sweiner
☆

2020-06-30 01:07
(566 d 05:23 ago)

Posting: # 21612
Views: 3,828

## higher order crossover methodology [General Sta­tis­tics]

Hi,
In a 5-way BE crossover trial, is there a methodology available that allows one to analyze 3 out of 5 sequences for BE in an interim fashion, and contingent upon results complete the remaining 2 sequences and analyze all 5 sequences at the conclusion?
Could you please provide references on this subject.

Thanks,
Sveta

Helmut
★★★

Vienna, Austria,
2020-06-30 12:19
(565 d 18:11 ago)

@ sweiner
Posting: # 21615
Views: 3,058

## All dosed subjects have to be analyzed

Hi Sveta,

» In a 5-way BE crossover trial, is there a methodology available that allows one to analyze 3 out of 5 sequences for BE in an interim fashion, and contingent upon results complete the remaining 2 sequences and analyze all 5 sequences at the conclusion?

Once you dosed subjects (i.e., in all 5 sequences), you have to analyze them. Anything else is not ethical. Along these lines the EMA’s BE-GL, section Subject accountability:

Ideally, all treated subjects should be included in the statistical analysis. […]
The data from all treated subjects should be treated equally. It is not acceptable to have a protocol which specifies that ‘spare’ subjects will be included in the analysis only if needed as replacements for other subjects who have been excluded. It should be planned that all treated subjects should be included in the analysis, even if there are no drop-outs.

(my emphasis)

Even if your protocol would be accepted by the IEC/IRB and agency (I hope not), your approach might lead to an inflated type I error (if you fail with 3 sequences and continue with the other 2) because at the end you will use 60% of the data twice. Calls for some kind of α-adjustment (narrower CI) due to multiplicity issues.

» Could you please provide references on this subject.

IMHO, nothing published – and for a reason. Don’t even think about the ‘magic’ α 0.0294. In short: Forget it.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
sweiner
☆

2020-06-30 17:08
(565 d 13:22 ago)

@ Helmut
Posting: # 21621
Views: 2,989

## All dosed subjects have to be analyzed

Thanks Helmut! This is very helpful.

sedhosen
☆

Iran,
2021-03-16 08:30
(306 d 21:00 ago)

@ Helmut
Posting: # 22270
Views: 1,150

## All dosed subjects have to be analyzed

Hi Helmut,

I read the mentioned guideline "BE-GL, section Subject accountability" and I have a question about replacing people in the study. Regarding the following sentence from the guideline:

"It is not acceptable to have a protocol which specifies that ‘spare’ subjects will be included in the analysis only if needed as replacements for other subjects who have been excluded."

Is it also not acceptable to replace subjects even if the number of excluded subjects is more than the number of considered dropouts?
For example, if 7 subjects are withdrawn during the study after receiving the dose, and the number of considered dropouts are 5 subjects, is there any way we can replace 2 subjects that were withdrawn over the number of dropouts?

Edit: Relax; see also this post #9. Another post with identical text deleted. [Helmut]
Helmut
★★★

Vienna, Austria,
2021-03-16 16:00
(306 d 13:29 ago)

@ sedhosen
Posting: # 22273
Views: 1,087

## All dosed subjects have to be analyzed

Hi sedhosen,

» Is it also not acceptable to replace subjects even if the number of excluded subjects is more than the number of considered dropouts?

Acc. to the GL, yes.

» For example, if 7 subjects are withdrawn during the study after receiving the dose, and the number of considered dropouts are 5 subjects, is there any way we can replace 2 subjects that were withdrawn over the number of dropouts?

Two points:
1. The impact of dropouts on power is overrated by many (you are not alone). An example with a dropout-rate of 15% and two less eligible subjects than anticipated:
Does it really matter? Remember that the sample estimation is based on assumptions – it’s not an exact calculation.
   CV  n  power dosed eligible power.actual  0.20 20 0.8347    24       18       0.7912  0.25 28 0.8074    34       26       0.7761  0.30 40 0.8158    48       38       0.7953  0.35 52 0.8075    62       50       0.7917
-scrip at the end.
2. If you would dose two subjects later, it could complicate the statistical model because periods differ. Of course, you could ignore that but I have seen nasty questions from regulators.

ibrary(PowerTOST) up2even <- function(n, ns = 2) {   return(as.integer(ns * (n %/% ns + as.logical(n %% ns)))) } nadj <- function(n, do.rate, ns = 2) {   return(as.integer(up2even(n / (1 - do.rate), ns))) } design  <- "2x2x2" CV      <- seq(0.2, 0.35, 0.05) theta0  <- 0.95 # T/R-ratio target  <- 0.80 # desired power do.rate <- 0.15 # 15% if (design == "parallel") {   ns <- 2L } else {   ns <- as.integer(substr(design, 3, 3)) } df <- data.frame(CV = CV, n = NA, power = NA,                  dosed = NA, eligible = NA, power.actual = NA) for (j in 1:nrow(df)) {   tmp <- sampleN.TOST(CV = CV[j], theta0 = theta0,                       targetpower = target,                       design = design, print = FALSE)   df[j, 2:3] <- tmp[7:8]   df[j, 4]   <- nadj(df[j, 2], do.rate, ns)   df[j, 5]   <- df[j, 2] - 2   df[j, 6]   <- power.TOST(CV = CV[j], theta0 = theta0,                            design = design, n = df[j, 5]) } print(signif(df, 4), row.names = FALSE)

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Relaxation
★

Germany,
2020-06-30 14:10
(565 d 16:19 ago)

@ sweiner
Posting: # 21618
Views: 3,030

## higher order crossover methodology

Dear Sveta.

I just want to ask, whether you may want to give some more background information on the situation.
Because I really was not able to think of a scenario, where considering only a part of the population based on the sequences could provide a benefit (unless its a sneaky two-stage, but then as said: nothing to add to the answer given).
Likely just a case of limited imaginative power on my side .

Best regards,

Steven.
sweiner
☆

2020-06-30 17:10
(565 d 13:20 ago)

@ Relaxation
Posting: # 21622
Views: 2,982

## higher order crossover methodology

» I just want to ask, whether you may want to give some more background information on the situation.
» Because I really was not able to think of a scenario, where considering only a part of the population based on the sequences could provide a benefit […]

Dear Steven, thanks for the feedback!
Here the first part of the 5-way crossover analysis (Part A- 3x3 analysis) would be under fasted conditions (Tablet formulation #1 vs. Tablet formulation #2 vs. Capsule formulation), and if BE is shown for either Tablet formulation vs. Capsule formulation, same subjects would complete in a 2x2 fashion under fed conditions (Tablet #1 or #2 [depending on BE result] vs. Capsule). In the final analysis, once all 5 sequences complete all 5 periods, fasted state vs. fed state to be compared for BE.

ElMaestro
★★★

Denmark,
2020-06-30 18:17
(565 d 12:13 ago)

@ sweiner
Posting: # 21623
Views: 2,984

## higher order crossover methodology

Ah,

it is all coming together now.

Thanks sweiner,
this is a great question.

» In a 5-way BE crossover trial, is there a methodology available that allows one to analyze 3 out of 5 sequences for BE in an interim fashion, and contingent upon results complete the remaining 2 sequences and analyze all 5 sequences at the conclusion?

Should read: "In a 5-way BE crossover trial, is there a methodology available that allows one to analyze 3 out of 5 periods for BE in an interim fashion, and contingent upon results complete the remaining 2 periods and analyze all 5 periods at the conclusion?"

This is an interesting idea, one that has not been explored much yet in the public domain; I think a few other sponsors have used it.
It is definitely doable as a single protocol. It is in fact what other companies / sponsors are achieving with two separate studies. Just do it, I think it may not require much operational adjustment or statistical fiddling, unless you can say you will switch to another formulation and re-start another 5-period trial once the first formulation fails after 3 periods.

A great post, in my opinion.

Pass or fail!
ElMaestro
Helmut
★★★

Vienna, Austria,
2020-07-01 11:14
(564 d 19:16 ago)

@ ElMaestro
Posting: # 21630
Views: 2,994

## Fed (IBD) → fasting

Hi ElMaestro & Sveta,

» Should read: "In a 5-way BE crossover trial, is there a methodology available that allows one to analyze 3 out of 5 periods for BE in an interim fashion, and contingent upon results complete the remaining 2 periods and analyze all 5 periods at the conclusion?"
»
» It is definitely doable as a single protocol. […] I think it may not require much operational adjustment or statistical fiddling, …

@ElMaestro: Operational – no. Statistical – likely yes.
The current regulatory thinking expressed at numerous conferences (nothing published) is that one still has to adjust α because in the first part one gets two chances to demonstrate BE (see this presentation, slide 23). I belonged to the first church (90% CI) for decades though now I’m a convert. I would not try to find an ‘optimal’ 1 adjusted α.

I regolatori stanno bene con Signor Bonferroni.

@Sveta: For most drugs it is more difficult to demonstrate BE in fed state (true food-drug interaction, higher variability) than in fasted state. Consider to switch your approach (fed followed by fast­ed). I haven’t seen a single case where the fed study passed and the fasted one failed, but a lot of cases the other way ’round – which required reformulation. Unfortunately many companies start with the fasted study (hey, that’s standard) only to be hit in fed state.
I recommend also to evaluate the first part according to the “Two at a Time” principle (two separate ana­lyses 2 as Incomplete Block Designs, i.e., T1 vs R and T2 vs R) and not “All at Once” (by one – pooled – ANOVA). 3 For details and references see the vignette of PowerTOST.
Note that the EMA’s BE-GL states:

In studies with more than two treatment arms (e.g. […] a four period study including test and reference in fed and fasted states), 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)

1. One would have to formalize the decision process in the selection of T1 or T2: GMR closer to one; if similar, the one with lower variability, etc. IMHO, not worth the efforts, since the average gain in sample sizes even for an optimistic α 0.0304 over Bonferroni’s 0.025 is just ~5%. Add the given reluctance of assessors towards simulation-based methods…

2. Since you will have only two treatments in each analysis, estimate the sample size for a 2×2×2 design and not for 3×3 Latin Squares. Requires sometimes slightly higher sample sizes.
library(PowerTOST) CV     <- seq(0.15, 0.3, 0.01) # Intra-subject CV theta0 <- 0.95                  # Assumed T/R-ratio target <- 0.80                  # Target (desired) power alpha0 <- 0.05                  # Nominal level k      <- 2                     # Number of tests alpha  <- alpha0/k              # Bonferroni-adjustment res    <- data.frame(CV = CV,                      design.3 = "3x3",   n.3 = NA, power.3 = NA,                      design.2 = "2x2x2", n.2 = NA, power.2 = NA) for (j in 1:nrow(res)) {   res[j, 3:4] <- signif(sampleN.TOST(alpha = alpha, CV = res$CV[j], theta0 = theta0, targetpower = target, design = res$design.3[j],                                      details = FALSE,  print = FALSE)[7:8], 3)   res[j, 6:7] <- signif(sampleN.TOST(alpha = alpha, CV = res$CV[j], theta0 = theta0, targetpower = target, design = res$design.2[j],                                      details = FALSE,  print = FALSE)[7:8], 3) } res$change <- sprintf("%+4.2f", 100*(res[, 6] - res[, 3])/res[, 3]) res$change[res\$change == "+0.00"] <- "±0.00" names(res)[2:6] <- rep(c("design", "n", "power"), 2) txt    <- paste0("Assumed \u03B8 ", theta0, ", target (desired) power ", target) if (alpha != 0.05) {   txt <- paste0(txt, ", adjusted \u03B1 ", alpha, " (", 100*(1-2*alpha), "% CI), ") } else {   txt <- paste0(txt, ", \u03B1 0.05 (conventional 90% CI), ") } txt    <- paste0(txt, "TIE \u2264", signif(1-(1-alpha)^k, 5), "\n") cat(txt); print(res, row.names = FALSE)

Peanuts:
Assumed θ 0.95, target (desired) power 0.8, adjusted α 0.025 (95% CI), TIE ≤0.049375    CV design  n power design  n power change  0.15    3x3 15 0.857  2x2x2 16 0.855  +6.67  0.16    3x3 15 0.808  2x2x2 16 0.806  +6.67  0.17    3x3 18 0.839  2x2x2 18 0.813  ±0.00  0.18    3x3 21 0.858  2x2x2 20 0.816  -4.76  0.19    3x3 21 0.817  2x2x2 22 0.817  +4.76  0.20    3x3 24 0.833  2x2x2 24 0.815  ±0.00  0.21    3x3 27 0.844  2x2x2 26 0.812  -3.70  0.22    3x3 27 0.808  2x2x2 28 0.808  +3.70  0.23    3x3 30 0.817  2x2x2 30 0.802  ±0.00  0.24    3x3 33 0.824  2x2x2 34 0.823  +3.03  0.25    3x3 36 0.828  2x2x2 36 0.816  ±0.00  0.26    3x3 39 0.830  2x2x2 38 0.808  -2.56  0.27    3x3 39 0.801  2x2x2 40 0.801  +2.56  0.28    3x3 42 0.803  2x2x2 44 0.813  +4.76  0.29    3x3 45 0.805  2x2x2 46 0.805  +2.22  0.30    3x3 48 0.805  2x2x2 50 0.814  +4.17

3. In the ANOVA you get only one – pooled – residual variance. Apart from problems with potentially biased estimates and inflated TIE, you could base your decision only on the T/R-ratios. If they are similar, which one will you select? Flip a coin? In the IBD-analyses you get two variance estimates, which in such a case would be helpful.

Dif-tor heh smusma 🖖
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
sweiner
☆

2020-07-07 03:00
(559 d 03:30 ago)

@ Helmut
Posting: # 21654
Views: 2,668

## Fed (IBD) → fasting

Helmut, thank you very much for the reply related to the adjustment of alpha, testing of fed vs. fasted, as well as the detailed sample size discussion with R code provided.

Can you please also provide feedback on one more scenario:
6-way crossover, with the following treatments A) tablet formulation #1 fasted, B) tablet formulation #2 fasted, C) capsule formulation fasted, D) tablet formulation #1 fed, E) tablet formulation #2 fed, F) capsule formulation fed.

Is there an approach that can be used to test only the first 3 treatments (all fasted) in an interim fashion, and contingent upon the results continue to test the final 3 treatments (all fed)? We are assuming a typical Latin Square such as:

A B C D E F
B C D E F A
C D E F A B
D E F A B C
E F A B C D
F A B C D E

To allow the proposed interim analysis (all fasted), is it possible to execute only the first necessary part of the Latin Square, such as:

A B C x x x
B C x x x A
C x x x A B
x x x A B C
x x A B C x
x A B C x x

With completion of the remaining parts (all fed) after the interim analysis:

x x x D E F
x x D E F x
x D E F x x
D E F x x x
x x A B C x
F x x x D E

Edit: Full quote removed. Please delete everything from the text of the original poster which is not necessary in understanding your answer; see also this post #5! [Ohlbe]