Ben ★ 20120804 15:46 (3948 d 19:19 ago) Posting: # 9037 Views: 5,530 

Dear all, Sorry for bringing up this topic again, I know there are lots of posts, but still I do not fully understand why a 3x6x3 crossover design is prefered compared to a 3x3x3 design (or in what cases the latter suffices) and hope you are willing to discuss again. The main argument is that it (the 3x6x3 design) is variance balanced and balanced for (first order) carryover effects. Let us consider the latter property: What exactly does it mean? Does it mean that on average the effect of carryover cancels out (as with the period effect)? Is that always the case or only in case carryover is incoprorated in the statistcal model? If it always holds true, then I wonder why people are concerned about carryover and incorporate washout (maybe otherwise a higherorder carryover will occur...)  at least for Williams designs. If it's only in case the effect is incorporated in the model (as I understood from this post this option is correct), then this advantage is actually useless because it is not recommended to incorporate this effect and in fact "nobody" does. Coming to the variance balanced property, this implies that all pariwise trt comparisons have the same variance and hence precision (> CI), but does it also imply the CIs are shorter compared to the ones obtained by 3x3x3? Probably not, it might be the case that one obtained by 3x3x3 is even shorter, but another is not. There are cases where we do not care about certain comparisons (e.g. T1, T2, R; here, T1 vs T2 may not be of interest). The problem though is that we cannot tell which comparison is/will be better or worse?! So instead of not knowing a priori we want to have the same variance for sure... Helmut mentioned in one of his posts that "The 3×3×3 does not give us unbiased estimates". But 3x6x3 does? For me, having variance balance does guarantee unbiased estimates (for the trt comparison). Can someone shed light on this? Another argument is that "Any Williams' design has the advantage that pairwise comparisons may be extracted (also recommended by Byron Jones in a personal communication at the BioInternational 2003)  which are also balanced (needed for nonparametric comparisons, which seems to be of historical interest in the EU...)" as posted by Helmut here. Is this extracting only useful for special additional analyses as e.g. the mentioned nonparametric comparisons or will this extracting 2x2 tables always be used when analysing/evaluating a crossover design (point estimate and CI)? If it's always used then clearly this property is an advantage, otherwise I wonder what these special analayses are good for and whether they are actually used. Regarding missing values/dropouts: In this case will the 3x3x3 design be more "vulnerable" when losing values? Meaning that also the 3x6x3 will get broader CIs but the 3x3x3 will get broader CIs more quickly (i.e. less dropouts needed in order to get broader CIs)? (The extent to what degree the CIs will get broader is not known) Side note and question regarding PowerTOST: When thinking of precision I was also thinking about the power. The common precision does not imply a greater power, though (leaving all other parameters unchanged); the property of having variance balance does not play a role when computing the power. Moreover, the only formal changes when changing a design are the degrees of freedom and the design constant bk (assuming equal allocation to sequences) and since both parameters are the same for 3x3x3 and 3x6x3 the powers should be the same. I used PowerTOST (v0.910) to verify but got
power.TOST(CV=0.2, design="3x3", n=18) Is that a bug in PowerTOST? When computing sample sizes the corresponding achieved powers coincide (although different from both values above):
sampleN.TOST(CV=0.2, design="3x3", print=FALSE) Btw, wrt power the 3x3x3 design seems to be less vulnerable than 3x6x3: In case of one dropout the power of 3x3x3 decreases about 3.8% whereas the power of 3x6x3 decreases about 5.1% (compare above results to power2.TOST(CV=0.2, design="3x3", n=c(5,6,6)) and power2.TOST(CV=0.2, design="3x6x3", n=c(2,3,3,3,3,3)), respectively). This might be an advantage of 3x3x3... Thanks, Ben 
d_labes ★★★ Berlin, Germany, 20120807 11:13 (3945 d 23:52 ago) @ Ben Posting: # 9045 Views: 4,197 

Dear Ben, ❝ Side note and question regarding PowerTOST: ❝ ... ❝ Is that a bug in PowerTOST? Shit happens. It is a bug . In the functions power.TOST(), power2.TOST() and power.noninf() the robust degrees of freedom are used although the argument robust=FALSE by default. Setting (falsely !) robust=TRUE will give the correct answer using the nonrobust df's: power.TOST(CV=0.2, design="3x3", n=18, robust=T) This bug affects all higher order crossover designs. But not the sample size estimation. Sorry for any inconvenience. A bug fix version is on the way via CRAN. — Regards, Detlew 
d_labes ★★★ Berlin, Germany, 20120807 21:14 (3945 d 13:52 ago) @ Ben Posting: # 9048 Views: 4,283 

Dear Ben, Lets come to your original questions (beside PowerTOST issues). I will only deal with carryover. Others may be more qualified for your other issues. ❝ The main argument is that it (the 3x6x3 design) is variance balanced and balanced for (first order) carryover effects. Let us consider the latter property: What exactly does it mean? Does it mean that on average the effect of carryover cancels out (as with the period effect)? For the latter you are mistaken. A period effect only cancels out if you have it in your evaluation model. Otherwise it will be incorporated into the treatment effect and into the error term (at least partially) and and will affect your 90% confidence intervals. The same applies to to carry over also. But carryover evaluation in crossover designs, if any, suffer from another fact. It is usually modeled as so called "first order" carryover or "simple carry over". The main feature of such modeling is that the carryover effect lasts only from the treatment given in the previous period. This model is extremely oversimplified and not in accordance with pharmacokinetic and pharmacodynamic conceptions. (see Steven Senn, Crossover Trials in Clinical Research, John Wiley & Sons, Chichester, chapter 10) Therefore Senn considered this model as obsolete. Thus the advantage of Williams designs, which are optimal with respect to such an carryover model, vanish as you by yourself noticed. Hope this helps. — Regards, Detlew 
ElMaestro ★★★ Denmark, 20120808 03:02 (3945 d 08:04 ago) @ d_labes Posting: # 9049 Views: 4,111 

Hi d_labes, ❝ But carryover evaluation in crossover designs, if any, suffer from another fact. It is usually modeled as so called "first order" carryover or "simple carry over". The main feature of such modeling is that the carryover effect lasts only from the treatment given in the previous period. ❝ This model is extremely oversimplified and not in accordance with pharmacokinetic and pharmacodynamic conceptions. (see Steven Senn, Crossover Trials in Clinical Research, John Wiley & Sons, Chichester, chapter 10) Has it ever been shown that this is not merely a theoretical problem? I understand why first order is not any order but really, is there an issue of practical importance here? — Pass or fail! ElMaestro 
d_labes ★★★ Berlin, Germany, 20120808 09:59 (3945 d 01:06 ago) @ ElMaestro Posting: # 9050 Views: 4,086 

Hi ElMaestro, ❝ Has it ever been shown that this is not merely a theoretical problem? I understand why first order is not any order but really, is there an issue of practical importance here? That was the meaning of my . As already said here: No practical importance to deal with carryover, exceptionally not with "first order". — Regards, Detlew 
Ben ★ 20120809 21:22 (3943 d 13:44 ago) @ d_labes Posting: # 9052 Views: 4,109 

Dear Detlew, First of all, I'm glad I could help improve PowerTOST! Thanks for your reply on the carryover issue. It helps, although this fact does not show that one should use Williams design. Best regards, Ben 
Ben ★ 20120819 14:54 (3933 d 20:12 ago) @ Ben Posting: # 9081 Views: 3,892 

❝ ... For me, having variance balance does guarantee unbiased estimates (for the trt comparison). ... 