%", 100*sum(PE.AUC > 1)/length(PE.AUC)), ">1\n"); summary(PE.AUC, digits=4)
58.9% >1
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
 0.7206  0.9804  1.0130  1.0260  1.0610  1.2850


» Same might apply for the vertical axis, not sure. Perhaps I am only hallewcinating?:-D
» Perhaps it is only a graphical artifact?

cat(sprintf("%.1f%%", 100*sum(PE.Cmax > 1)/length(PE.Cmax)), ">1\n"); summary(PE.Cmax, digits=4)
58.2% >1
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
 0.6692  0.9579  1.0340  1.0260  1.0950  1.3800


» Or something with the selection of studies plotted?

Duno. In the meantime I removed two studies (one was a tablet vs. a suspension and the other one from Hauschke/Steinijans – two MRs but the target was “flatter is better”) and added a couple more. It was easy to import the data but now I have to check in the protocols what the target was. Not always stated in the title and boring reading matter. Will take a while.

» If not, then I wonder why that wold be so? I imagine you don't really have the answer but I am curious about any speculation from your side or from others. My head is already abuzz with stabilities and overages and much other weird stuff from the odd sock drawer of speculation.

You are not alone. I have more questions than answers myself. Most circle around which ones to include:
  • Some studies failed completely and passed after re-formulation. Shall I exclude the former?
  • Shall I keep failed ones but with PEs within the AR (indecisive)?
  • etc.
  Basic research is what I’m doing
when I don’t know what I’m doing.
    Wernher von Braun

[image]Regards,
Helmut Schütz 
[image]

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Astea
Regular

Russia,
2018-02-09 20:47

@ Helmut
Posting: # 18392
Views: 1,510
 

 If u see a BEar don't forget to R oaρ

Dear all!
I tried to repeat the Helmut's conclusions. I used 70 datasets from 63 2x2x2 studies. My rho is 0.797 for untransformed data and 0.799 for log-transformed. Meanwhile it tends to have the inverse situation, that is there are more values lower than 100 comparing to those uper than 100 (66% for Cmax and 63% for AUC). Suppose it is just an artefact of low sample size..
[image]

P.S. One dummy's question about PowerTOST from GitHub: while trying to do devtools::install_github("Detlew/PowerTOST") I got:
package ‘mvtnorm’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘mvtnorm’
In R CMD INSTALL
Installation failed: Command failed (1)

Should I delete it manually or ...what to do?
d_labes
Hero

Berlin, Germany,
2018-02-11 13:23

@ Astea
Posting: # 18394
Views: 1,321
 

 package install failed

Dear Nastia

» P.S. One dummy's question about PowerTOST from GitHub: while trying to do devtools::install_github("Detlew/PowerTOST") I got:
» » package ‘mvtnorm’ successfully unpacked and MD5 sums checked
» Warning: cannot remove prior installation of package ‘mvtnorm’
» In R CMD INSTALL
» Installation failed: Command failed (1)
»


That error occasionally occures if you try to install a package which is in use, especially some dll (compiled code) is in use.

» Should I delete it manually or ...what to do?

Yes my dear, delete any leftover of ‘mvtnorm’ and install again, of course in a fresh R console.

Regards,

Detlew
d_labes
Hero

Berlin, Germany,
2018-02-11 15:57

@ Helmut
Posting: # 18395
Views: 1,309
 

 The mysterious ρ -between or within studies

Dear Helmut! Dear all!

» Current state of affairs about the mysterious ρ

That's not the whole truth about the state of affairs :no:.
The question is: Correlation of treatment differences between studies or within studies.

For the latter I recall you two references:
Phillips KF.
Power for Testing Multiple Instances of the Two One-Sided Tests Procedure
Int J Biostat. 2009;5(1):Article 15. doi: 10.2202/1557-4679.1169

Hua SY, Xu S, D’Agostino RB Sr.
Multiplicity adjustments in testing for bioequivalence
Stat Med. 2015;34(2):215–31. doi: 10.1002/sim.6247

Quote from Kem Phillips:
"The correlation will usually be difficult to estimate, unless a similar experiment has been conducted...".
That smells for me like within a study.

Definition in Hua et al., Section 2, clearly is within a study. Although Ben doubt the correctness of the formulas for ρ given in that paper. Note further that the observed alpha-inflation in that paper is crap. The IUT principle protects us from the need to adjust alpha.

Regards,

Detlew
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-11 17:31

@ d_labes
Posting: # 18396
Views: 1,292
 

 The mysterious ρ -between or within studies

Dear Detlew

» That's not the whole truth about the state of affairs :no:.
» The question is: Correlation of treatment differences between studies or within studies.
»
» For the latter I recall you two references:
» Phillips KF.
» Power for Testing Multiple Instances of the Two One-Sided Tests Procedure
» Int J Biostat. 2009;5(1):Article 15. doi: 10.2202/1557-4679.1169
»
» Quote from Kem Phillips:
» "The correlation will usually be difficult to estimate, unless a similar experiment has been conducted...".
» That smells for me like within a study.

Yep, then I expect a high correlation (based on my limited knowledge of PK). For my data sets I get with
pearson <- cor.test(log(study$AUC), log(study$Cmax))
rho[set, "estimate"] <- pearson$estimate
rho[set, "lower"]    <- as.numeric(pearson$conf.int)[[1]]
rho[set, "upper"]    <- as.numeric(pearson$conf.int)[[2]]
summary(rho, digits=5)

    estimate           lower               upper     
 Min.   :0.20123   Min.   :-0.094923   Min.   :0.40858
 1st Qu.:0.70335   1st Qu.: 0.516584   1st Qu.:0.82721

 Median :0.81713   Median : 0.677607   Median :0.89959
 Mean   :0.77413   Mean   : 0.633948   Mean   :0.86544
 3rd Qu.:0.90122   3rd Qu.: 0.827646   3rd Qu.:0.94461
 Max.   :0.98928   Max.   : 0.977336   Max.   :0.99494


IIRC, Chow & Liu state somewhere that the mean of within-subject ratios is a biased estimate…

Another question: What is a “similar experiment”? Given the dispersion in the summary above I would rather say that trying to get an estimate across different drugs (or even for the same drug but IR/IR and MR/MR) is futile. Will try again with the analyte as a factor.

[image]Regards,
Helmut Schütz 
[image]

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
d_labes
Hero

Berlin, Germany,
2018-02-11 19:21

@ Helmut
Posting: # 18399
Views: 1,271
 

 The mysterious ρ -between or within studies

Dear Helmut,

» Yep, then I expect a high correlation (based on my limited knowledge of PK). For my data sets I get with
» pearson <- cor.test(log(study$AUC), log(study$Cmax))
» rho[set, "estimate"] <- pearson$estimate
» rho[set, "lower"]    <- as.numeric(pearson$conf.int)[[1]]
» rho[set, "upper"]    <- as.numeric(pearson$conf.int)[[2]]
» summary(rho, digits=5)
»
»     estimate           lower               upper     
»  Min.   :0.20123   Min.   :-0.094923   Min.   :0.40858
»  1st Qu.:0.70335   1st Qu.: 0.516584   1st Qu.:0.82721

»  Median :0.81713   Median : 0.677607   Median :0.89959
»  Mean   :0.77413   Mean   : 0.633948   Mean   :0.86544
»  3rd Qu.:0.90122   3rd Qu.: 0.827646   3rd Qu.:0.94461
»  Max.   :0.98928   Max.   : 0.977336   Max.   :0.99494

Thanks for that numbers.

» Another question: What is a “similar experiment”?

Good question. Another question. Duno. Pilot study?
For me the hole rho story is a mystery. And rather academic.
Nobody is acting according to that theory. Not at least because the understanding of what value rho shoud have or how it could be estimated.

Taking the greater sample size from the two estimations worked in the past. So what?
Theoretically this would be the case if rho ~ 1 since then the combined power of the two TOST is the minimum of the individual powers. That would be my favoured candidate :-D.
Be warned: rho=1 throws an error in sampleN.2TOST().

Regards,

Detlew
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-11 22:44

@ d_labes
Posting: # 18401
Views: 1,253
 

 The mysterious ρ within studies

Dear Detlew,

» » Another question: What is a “similar experiment”?
»
» Good question. Another question. Duno. Pilot study?
» For me the hole rho story is a mystery. And rather academic.
» Nobody is acting according to that theory. Not at least because the understanding of what value rho shoud have or how it could be estimated.
»
» Taking the greater sample size from the two estimations worked in the past. So what?
» Theoretically this would be the case if rho ~ 1 since then the combined power of the two TOST is the minimum of the individual powers. That would be my favoured candidate :-D.

Yep. Slowly I get the impression it is a dead end.

Below “within study rhos” aggregated by analyte. Within each study rho and its 95% CI was calculated. Then I calculated their medians (n is the number of studies), sorted by rho:

                    analyte n med.rho med.lower med.upper
       Acetylsalicylic acid 1  0.2012   -0.0879    0.4591
                   Losartan 1  0.2267    0.0274    0.4086
                 Torasemide 1  0.3036   -0.0949    0.6182
                Linsidomine 1  0.3581    0.0823    0.5829
       2-pyridylacetic acid 1  0.4593    0.2131    0.6506
                Simvastatin 1  0.4871    0.3505    0.6034
              Acetazolamide 3  0.5157    0.3901    0.6308
        Hydrochlorothiazide 1  0.5340    0.2946    0.7104
           Sulfamethoxazole 2  0.5658    0.3180    0.7496
                  Ibuprofen 1  0.5714    0.2781    0.7672
               Dexamfetamin 1  0.5760    0.3052    0.7606
              Metronidazole 2  0.5987    0.3362    0.7783
               Galanthamine 1  0.6091    0.3507    0.7813
                 Omeprazole 1  0.6373    0.4258    0.7828
          Ezetimibe (total) 1  0.6388    0.5299    0.7271
             Chlorthalidone 1  0.6450    0.4174    0.7965
                    Celoxib 1  0.6542    0.5413    0.7440
               Prednisolone 3  0.6585    0.3784    0.8281
              Nortriptyline 2  0.6643    0.4328    0.8141
                       NA.2 1  0.6751    0.5631    0.7628
                   Tramadol 1  0.6810    0.3828    0.8507
       2-phenylbutiric acid 2  0.6817    0.4740    0.8206
                 Diclofenac 1  0.7082    0.5396    0.8222
                Amoxicillin 3  0.7092    0.4686    0.8518
                  Valsartan 2  0.7232    0.6058    0.8128
          Omeprazolesulfone 1  0.7240    0.5492    0.8382
                   Cefaclor 2  0.7272    0.5296    0.8499
                Telmisartan 1  0.7425    0.6488    0.8140
         Methylprednisolone 2  0.7490    0.5514    0.8676
                  Captopril 1  0.7521    0.5952    0.8538
               Pantoprazole 1  0.7584    0.6390    0.8421
              Valproic acid 4  0.7586    0.5662    0.8725
               Trimethoprim 2  0.7588    0.5581    0.8757
                 Olmesartan 1  0.7723    0.6584    0.8516
           Ezetimibe (free) 1  0.7752    0.6998    0.8335
     7-alpha-Spironolactone 2  0.7757    0.6211    0.8724
 Docorubicin (encapsulated) 1  0.7773    0.5693    0.8917
                 Quetiapine 2  0.7786    0.6691    0.8551
                  Enalapril 4  0.7793    0.6368    0.8706
                  Canrenone 4  0.7810    0.6386    0.8717
                 Furosemide 4  0.7858    0.6366    0.8784
                Doxycycline 2  0.7860    0.6123    0.8873
              Ciprofloxacin 1  0.7950    0.5765    0.9074
              Amitriptyline 2  0.8000    0.6487    0.8919
              Glibenclamide 1  0.8088    0.6813    0.8887
                Clindamycin 5  0.8149    0.6697    0.9001
               Methotrexate 1  0.8162    0.6767    0.8991
                 Ranitidine 1  0.8166    0.6541    0.9070
                  Gefitinib 1  0.8214    0.7524    0.8726
         Docorubicin (free) 1  0.8236    0.6504    0.9154
             Salicylic acid 1  0.8377    0.7265    0.9062
        Docorubicin (total) 1  0.8392    0.6785    0.9232
               Lansoprazole 1  0.8403    0.7686    0.8912
                  Metformin 1  0.8513    0.7148    0.9253
   5-Methyltetrahydrofolate 2  0.8560    0.7557    0.9170
              Pentoxifyllin 1  0.8621    0.7654    0.9207
                  Carvediol 1  0.8636    0.7704    0.9206
                    Doxepin 1  0.8645    0.7786    0.9186
            Methylphenidate 7  0.8691    0.7263    0.9382
                       NA.1 1  0.8721    0.8272    0.9060
                   Ambroxol 1  0.8835    0.7892    0.9371
                  Oxycodone 2  0.8873    0.8398    0.9214
   Isosorbide 5-mononitrate 3  0.8944    0.8256    0.9370
                  11-OH THC 1  0.8950    0.7942    0.9479
               Norverapamil 1  0.9009    0.8290    0.9435
                Abiraterone 1  0.9021    0.8686    0.9275
                   Atenolol 1  0.9044    0.8254    0.9486
                  Acyclovir 1  0.9063    0.8601    0.9377
                 Amiodarone 1  0.9067    0.8200    0.9528
                 Dronabinol 1  0.9103    0.8228    0.9557
                  Sulpiride 2  0.9134    0.8618    0.9463
     O-Desmethylvenlafaxine 1  0.9178    0.8619    0.9517
                  Verapamil 1  0.9191    0.8595    0.9541
               Theophylline 1  0.9202    0.8595    0.9553
                   Bosentan 2  0.9206    0.8865    0.9448
                Molsidomine 1  0.9214    0.8633    0.9554
                 Propafenon 1  0.9338    0.8776    0.9647
                 Lisinopril 4  0.9373    0.8895    0.9648
            Clavulanic acid 3  0.9442    0.8878    0.9727
         Desethylamiodarone 1  0.9545    0.9081    0.9778
                Venlafaxine 1  0.9546    0.9226    0.9735
               Folinic acid 1  0.9633    0.9351    0.9794
             Acetylcysteine 1  0.9637    0.9358    0.9796
               Nicotinamide 1  0.9725    0.9581    0.9820
              Docorubicinol 1  0.9741    0.9423    0.9885
                 Paroxetine 5  0.9788    0.9649    0.9872


Some results are crazy. ASA has a very low rho and its lower CL is negative…

The extremes:

[image]    [image]

Example of concordant outliers (poor metabolizers):

[image]

» Be warned: rho=1 throws an error in sampleN.2TOST().

I know. Maybe:

if (rho == 1) rho <- 1-.Machine$double.eps

and throwing a message?

[image]Regards,
Helmut Schütz 
[image]

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nobody
Senior

2018-02-12 09:08

@ Helmut
Posting: # 18403
Views: 1,205
 

 The mysterious ρ within studies

...maybe bring Clearance into the picture?

Kindest regards, nobody
Astea
Regular

Russia,
2018-02-11 17:34
(edited by Astea on 2018-02-11 17:54)

@ d_labes
Posting: # 18397
Views: 1,291
 

 The mysterious ρ -between or within studies

Dear Detlew!

Thank you very much for the explanation! Now I've got it! (Yes, multitasks harm..)

So, for ρ=0.8 I get now
type1error.2TOST(CV=c(0.3, 0.25), n=122, theta1=c(0.8, 0.9),
                 theta2=c(1.25, 1/0.9), rho=0.8, details=TRUE)
1e+06 simulations. Time consumed (secs)
   user  system elapsed
 979.75   95.18 1115.72
  Intersection null P(Type I Error) theta0 #1 theta0 #2
1      H_A01 n H_Ca        0.048904 0.8000000  0.953047
2      H_A02 n H_Ca        0.049193 1.2500000  1.056122
3      H_Aa n H_C01        0.049620 0.9709781  0.900000
4      H_Aa n H_C02        0.050024 1.0000000  1.111111
5     H_A01 n H_C01        0.024479 0.8000000  0.900000
6     H_A01 n H_C02        0.000000 0.8000000  1.111111
7     H_A02 n H_C01        0.000000 1.2500000  0.900000
8     H_A02 n H_C02        0.024451 1.2500000  1.111111
and inflation at 0.050024

As I understand estimated from several datasets ρ is "between studies" ρ while in Power2TOST we need to use "within studies ρ". But it may be interesting while discussing multicomponent drugs. Especially question may arise for two components, when one of them is HVD while other is not and different approaches or CI limits involved (like telmisartan+amlodipine).
The intermediate question is mentioned situation about parent drug and its metabolite - logically, the correlation for the parameters of parent drug and its metabolite should be stronger than for two different drugs tested on the same population?

Dear Helmut!
Do I understand correctly that in your latest code you exployed all AUC and Cmax individual data for each study in order to estimate "within rho"?
d_labes
Hero

Berlin, Germany,
2018-02-11 19:36

@ Astea
Posting: # 18400
Views: 1,270
 

 The mysterious ρ -between or within studies

Dear Nasty!

» So, for ρ=0.8 I get now
» > type1error.2TOST(CV=c(0.3, 0.25), n=122, theta1=c(0.8, 0.9),
» +                        theta2=c(1.25, 1/0.9), rho=0.8, details=TRUE)
» 1e+06 simulations. Time consumed (secs)
»    user  system elapsed
»  979.75   95.18 1115.72
»   Intersection null P(Type I Error) theta0 #1 theta0 #2
» 1      H_A01 n H_Ca        0.048904 0.8000000  0.953047
» 2      H_A02 n H_Ca        0.049193 1.2500000  1.056122
» 3      H_Aa n H_C01        0.049620 0.9709781  0.900000
» 4      H_Aa n H_C02        0.050024 1.0000000  1.111111
» ...
»
and inflation at 0.050024
Don't forget, that type1error.2TOST() is based on simulations now.
That's mainly because our (authors team of PowerTOST) brain is to small to derive an analytical solution. And simulations have an error. F.i. in setting the seed for the random number generator.
type1error.2TOST(CV=c(0.3, 0.25), n=122, theta1=c(0.8, 0.9), theta2=c(1.25, 1/0.9), rho=0.8, setseed=F)
[K1] 0.049977

In that light 0.050024 is of course not an alpha-inflation. As we know from theory, the IUT principle, the combination of two TOST via and assures alpha<=0.05. Thus interpret the result as 0.05. Full stop

» As I understand estimated from several datasets ρ is "between studies" ρ while in Power2TOST we need to use "within studies ρ".

I myself don't know what ρ or what value of ρ should be used. As I wrote above: It's under discussion now.

Thus if you or somebody else have an opinion or arguments: Give it to me.

Regards,

Detlew
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-11 22:48

@ Astea
Posting: # 18402
Views: 1,252
 

 The mysterious ρ within studies

Dear Astea!

» The intermediate question is mentioned situation about parent drug and its metabolite - logically, the correlation for the parameters of parent drug and its metabolite should be stronger than for two different drugs tested on the same population?

Oh dear, yes! But that’s yet another story.

» 1e+06 simulations. Time consumed (secs)
»    user  system elapsed
» 979.75   95.18 1115.72

Get a faster machine! My almost three years old can:
1e+06 simulations. Time consumed (secs)
   user  system elapsed
 208.64   31.17  240.75


» Do I understand correctly that in your latest code you exployed all AUC and Cmax individual data for each study in order to estimate "within rho"?

Yes. Apples and oranges. See there for a breakdown by analyte.

[image]Regards,
Helmut Schütz 
[image]

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ElMaestro
Hero

Denmark,
2018-02-11 18:43

@ Louis52
Posting: # 18398
Views: 1,276
 

 Power and type I adjustment for 222BE when several PK prams considered

Hi all,

can we take one step back and discuss what we are really trying to achieve here? Forget for a moment the famous theorems and equations and think practically.

Ratios of Cmax and AUC are correlated within studies. No doubt about it. Increase Cmax and AUC increases, all other factors equal, and vice versa.
It would perhaps be good if we could some model that via some rho so that we could calculate a slightly higher sample size to take into consideration that both ratios must pass the 80.00%- 125.00% criterion.

But I never heard a sponsor or CRO really express that need in a specific manner. I never saw a study failing on one metric and passing on the other and where someone afterwards thought she/he should have taken the correlation into consideration and all would have been good.

Bear in mind also that both test and reference batches of any product have variation within and between. The latter is almost never explored but it is generally higher than within, as usual. Both, whether estimable/measurable or not, will be reflected in the rho we can observe and thus we may have to take that into consideration too.
The exact same issue applies to the papers that try to express a general opinion about product A vs product C, when product A has been tested against product B, and B has been tested against C. It is a matter authors have managed to tiptoe elegantly around and certainly not something anyone wants to get mixed up in.

What I mean here is I am in doubt if we are discussing a practical problem. And secondly if this problem is practical, I am wondering if the treatment we can realistically give the issue changes much given all the underlying complexity.:-):confused:

I could be wrong, but…


Best regards,
ElMaestro

- Bootstrapping for dissolution data is a relatively new hobby of mine.
nobody
Senior

2018-02-12 09:10

@ ElMaestro
Posting: # 18404
Views: 1,200
 

 Power and type I adjustment for 222BE when several PK prams considered

General approach: problem -> solution.

;-)

Don't try to find nails everywhere, just 'cause you brought your hammer with you that day...

Kindest regards, nobody
ElMaestro
Hero

Denmark,
2018-02-12 09:18

@ nobody
Posting: # 18405
Views: 1,195
 

 reminds me of something

Hi nobody,

this reminds me of a quote I stumbled upon the other day, attributed to Paul Dirac who was known for being a stone-cold scientist:
"The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible."

I could be wrong, but…


Best regards,
ElMaestro

- Bootstrapping for dissolution data is a relatively new hobby of mine.
nobody
Senior

2018-02-12 09:38

@ ElMaestro
Posting: # 18406
Views: 1,191
 

 reminds me of something

Yeah, but both aim at new "fairy tales", new "lingual pictures", new parables, one side based on facts and numbers, the other side based on emotions.

But where does the hammer come from? :-D

Kindest regards, nobody
Astea
Regular

Russia,
2018-02-12 18:37

@ nobody
Posting: # 18408
Views: 1,136
 

 Hammer to fall

Dear all!

They say that upon hearing that one of his students had dropped out to study poetry David Hilbert said: "Good, he did not have enough imagination to become a mathematician".


Dear Detleww!

» In that light 0.050024 is of course not an alpha-inflation. As we know from theory, the IUT » principle, the combination of two TOST via and assures alpha<=0.05. Thus interpret the result as 0.05. Full stop

Thank you very much for clarification! So then it's just a numeric atavism and misinterpretation.


Dear Helmut!

» Get a faster machine!
I've got one, but currently my car do not provide calculations in PowerTost :-)

I may suppose there were some problems in your Acetylsalicylic acid study - may be in stability or inadequate sampling time, did you measure also salicylic acid? What was the dose? Below are my results on ASA study (data of reference drug only): rho is 0.667, 95% CI: [0.361; 0.844]. For Test+Ref data rho is about 0.755.

[image]
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-13 01:37

@ Astea
Posting: # 18409
Views: 1,117
 

 Put the sickle to the corn

Dear Astea,

may I call you Nasty as well?

» I may suppose there were some problems in your Acetylsalicylic acid study - may be in stability…

Samples put in an ice-bath until centrifugation, plasma diluted 1:1 with 5% o-phosphoric acid for stabilization.

» … or inadequate sampling time,

Hhm. 00:05, 00:10, 00:15, 00:20, 00:30, 00:40, 00:50, 01:00, 01:20, 01:40, 02:00, 02:30, 03:00, 04:00, 06:00, 08:00, 10:00, 12:00 hours. No problems with "first point Cmax".


» … did you measure also salicylic acid?

Sure. Nice rho (see in the list above).

[image]

» What was the dose?

100 mg ASA (actually 5 × 20 mg tablets).

I recalculated the entire study (October 1997!) from raw-data (nasty!). Maximum residual AUC in any of the 24 subjects 7%. CVintra: ASA 18.3% (Cmax), 11.8% (AUC0–t); SA 9.58% (Cmax), 8.71% (AUC0–t). Both ASA and SA passed BE with flying colors.

[image]Regards,
Helmut Schütz 
[image]

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Astea
Regular

Russia,
2018-02-13 08:27

@ Helmut
Posting: # 18410
Views: 1,096
 

 Put the sickle to the corn

Dear Helmut!

» I recalculated the entire study (October 1997!) from raw-data (nasty!).

Very sorry about that! I didn't want to doubt into results of your study, just wanted to point out that the results of miscorrelation between Cmax and AUC should be more dependent on the features of the specific study than on the drug itself.

» Maximum residual AUC in any of the 24 subjects 7%.

Is it possible to expect large AUCresid for ASA when sampling time is 12 hours?

» Hhm. 00:05, 00:10...

That's perfect. But than may it be connected with fast elimination of ASA? That is sampling time is adequate but comparing to the less rapid drug the difference in 30 minutes in elimination phase could give sufficiently more error in calculating AUC (one day for jinn as thousand years for man)?
The last question about it: what was the mean T1/2 (from these abrupt profiles I may suspect less than known in literature 20 minutes)?

» may I call you Nasty as well?

I would prefer Nastia (that was Nabokov who first pointed out the similar pronounciation)
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-13 12:45

@ Astea
Posting: # 18412
Views: 1,052
 

 Put the sickle to the corn

Hi Nastia,

» I didn't want to doubt into results of your study,

No worries. I was curious myself.

» just wanted to point out that the results of miscorrelation between Cmax and AUC should be more dependent on the features of the specific study than on the drug itself.

Or the dose regimen (as Relaxation suspected)?

» […] is sampling time is adequate but comparing to the less rapid drug the difference in 30 minutes in elimination phase could give sufficiently more error in calculating AUC (one day for jinn as thousand years for man)?

:-D

» The last question about it: what was the mean T1/2 (from these abrupt profiles I may suspect less than known in literature 20 minutes)?

Linear plot with enlarged time scale:

[image]

Your wish is my command. Data cemetery:
     tlag        tmax        t½         Cmax          AUCt           AUCext
T  5 (0, 5)  40 (30, 57) 
22 (6.5)  5.36 (24.3)  6.03 (15.6)  6.37 (5.81, 6.94)
R  5 (0, 5)  35 (30, 50) 
23 (5.2)  5.03 (25.9)  5.48 (21.0)  5.45 (4.89, 6.73)

tlag, tmax minutes; median (quartiles)
t½         minutes; harmonic mean (jacknife SD)
Cmax       µg/mL; geometric mean (CV%)
AUCt       h×µg/mL; geometric mean (CV%)
AUCext     100(AUCt–AUC)/AUC; median (quartiles)

[image]Regards,
Helmut Schütz 
[image]

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Relaxation
Junior

Germany,
2018-02-13 10:37

@ Helmut
Posting: # 18411
Views: 1,064
 

 Put the sickle to the corn

Dear All.

Although I am lost regarding most of the mathematical discussion, I understood, that ASA seems to misbehave - and this information catched my exe.

» 100 mg ASA (actually 5 × 20 mg tablets).

To be very brief, from my experience, a study with multiple tablets under fasted conditions always could/might (will) result in an increase in variability in particular in Cmax.
To visualize, imagine all tablets reach the stomach immediately, but every tablet has an individual chance to be emtyied into the intestine.
So we might see subjects with one single peak (5 tablets at once) or subjects with 5 individual peaks (blurred together) and anything in between. These subjects will then show artificially low Cmax values that will not correlate at all.

Looking at the individual profiles posted above I think I see at least a few "irregular profiles" even including late tmax at about 2 hours.

I am not saying that this is a/the solution, but this may be one aspect helping to understand, why the correlation for ASA is lousy in comparison with other APIs and drug products.

Just flashed through my mind.

Best regards,

Relaxation.
Helmut
Hero
Homepage
Vienna, Austria,
2018-02-13 13:31

@ Relaxation
Posting: # 18413
Views: 1,050
 

 multiple units, lag-times

Hi Relaxation,

» Although I am lost regarding most of the mathematical discussion,…

You are not alone.

» … ASA seems to misbehave - and this information catched my exe.
»
» » 100 mg ASA (actually 5 × 20 mg tablets).

I have to correct myself: 500 mg ASA (5 × 100 mg tablets; reference Bayer’s Aspirin® 100 N). Don’t know why such a high dose was chosen (sponsor’s wish ;-)); we had another analytical method sensitive enough for 100 mg as well. It is well known that at ~200 mg we are deep in saturation…

» To be very brief, from my experience, a study with multiple tablets under fasted conditions always could/might (will) result in an increase in variability in particular in Cmax.
» To visualize, imagine all tablets reach the stomach immediately, but every tablet has an individual chance to be emtyied into the intestine.
» So we might see subjects with one single peak (5 tablets at once) or subjects with 5 individual peaks (blurred together) and anything in between. These subjects will then show artificially low Cmax values that will not correlate at all.

You are absolutely right in general. Below simulations (n=24) with a lag-time of 1±1 h. Lag-times simulated with a truncated normal distribution [0, 2]. V, ka, CL simulated with a lognormal. CVintra 12%, CVinter 21%, GMR 100%.

Single dose of 500 mg:

[image]

AUC  GMR (90% CI):  97.59% (74.93% – 127.11%)
Cmax GMR (90% CI):  97.60% (74.92% – 127.15%)


Single dose of five units à 100 mg:

[image]

AUC  GMR (90% CI):  98.96% (87.28% – 112.22%)
Cmax GMR (90% CI): 100.81% (70.93% – 143.26%)


As you rightly assumed, the intra-subject variability of Cmax substantially increases (whereas the one of AUC decreases).

» Looking at the individual profiles posted above I think I see at least a few "irregular profiles" even including late tmax at about 2 hours.

See a more telling plot above. Latest tmax at 1:40. But: In 4/24 subjects tmax was >50 minutes (R) and in 6/24 (T). Too stupid to simulate that.

» I am not saying that this is a/the solution, but this may be one aspect helping to understand, why the correlation for ASA is lousy in comparison with other APIs and drug products.

Maybe. Maybe not. There are many examples very the variability in Cmax is much higher than the one of AUC and the correlation is pretty good. Here both behave nicely (CVintra 18.3% and 11.8%). Still fail to understand why I saw such a low correlation in my study (and Astea a much higher one in hers).

[image]Regards,
Helmut Schütz 
[image]

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