Bioequivalence and Bioavailability Forum • Partial replicate design

Silva
☆

Portugal,
2013-12-31 13:17
(4537 d 09:20 ago)

Posting: # 12118
Views: 7,517

Partial replicate design [RSABE / ABEL]

Dear Helmut
This is my first post, although I've been an enthusiastic reader of the forum and a student with your presentations. Congratulations for this site/forum, which I think is very usefull for BE community.

A question that I've been tried to understand is that in a partial replicate design (for EU Scaled Average BE), is it possible to use 2 sequence (RTR and TRR, for example) or the study has to be design in a 3 sequence (RTR, TRR and RRT, according to FDA's progesterone example or according to data set on EMA Q&A document)? Does Phoenix enable a correct estimation of within subject variability and the calculation of the 90% CIs with this study design, or only SAS is possible to be used?

Best regards and wish of a pleasant 2014
Silva

Helmut
★★★

Vienna, Austria,
2013-12-31 15:28
(4537 d 07:09 ago)

@ Silva
Posting: # 12119
Views: 6,598

extra-reference design

Post reply

Hi Silva,

THX for the :flower:

❝ A question that I've been tried to understand is that in a partial replicate design (for EU Scaled Average BE), is it possible to use 2 sequence (RTR and TRR, for example)

Well, that’s the so-called “extra-reference design” (a conventional 2×2 cross-over with an additional period where only R is administered). There are some publications (dealing with PBE/IBE) and it is covered in Chow/Liu – I guess. See also this post. I don’t like it because T is never administered in period 3; personally I haven’t seen one in the context of RSABE yet. If you want to opt for three periods, I would suggest a fully replicated design instead: RTR|TRT (we get not only CV_WR, but CV_WT as well – nice to know).

❝ or the study has to be design in a 3 sequence (RTR, TRR and RRT, according to FDA's progesterone example or according to data set on EMA Q&A document)?

EMA doesn’t recommend a specific design in the GL (3- or 4-period replicates are acceptable). IMHO, the codes in the Q&A are just examples.

❝ Does Phoenix enable a correct estimation of within subject variability and the calculation of the 90% CIs with this study design, or only SAS is possible to be used?

We have cross-validated RTRT|TRTR and RTR|TRR|RRT in Phoenix/WinNonlin against SAS. See my whitepaper. Recently we showed how to use templates, which is much more comfortable.* Download the examples for PHX/WNL v6.3 here (sorry, 20.9MB…).
When it comes to RTR|TRR: Like the partial replicate it might be problematic, since T is not replicated. However, since EMA’s model is “crippled” you would not face problems like with FDA’s mixed-effects model (if CV_WR <30%). You can use the code in the Q&A document without modifications (both in SAS and PHX).

Henry A, Schütz H, Hughes L, Davis S. Performing Reference-Scaled Average Bioequivalence (RSABE) in Phoenix^® WinNonlin^®. AAPS Annual Meeting (November 12, 2013), Poster: T2350.
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Helmut Schütz

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BEQool
★

2025-12-22 19:27
(163 d 03:11 ago)

@ Helmut
Posting: # 24535
Views: 1,834

extra-reference design

Post reply

Hello all,

what is the correct ANOVA model for extra-reference design?

Lets have a standard 2×2×2 crossover study that is extended by adding a third period in which all subjects receive the reference formulation only (i.e., extra-reference data in Period 3). If I analyze the data using the conventional ANOVA model with the four fixed effects (subject, sequence, period, and treatment), I obtain exactly the same PE as in the original 2×2×2 design. This remains true even when I simulate the reference data in Period 3 to be significantly lower (e.g., approximately half of the values observed in Periods 1 and 2).

This seems unreasonable, because the additional reference data contain some additional information about reference treatment and should therefore probably influence the PE? :confused:

So my question would be what is the correct statistical ANOVA model for an extra-reference design?

Best regards
BEQool

mittyri
★★

Russia,
2025-12-23 12:58
(162 d 09:39 ago)

(edited on 2025-12-23 15:33)
@ BEQool
Posting: # 24537
Views: 1,836

extra-reference design: R code

Post reply

Dear BEQool!

❝ This seems unreasonable, because the additional reference data contain some additional information about reference treatment and should therefore probably influence the PE? :confused: So my question would be what is the correct statistical ANOVA model for an extra-reference design?

The model is correct. See explanations in the code below

library(PowerTOST)

library(emmeans)



set.seed(1234)



seqs  <- c("TRR", "RTR")

nseq  <- c(12, 12)

ldiff <- log(0.95)

s2wT  <- 0.2^2

s2wR  <- 0.1^2



d0 <- PowerTOST:::prep_data2(

  seqs = seqs,

  nseq = nseq,

  muR  = log(10),

  ldiff = ldiff,

  s2wT  = s2wT,

  s2wR  = s2wR

)



d0$subject  <- factor(d0$subject)

d0$sequence <- factor(d0$sequence)

d0$period   <- factor(d0$period)

d0$tmt      <- factor(d0$tmt, levels = c("R", "T"))



# Say you want period 3 R values about 50% lower

eff.per <- c("1" = 0,

             "2" = 0,

             "3" = -log(2)) 



d0$logCmax <- d0$logval + eff.per[d0$period]


sd.subj <- 0.3 # BSW



n.subj <- nlevels(d0$subject)

b.subj <- rnorm(n.subj, mean = 0, sd = sd.subj)



d0$logCmax <- d0$logCmax + b.subj[d0$subject]


d0$Cmax <- exp(d0$logCmax)



# in period 3, the model can only estimate the sum of Period3 and TreatmentR

m.full <- lm(logCmax ~ sequence + subject %in% sequence + period + tmt, data = d0)

# There is no information at all to distinguish Period and Reference

# (no T to distinguish)

# due to residual DF changes F for trt effects is also changed

anova(m.full)

# Analysis of Variance Table

# 

# Response: logCmax

# Df Sum Sq Mean Sq  F value    Pr(>F)    

# sequence          1 0.0010  0.0010   0.0345    0.8534    

# period            2 8.6272  4.3136 146.8938 < 2.2e-16 ***

#   tmt               1 0.0673  0.0673   2.2932    0.1369    

# sequence:subject 22 7.1322  0.3242  11.0398 1.124e-11 ***

#   Residuals        45 1.3215  0.0294                       

# ---

#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1



emm.full <- emmeans(m.full, ~ tmt)

# The least-squares estimate of the treatment contrast (T–R) uses only periods 1 and 2

# Any change in period 3’s reference mean simply changes the period 3 effect, not the treatment effect

# Therefore the PE (T/R ratio) is exactly the same as in the pure 2-period 2×2 core:

PE.full  <- summary(contrast(emm.full, "revpairwise"), infer = TRUE)

PE.full

# contrast estimate     SE df lower.CL upper.CL t.ratio p.value

# T - R     -0.0749 0.0495 45   -0.175   0.0247  -1.514  0.1369

# 

# Results are averaged over the levels of: subject, period, sequence 

# Confidence level used: 0.95 



emm.period.full <- emmeans(m.full, ~ period)

summary(contrast(emm.period.full, "revpairwise"), infer = TRUE)

# contrast          estimate     SE df lower.CL upper.CL t.ratio p.value

# period2 - period1 -0.00261 0.0495 45   -0.123    0.117  -0.053  0.9985

# period3 - period1 -0.77306 0.0553 45   -0.907   -0.639 -13.978 <0.0001

# period3 - period2 -0.77045 0.0553 45   -0.904   -0.636 -13.930 <0.0001

# 

# Results are averaged over the levels of: subject, tmt, sequence 

# Confidence level used: 0.95 

# Conf-level adjustment: tukey method for comparing a family of 3 estimates 

# P value adjustment: tukey method for comparing a family of 3 estimates

# Periods comparison:

dtwo <- d0[d0$period != 3,]


m.two <- lm(logCmax ~ sequence + subject %in% sequence + period + tmt, data = dtwo)

anova(m.two)

# Analysis of Variance Table

# 

# Response: logCmax

# Df Sum Sq  Mean Sq F value    Pr(>F)    

# sequence          1 0.0040 0.004008  0.1146    0.7382    

# period            1 0.0001 0.000082  0.0023    0.9619    

# tmt               1 0.0673 0.067342  1.9249    0.1792    

# sequence:subject 22 5.1212 0.232782  6.6538 1.877e-05 ***

#   Residuals        22 0.7697 0.034985                      

# ---

#   Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

emm.two <- emmeans(m.two, ~ tmt)

PE.two  <- summary(contrast(emm.two, "revpairwise"), infer = TRUE)

PE.two

# contrast estimate    SE df lower.CL upper.CL t.ratio p.value

# T - R     -0.0749 0.054 22   -0.187   0.0371  -1.387  0.1792

# 

# Results are averaged over the levels of: subject, period, sequence 

# Confidence level used: 0.95 



# just for reference, period comparison:

emm.period.two <- emmeans(m.two, ~ period)

summary(contrast(emm.period.two, "revpairwise"), infer = TRUE)

# contrast          estimate    SE df lower.CL upper.CL t.ratio p.value

# period2 - period1 -0.00261 0.054 22   -0.115    0.109  -0.048  0.9619

# 

# Results are averaged over the levels of: subject, tmt, sequence 

# Confidence level used: 0.95 



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#  Merry Christmas!

—
Kind regards,
Mittyri

BEQool
★

2025-12-29 15:17
(156 d 07:20 ago)

@ mittyri
Posting: # 24541
Views: 1,688

extra-reference design: R code

Post reply

Dear Mittyri!

Thank you very much for the clear explanation and example, this is a wonderful Christmas present :-D

❝ # in period 3, the model can only estimate the sum of Period3 and TreatmentR

❝ m.full <- lm(logCmax ~ sequence + subject %in% sequence + period + tmt, data = d0)

❝ # There is no information at all to distinguish Period and Reference

❝ # (no T to distinguish)

❝ # due to residual DF changes F for trt effects is also changed

❝ # The least-squares estimate of the treatment contrast (T–R) uses only periods 1 and 2

❝ # Any change in period 3’s reference mean simply changes the period 3 effect, not the treatment effect

❝ # Therefore the PE (T/R ratio) is exactly the same as in the pure 2-period 2×2 core:

So the values of reference treatment in period 3 are irrelevant for estimating the treatment effect (and therefore for calculating PE and 90% CI)? If I understand correctly, the model assumes that these "lower" values of reference treatment in period 3 are a " consequence" of the period (period effect) rather than the reference product itself? Consequently, the model assumes that if. e.g. the values of reference treatment are about 50% lower in period 3, the values of test treatment would also be about 50% lower if administered in the same period 3? Therefore, the PE remains unchanged?
What does one then even gain with this design (with additional 3rd period) - I assume this design is more powerful than 2x2x2 design and therefore the 90% CIs should be narrower compared to 2x2x2?

Additional question; for this extra-reference design I have seen statements such as
- "Since the Test is not administered in all periods, lacking period effects must be assumed" [link],
- "The three period partial replicate design with two sequences TRR|RTR (a.k.a. extra-reference design) should be avoided because it is biased in the presence of period effects" [link] and
- "[...] a true period effect may give a biased treatment estimate in the extra-reference design"[link].
I am a little confused about what this means exactly. Here in our example we have period effect - does that mean that the calculated PE is biased?
Or is PE (treatment effect) biased only if we exclude factor period (i.e. using ANOVA model with treatment, sequence and subject(sequence)) in the presence of period effect (as is the case here)? Namely if we exclude factor period from the ANOVA model, I think it would then also take into account reference data in period 3 for the calculation of treatment effect (PE) and the results would differ from that based on the first 2 periods?
Therefore, would ANOVA model without factor period (i.e. ANOVA model of treatment, sequence and subject(sequence) also suffice if there is no period effect?

Merry Christmas to everyone!
BEQool

mittyri
★★

Russia,
2025-12-29 23:16
(155 d 23:21 ago)

@ BEQool
Posting: # 24542
Views: 1,636

extra-reference design: no period effect assumption

Post reply

Dear BEQool!

❝ So the values of reference treatment in period 3 are irrelevant for estimating the treatment effect (and therefore for calculating PE and 90% CI)?

Yes - for PE under the standard fixed effects model with period (any systematic shift in period 3, R is absorbed by the period 3 effect)
No - for CI (see residual MS and residual df)

❝ If I understand correctly, the model assumes that these "lower" values of reference treatment in period 3 are a " consequence" of the period (period effect) rather than the reference product itself? Consequently, the model assumes that if. e.g. the values of reference treatment are about 50% lower in period 3, the values of test treatment would also be about 50% lower if administered in the same period 3? Therefore, the PE remains unchanged?

Yes, if period 3 R values are lower and period 3 contains only R, then the model can only represent that systematic shift through the period 3 coefficient (if period term is included in the model, see below)

❝ What does one then even gain with this design (with additional 3rd period) - I assume this design is more powerful than 2x2x2 design and therefore the 90% CIs should be narrower compared to 2x2x2?

Some CI tightening via more df/different MSE, but that’s typically not the reason to choose the design. It allows reference-scaled BE - that's the only real gain we can guess. Although more affordable designs exist (TRT|RTR, TRR|RRT|RTR)

❝ I am a little confused about what this means exactly. Here in our example we have period effect - does that mean that the calculated PE is biased?

With the model including period, treatment estimate is not biased in the way you are worried about, because the period 3 shift is absorbed by the period term, and the contrast is identified from where both treatments occur.
The bias warnings mainly target analysis without period term included (i.e. assuming no period effect). So there's a trade-off: include period term and get PE driven by periods 1-2 or omit period term and assume no period effect at all.

❝ Or is PE (treatment effect) biased only if we exclude factor period (i.e. using ANOVA model with treatment, sequence and subject(sequence)) in the presence of period effect (as is the case here)? Namely if we exclude factor period from the ANOVA model, I think it would then also take into account reference data in period 3 for the calculation of treatment effect (PE) and the results would differ from that based on the first 2 periods? Therefore, would ANOVA model without factor period (i.e. ANOVA model of treatment, sequence and subject(sequence) also suffice if there is no period effect?

Correct. But no period effect is a strong assumption you typically cannot justify. I guess this is the reason why extra-reference design is widely discouraged

—
Kind regards,
Mittyri

BEQool ★ 2025-12-31 17:42 (154 d 04:55 ago) @ mittyri Posting: # 24546 Views: 1,570	extra-reference design: no period effect assumption Post reply
	Hey Mittyri, ❝ Yes - for PE under the standard fixed effects model with period (any systematic shift in period 3, R is absorbed by the period 3 effect) ❝ No - for CI (see residual MS and residual df) Yes of course CI is affected, I wanted to say only PE - thanks! Thank you for all your answers and explanations, everything is clear now All the best for the new year BEQool