acfalcao
●

2007-09-16 22:34
(5126 d 05:25 ago)

Posting: # 1082
Views: 26,613

## Three-way crossover BABE Studies [Design Issues]

Dear All

My name is Amilcar Falcão and I am professor at the University of Coimbra, Portugal. I am looking for help regarding the execution and interpretation of three-way cross-over bioequivalence trials (for academic purposes).

I would like to have an example (explicit) where a three-way crossover study is appropriately analysed. Namely, I would like to know what kind of ANOVA should be performed and how the 90%CI should be calculated for the different combinations of study formulations (i.e. A vs B and A vs C). Perhaps this is a naive question, but in fact I am trying to look for it in literature and I have seen different approaches always without the adopted statistical pathway. Therefore, I would like to be sure regarding the correct way to deal with.

Best regards; Amílcar Falcão
Laboratory of Pharmacology
Faculty of Pharmacy
University of Coimbra
Coimbra - Portugal
Helmut
★★★  Vienna, Austria,
2007-09-17 20:47
(5125 d 07:13 ago)

@ acfalcao
Posting: # 1088
Views: 24,699

## Three-way crossover example data set

Dear Amilcar!

» I would like to have an example (explicit) where a three-way crossover study is appropriately analysed.

You have to use a Williams’ design (three period, six sequences); the topic was covered in previous threads.

A 6×3 design is needed in order to ‘extract’ two 2×2 tables, which are also balanced. Although the full 6×3 table will be used in the analysis of AUC and Cmax, you will need these 2×2s for the nonparametric analysis of tmax (un­fortunately there’s no confidence interval based nonparametric method available for more than two formulations/periods). The asterisks * denote pseudo-sequences and pseudo-periods, e.g. P1* means only that the treatment was given in a period prior to P2* – irrespective of the true study period:
+----+------------+  -->  +----+--------+  and  +----+--------+ |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*| +----+------------+       +----+--------+       +----+--------+ | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 | | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  | | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  | | S4 | T   R2  R1 |       | S1*| T   R1 |       | S1*| T   R2 | | S5 | R1  T   R2 |       | S2*| R1  T  |       | S1*| T   R2 | | S6 | R2  R1  T  |       | S2*| R1  T  |       | S2*| R2  T  | +----+------------+       +----+--------+       +----+--------+                             ^   balanced          ^   balanced
A common mistake is to design the study as a set of 3×3 latin squares, which will lead (especially if the sample size is small and in the case of drop outs) to extremely imbalanced data sets:
+----+------------+  -->  +----+--------+  and  +----+--------+ |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*| +----+------------+       +----+--------+       +----+--------+ | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 | | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  | | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  | +----+------------+       +----+--------+       +----+--------+                             ^ imbalanced          ^ imbalanced

» Namely, I would like to know what kind of ANOVA should be performed and how the 90%CI should be calculated for the different combinations of study formulations (i.e. A vs B and A vs C).

For the design see Chapter 10 of

Chow S-C, Liu J-p.
Design and Analysis of Bioavailability and Bioequivalence Studies. New York: Marcel Dekker; 2nd ed. 2001, p. 302–32.

For a detailed discussion of variance balanced designs see Chapter 4 of

Jones B, Kenward MG.
Design and Analysis of Cross-over Trials. Boca Raton: Chapman & Hall/CRC; 2nd ed. 2003, p. 151–204.

For an example data set see Chapter 4 of

Patterson S, Jones B.
Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: Chapman & Hall/CRC; 2006, p. 79–132.

Their example 4.5 matches your design (one test formulation is compared to two reference formulations)
You may download zipped datasets and programs (SAS/S+) from CRC’s website. If you don’t have access to commercial software, S+ code will run with open-source R with minor modifications.

Patterson/Jones give results in Table 4.12 (p. 105) with:
T/R (% Test vs. Reference 1)
+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      | 116.2 | 109.0 , 123.9 | | Cmax     | 130.0 | 119.1 , 141.8 | +----------+-------+---------------+
T/S (% Test vs. Reference 2)
+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      |  82.8 |  77.6 ,  88.3 | | Cmax     |  81.5 |  74.7 ,  89.0 | +----------+-------+---------------+

WinNonlin 5.2 comes up with:
T/R (% Test vs. Reference 1)
+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      | 116.2 | 109.0 , 123.8 | | Cmax     | 129.7 | 118.4 , 141.5 | +----------+-------+---------------+
T/S (% Test vs. Reference 2)
+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      |  82.6 |  77.5 ,  88.1 | | Cmax     |  81.2 |  74.3 ,  88.6 | +----------+-------+---------------+

Results are slightly different (although both WinNonlin and SAS use GLM – not ANOVA; implementation, rounding, etc. is different – and of course ‘proprietary information’ and not documented). I assume Patterson/Jones' results were produced by SAS; I will check the results from their S+ code the next days.

» Perhaps this is a naive question,...

Not at all; little is published - and no worked examples at all.

There’s another point which is a little bit tricky: multiplicity.
If you are testing one test (A) against two references (B, C), any impression of ‘data dredging’ must be avoided, e.g., calculation 90% CIs of A/B and A/C - and only picking out the best getting an approval.
Since in the EU the Innovator's product from any European country may be used as the reference, you may run into problems (A vs B is BE, whereas A vs C is not BE). It may be wise to use 95% CIs instead (overall Bonferroni-corrected patient’s risk: α = 1–(1–0.05/k)k, where k is the number of simultaneous comparisons).
95% CI should also be applied in testing for dose proportionality of three dose levels (or 96.67% for four levels).
IMHO the only case where 90% CIs should be used is the comparison of two test formulations against one reference, and only one of them will be further used in the approval process.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
acfalcao
●

2007-09-17 23:02
(5125 d 04:58 ago)

@ Helmut
Posting: # 1089
Views: 23,652

## Three-way crossover example data set

Dear Helmut

Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future.

Best regards;

Amílcar Helmut
★★★  Vienna, Austria,
2007-09-18 12:47
(5124 d 15:12 ago)

@ acfalcao
Posting: # 1092
Views: 23,685

## Three-way crossover example data set

Dear Amilcar!

» Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future.

Please try it again (it worked yesterday, and today as well): CRC’s website
Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find:
Example4.5.sas (the SAS code)
Example45.SSC (the S+ code)
exam45.dat (the raw data for S+)

As I get from the SAS-code a mixed model (degrees of freedom according to Kenward) was used, whereas WinNonlin uses Satterthwaite's method.
The S+ code does not calculate BE, but is used for the example’s figures only.

Edit: Wrong link corrected in both posts. [Helmut]

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Nirali
★

India,
2007-09-21 06:49
(5121 d 21:10 ago)

@ Helmut
Posting: # 1106
Views: 23,553

## Three-way crossover (WinNonlin)

Dear Helmut,

Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)?
Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"?
In threeway crossover study, does we try to compare Equivalence of all three product together?

Thanks & Regards,
Nirali
Helmut
★★★  Vienna, Austria,
2007-09-21 13:04
(5121 d 14:55 ago)

@ Nirali
Posting: # 1109
Views: 23,565

## Three-way crossover (WinNonlin)

Dear Nirali,

» Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)?
» Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"?
» In threeway crossover study, does we try to compare Equivalence of all three product together?

Comparisons are based on paired differences; IMHO a simultaneous test of T1=T2=R is not possible.
WinNonlin always uses one reference in a run, i.e., T1=R (one test), and T2=R (the second test).
If you are interested in T1=T2 you have to start a second run (defining T2 as the reference), and will obtain two tests (T1=T2 and R=T2) within.

BTW, why are you interested in T1=T2?

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Nirali
★

India,
2007-09-25 08:05
(5117 d 19:54 ago)

@ Helmut
Posting: # 1127
Views: 23,487

## Three-way crossover (WinNonlin)

Dear Helmut,

Actully, I was under impression that WinNonlin is using the hypo:
"MEAN(Test1)=MEAN(Test2)=MEAN(Ref)".

(A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo:
(1) R1=T and (2) R1=R2.

same way,
(B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2

pls do correnct me if i am wrong to understand.

Thanks & Regards,
Nirali
Helmut
★★★  Vienna, Austria,
2007-09-25 13:15
(5117 d 14:45 ago)

@ Nirali
Posting: # 1132
Views: 23,627

## Three-way crossover (WinNonlin)

Dear Nirali!

» Actully, I was under impression that WinNonlin is using the hypo:
» "MEAN(Test1)=MEAN(Test2)=MEAN(Ref)".

Again, we obtain only pairwise comparisons, not simultaneous ones.

» (A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo:
» (1) R1=T and (2) R1=R2.

Not quite, but very close. If you run the BE wizard (R1=reference), you will get
(1a) T vs. R1, and
(1b) R2 vs. R1.
You have to run the BE wizard again (this time with R2 as reference), getting
(2a) T vs. R2, and
(2b) R1 vs. R2.
In order to get comparisons of T=R1 and T=R2, you have to run the wizard twice (results 1a and 2a), omitting 1b/2b. Of course PE and CL of 2b=1/1b.

» same way,
» (B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2

Yes this is correct; in this case only one run of the BE wizard is necessary, getting:
(1a) T1 vs. R, and
(1b) T2 vs. R.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Irene_I
☆

Indonesia,
2018-06-07 11:09
(1209 d 16:51 ago)

@ Helmut
Posting: # 18861
Views: 15,726

## Three-way crossover example data set

Dear Helmut,

» Please try it again (it worked yesterday, and today as well): CRC's website
» Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find:
» Example4.5.sas (the SAS code)
» Example45.SSC (the S+ code)
» exam45.dat (the raw data for S+)

I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this. Thank you. Best Regards,

Irene I
Helmut
★★★  Vienna, Austria,
2018-06-07 13:04
(1209 d 14:56 ago)

@ Irene_I
Posting: # 18862
Views: 15,794

## Three-way crossover example data set

Hi Irene,

» I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this.

The files of the 2005 edition are gone. However, here is the link for the 2016 edition.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Irene_I
☆

Indonesia,
2018-06-08 11:09
(1208 d 16:51 ago)

@ Helmut
Posting: # 18869
Views: 15,664

## Three-way crossover example data set

Dear Helmut,

Thanks a lot! It is really a great help to me.. Regards,

Irene I
Irene_I
☆

Indonesia,
2018-06-12 09:21
(1204 d 18:39 ago)

@ Helmut
Posting: # 18886
Views: 15,614

## Three-way crossover example data set

Dear Helmut,

After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intra-subject CV from this value, right? So, My question is, Do we only get one Intra-subject CV from three-comparison of the treatment (T-R, S-T, S-R) while we got three CI (T-R, S-T, S-R)?

I am looking forward for your help.
Thank you very much.

Best Regards,

Irene I
Helmut
★★★  Vienna, Austria,
2018-06-12 12:42
(1204 d 15:17 ago)

@ Irene_I
Posting: # 18888
Views: 16,080

## Leave-One-Out (IBD)

Hi Irene,

» After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intra-subject CV from this value, right? So, My question is, Do we only get one Intra-subject CV from three-comparison of the treatment (T-R, S-T, S-R) while we got three CI (T-R, S-T, S-R)?

I’m not equipped with … Hence, my understanding of the code is limited.
You are right, in the model you get only one (pooled) estimate of the variance. That’s not a good idea, since it will “work” only if intra-subject variances would be identical and the treatment differences at least very similar. Otherwise, the treatment estimates will be biased and the type I error is not controlled. See also this post.
At the 2nd Workshop of the Global Bioequivalence Harmonisation Initiative (Rockville, September 2016) Pina D’Angelo gave a presentation “Testing for Bioequivalence in Higher‐Order Crossover Designs: Two‐at‐a‐Time Principle Versus Pooled ANOVA” showing exactly that. Here some of her slides:

Purpose

• To determine which method of statistical analysis
is more appropriate to conclude bioequivalence in
higher‐order crossover studies:
1. The two‐at‐a‐time principle using two separate
incomplete block design ANOVAs
2. A pooled approach using one ANOVA and a common
error term for the two contrasts

Introduction

Statistical Concerns:

1. Different means (point estimates) between formulations
2. Different variances between formulations
• If either situations exist, which method of analysis
reduces bias the most: two‐at‐a‐time principle or
pooled ANOVA?

Simulated Data: Summary

• When all three treatments have similar means and there is
homogeneity of variances, both methods give very similar
results.
• When treatment means differ but there is homogeneity of
variances, both methods give very similar results. With higher
variability, the power is slightly increased when using the pooled
ANOVA method.
• When treatment means are similar but variances are not
homogeneous, the two‐at‐a‐time method gives higher power to
detect BE for the treatment with lower variability
• When treatment means differ and variances are not
homogeneous, the two‐at‐a‐time method increases power to
detect BE for the treatment with lower variability. Moreover,
type I error is higher when using the pooled ANOVA method.

Closing Remarks

• Using a two‐at‐a‐time principle for statistical analysis of a
higher‐order pilot study will have more value for decision-
making on which multiple tests lots will be selected for use
in a pivotal study based on the pilot study results. The
intra‐subject variability of a specific test‐to‐reference
comparison can be determined using the two‐at‐a‐time
principle, which may be an important factor in selecting a
test product considering test products are generally
formulated to show different characteristics for testing in a
pilot study.
(my emphasis)

If you are interested in all pairwise comparisons, generate three data sets (values or T&R, S&R, T&S, i.e., exclude all values of the respective other treatment S, T, R) whilst keeping the codes for sequence and period. This will give you data sets which represent an IBD (incomplete block design). Run the usual model on them.
This approach is also recommended in the EMA’s BE-GL:

In studies with more than two treatment arms (e.g. a three period study including two references, one from EU and another from USA […], 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)
Example 4.5 (Phoenix/WinNonlin 8, mixed effects, Satterthwaite’s degrees of freedom):
                           PE (%)    90% CI (%)       s²w    CVw (%) pooled model T vs. R  AUC  116.15 (108.97, 123.81)  0.043954  21.20              T vs. R  Cmax 129.65 (118.84, 141.45)  0.084569  29.71              S vs. R  AUC  140.63 (131.93, 149.90)  0.043954  21.20              S vs. R  Cmax 159.75 (146.42, 174.30)  0.084569  29.71              T vs. S  AUC   82.60 ( 77.46,  88.07)  0.043954  21.20              T vs. S  Cmax  81.16 ( 74.35,  88.56)  0.084569  29.71 IBD models   T vs. R  AUC  116.05 (108.92, 123.65)  0.042525  20.84              T vs. R  Cmax 129.54 (119.26, 140.71)  0.074744  27.86              S vs. R  AUC  141.09 (131.39, 151.51)  0.053706  23.49              S vs. R  Cmax 160.48 (145.20, 177.36)  0.109492  34.02              T vs. S  AUC   83.12 ( 78.37,  88.15)  0.035788  19.09              T vs. S  Cmax  81.63 ( 75.33,  88.46)  0.069372  26.80
Evidently variances are not identical. The ones of T/R are smaller than the ones of S/R. If we apply the pooled model the CIs of T/R will be wider than in the IBD model and the CIs of S/R narrower.

BTW, I don’t understand what the purpose of the carry variable in Byron’s code is.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Irene_I
☆

Indonesia,
2018-06-13 05:02
(1203 d 22:57 ago)

@ Helmut
Posting: # 18896
Views: 15,600

## Leave-One-Out (IBD)

Dear Helmut,

Thank you for your explanation. I am sure this will be helpful not only for me, but also others who looking for information about this topic too. Best Regards,

Irene I
Helmut
★★★  Vienna, Austria,
2018-06-13 15:02
(1203 d 12:58 ago)

@ Irene_I
Posting: # 18898
Views: 15,489

## Impact of pooled variance (bias, CI)

Hi Irene,

» Thank you for your explanation.

You are welcome. A picture tells more than a thousand words (AUC only).  The PE and its CI of T/R are similar. The small bias is caused by the fact that the average PE of S/R and T/S ($$\sqrt{141.09\times83.12}$$) is close to the T/R with 116.05. The other comparisons show negative biases. The CI of S/R is narrower in the pooled model and the CI of T/S is wider.
We don’t want to go there.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
d_labes
★★★

Berlin, Germany,
2018-06-13 15:31
(1203 d 12:28 ago)

@ Helmut
Posting: # 18899
Views: 15,517

## carry (over?)

Dear Helmut,

» BTW, I don’t understand what the purpose of the carry variable in Byron’s code is.

As far as I see it: It serves for nothing because it is not used except in the class statement of Proc Mixed.
The class statement serves the purpose of declaring factors even if a variable is numeric. Nothing more.

The carry variable seems to me a left over from old times where carry-over was an issue.

Regards,

Detlew
ElAlumno
☆

2019-03-14 23:40
(929 d 03:19 ago)

@ Helmut
Posting: # 20036
Views: 12,220

## Pseudo-periods

Dear Helmut,

In your old but still-useful post (excerpt below), you describe generating pseudo-sequences and pseudo-periods when assessing ABE.

» A 6×3 design is needed in order to ‘extract’ two 2×2 tables, which are also balanced. Although the full 6×3 table will be used in the analysis of AUC and Cmax, you will need these 2×2s for the nonparametric analysis of tmax (unfortunately there’s no confidence interval based nonparametric method available for more than two formulations/periods). The asterisks * denote pseudo-sequences and pseudo-periods, e.g. P1* means only that the treatment was given in a period prior to P2* – irrespective of the true study period:

Your post preceded the EMA's Guideline on the Investigation of Bioequivalence by a few years, but it appears to agree with their stipulation:

In studies with more than two treatment arms (e.g. a three period study including two references, one from EU and another from USA, or 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.

I completely understand the use of pseudo-sequences in this situation. But what about pseudo-periods? The EMA guidelines aren't too specific.

On the one hand, if you wish to display your data with pseudo-sequences, you would want to use pseudo-periods too (so that the timing of treatments is consistent within pseudo-sequence). On the other hand, the ANOVA doesn't care whether your pseudo-sequences are logically consistent with your period.

If there is no period effect, the difference between using period and pseudo-period will be minimal (use a few more df in the ANOVA table with period than pseudo-period). But what if there is a serious period effect? You would miss it (wrong sum of squares for period) by using pseudo-periods.

Opinions?
Helmut
★★★  Vienna, Austria,
2019-03-15 00:47
(929 d 02:13 ago)

@ ElAlumno
Posting: # 20037
Views: 12,135

## Pseudo-periods

Hi ElAlumno (nice nick!),

» In your old but still-useful post (excerpt below), you describe generating pseudo-sequences and pseudo-periods when assessing ABE.
»
» » A 6×3 design is needed in order to ‘extract’ two 2×2 tables, [ ]
»
» Your post preceded the EMA's Guideline on the Investigation of Bioequivalence by a few years, but it appears to agree with their stipulation: […]

Not quite.

» I completely understand the use of pseudo-sequences in this situation. But what about pseudo-periods? The EMA guidelines aren't too specific.

The EMA means the ‘Two‐at‐a‐Time Principle’ (see above) and not my recoding (which I abandoned a good while ago as well).
David Brown (MHRA, member the EMA’s Biostatistics Working Party) gave a clarification at the 3rd EGA Symposium on Bioequivalence (London, June 2010). The Q&A document states:

However, the treatment, groups, sequences and periods should have their original values maintained in the analysis, and not have the values modified. For example an observation made in period 3 should still be coded as period 3, not have the period changed to “2” because the results for that subject in one of the earlier periods has now be removed.

(my emphasis)

» On the one hand, if you wish to display your data with pseudo-sequences, you would want to use pseudo-periods too …

Nope. Relevant slides: GL Q Q A A

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
ElAlumno
☆

2019-03-22 21:59
(921 d 05:00 ago)

@ Helmut
Posting: # 20069
Views: 11,822

## Two‐at‐a‐Time analysis in R

Thank you Helmut for your quick and detailed response!

I'm glad to hear that pseudo-periods have gone the way of the dodo. I agree with the EMA Biostatistics Working Party that it makes more sense (statistically) to maintain the original codings when leaving out data from irrelevant treatments (not to mention being less prone to human error than recoding). Also, I love their name; I bet the EMA Biostatistics Working Party could form a coalition government with the Slightly Silly Party, the Surprise Party, and the Rent Is Too Damn High Party. But I digress.

I don't have WinNonlin, but I tried analyzing Patterson & Jones' Example 4.5 with the Two‐at‐a‐Time Principle using R 1.1.456. My results were similar to the results you posted above, but not as close as I would have expected. I will post the code & table below. Do you see anything I am doing wrong or any obvious explanation of the differences? I don't know how WinNonlin handles missing values, so I tried a few variations (subjects with complete cases in the full dataset, and subjects with complete cases in each of the 3 incomplete-block subsets), but the differences persisted.

 # 3-treatment design from Patterson & Jones (2017) dta = read.table("exam45.dat", header=T) dta[dta==99999] = NA dta$Subj = factor(dta$subject) dta$Per = factor(dta$period) dta$Seq = factor(dta$sequence) dta$Trt = factor(dta$formula) options(contrasts=c("contr.treatment","contr.poly"), digits=4) # Data frame to store results IBD = data.frame(Test = paste(rep(c("T vs. R", "S vs. R", "T vs. S"), each=2),                               rep(c(" AUC", "Cmax"), times=3)),                  PE = NA, LCI = NA, UCI = NA, CV = NA) # AUC T vs R muddle = lm(log(AUC)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="S",]) IBD[1, 2:5] = c(100*exp(coef(muddle)["TrtT"]), 100*exp(confint(muddle,c("TrtT"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # Cmax T vs R muddle = lm(log(CMAX)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="S",]) IBD[2, 2:5] = c(100*exp(coef(muddle)["TrtT"]), 100*exp(confint(muddle,c("TrtT"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # AUC S vs R muddle = lm(log(AUC)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="T",]) IBD[3, 2:5] = c(100*exp(coef(muddle)["TrtS"]), 100*exp(confint(muddle,c("TrtS"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # Cmax S vs R muddle = lm(log(CMAX)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="T",]) IBD[4, 2:5] = c(100*exp(coef(muddle)["TrtS"]), 100*exp(confint(muddle,c("TrtS"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # AUC T vs S muddle = lm(log(AUC)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="R",]) IBD[5, 2:5] = c(100*exp(coef(muddle)["TrtT"]), 100*exp(confint(muddle,c("TrtT"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # Cmax T vs S muddle = lm(log(CMAX)~Trt+Per+Seq+Subj, data=dta[dta$Trt!="R",]) IBD[6, 2:5] = c(100*exp(coef(muddle)["TrtT"]), 100*exp(confint(muddle,c("TrtT"), level=.9)), 100*sqrt(exp(summary(muddle)$sigma^2)-1)) # Print table print(IBD) 

This produced the following table:

           Test     PE    LCI    UCI    CV 1 T vs. R  AUC 116.14 108.98 123.78 20.88 2 T vs. R Cmax 129.82 119.50 141.04 27.87 3 S vs. R  AUC 140.73 131.02 151.15 23.50 4 S vs. R Cmax 160.04 144.76 176.94 34.06 5 T vs. S  AUC  83.51  78.73  88.58 19.07 6 T vs. S Cmax  82.30  75.96  89.18 26.67 
Helmut
★★★  Vienna, Austria,
2019-03-23 01:03
(921 d 01:57 ago)

@ ElAlumno
Posting: # 20070
Views: 11,902

## fixed & mixed (dammit!) and a request to SASians

Hi ElAlumno,

» I agree with the EMA Biostatistics Working Party […]. Also, I love their name; I bet the EMA Biostatistics Working Party could form a coalition government with the Slightly Silly Party, the Surprise Party, and the Rent Is Too Damn High Party. It is really existing though David Brown will be brexited (is this already a word?) unceremoniously.

» But I digress.

That’s fine. Adds some spice to that dry stuff.

» […] I tried analyzing Patterson & Jones' Example 4.5 with the Two‐at‐a‐Time Principle using R 1.1.456.

What? 1.1.456? 1.1.4 was the first one I used in 2002. Current is 3.5.3!

» My results were similar to the results you posted above, but not as close as I would have expected. I will post the code & table below. Do you see anything I am doing wrong or any obvious explanation of the differences? I don't know how WinNonlin handles missing values,…

That’s not the point. The devil is in the details. I wrote above:

» » » » Phoenix/WinNonlin 8, mixed effects

Why? Well, Patterson & Jones analyzed the data with SAS Proc MIXED and I wanted to reproduce it in PHX/WNL (restricted maximum likelihood). OK, they used in SAS ddfm=KENWARDROGER1 and yours truly Satterthwaite’s df.
When I switched in PHX/WNL to a fixed effects model (i.e., like lm() in R and SAS Proc GLM) I got exactly your results. Hence, your code is fine though you can streamline it (specify NAs and classes):

dta <- read.table("exam45.dat", header=TRUE, na.strings="99999",                   colClasses=c(rep("factor", 4), rep("numeric", 2)))

Checks:

sum(is.na(dta$AUC))  6 sum(is.na(dta$CMAX))  2 str(dta) 'data.frame':   186 obs. of  6 variables:  $subject : Factor w/ 62 levels "001","002","004",..: […]$ sequence: Factor w/ 6 levels "1","2","3","4",..: […]  $period : Factor w/ 3 levels "1","2","3": […]$ formula : Factor w/ 3 levels "R","S","T": […]  $AUC : num 4089 7411 5513 2077 3684 ...$ CMAX    : num  906 1711 1510 504 845 ...

I would not use option(..., digits=4) cause IMHO, this may affect also the precision of internal calculations. Better to round yourself at the end, f.i.

IBD[1, 2:5] <- round(c(foo, bar, baz), 2).

So what do we have (T/R of AUC)?
$$\small{\begin{array}{llccc}\hline \text{Method} & \text{Setup} & \text{PE} & \text{90% CI} & \text{CV (%)} \\ \hline \text{pooled} & \text{SAS Proc MIXED (KENWARDROGER)} & \text{116.2 } & \text{109.0 , 123.9 } & \text{21.3}^2 \\ & \text{SAS Proc MIXED (SATTERTHWAITE)} & 116.15 & \text{108.97, 123.81} & 21.20 \\ & \text{PHX/WNL REML (Satterthwaite)} & 116.15 & \text{108.97, 123.81} & 21.20 \\ & \text{R lmer() Satterthwaite} & 116.15 & \text{108.97, 123.81} & 21.20 \\ & \text{R lmer() Kenward-Roger} & 116.15 & \text{108.97, 123.81} & 21.20 \\\hline & \text{PHX/WNL LM (all fixed)} & 116.24 & \text{109.04, 123.92} & 21.22 \\ & \text{R lm()} & 116.24 & \text{109.04, 123.92} & 21.22 \\ \hline \text{IBD} & \text{SAS Proc MIXED (SATTERTHWAITE)} & 116.05 & \text{108.92, 123.65} & 20.84 \\ & \text{PHX/WNL REML (Satterthwaite)} & 116.05 & \text{108.92, 123.65} & 20.84 \\\hline & \text{R lmer() Satterthwaite} & 116.05 & \text{108.97, 123.58} & 20.84 \\ & \text{R lmer() Kenward-Roger} & 116.05 & \text{108.97, 123.58} & 20.84 \\\hline & \text{PHX/WNL LM (all fixed)} & 116.14 & \text{108.98, 123.78} & 20.88 \\ & \text{R lm()} & 116.14 & \text{108.98, 123.78} & 20.88 \\ \hline \end{array}}$$If Satterthwaite’s df are used, the mixed effects model in SAS and PHX/WNL agree. R lm() agrees with PHX/WNL’s fixed effects model. A little bit surprising to me that Proc MIXED with ddfm=KENWARDROGER3 gives the same result as a fixed effects model. OK, but the FDA recommends Satterthwaite’s df for ages.4 A mixed effects model may give a lower residual variance – like in this case – because information of incomplete data is recovered. R lmer() is set up according to the EMA’s ‘Method B’ which treats subject as a random effect but is not a true mixed effects model. Given only for completeness (for the code see below).
A request to the SASians of the forum: Please run Proc MIXED with DDFM=SATTERTHWAITE of the pooled data and both DDFM-options on the data where S is excluded. THX to mittyri!

Mixed effects in R are a swamp. The closest I could get was this:

library(nlme) muddle1 <- lme(log(AUC) ~ Trt + Per + Seq, data = dta[dta$Trt != "S", ], random = ~Trt-1|Subj, weights = varIdent(form = ~1|Trt), method = "REML", na.action = na.exclude, control = lmeControl(opt = "optim", msMaxIter = 200)) anova(muddle1, type="marginal") summary(muddle1) Extracting the required stuff is beyond me. We had many discussions already in the forum and the verdict was “mixed effects models for BE are not possible in R”. So if you are happy with a fixed effect model (many jurisdictions like the EMA), fine. If you have to deal with the FDA or Health Canada, maybe not. 1. Jones B, Kenward MG. Design and Analysis of Cross-Over Trials. Boca Raton: CRC Press; 3rd ed. 2015. 2. Limited precision of results in the log-domain of the reference, CV not given. Estimated: library(PowerTOST) round(100*CVfromCI(lower = 1.09, upper = 1.239, design = "3x6x3", n = c(8, 11, 11, 10, 10, 10)), 1) 3. SAS tells me: This method changes output in the following tables: Contrast, CorrB, CovB, Diffs, Estimates, InvCovB, LSMeans, Slices, SolutionF, SolutionR, Tests1–Tests3. The OUTP= and OUTPM= data sets are also affected. 4. FDA/CDER. Guidance for Industry. Statistical Approaches to Establishing Bioequivalence. January 2001. download. Dif-tor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes mittyri ★★ Russia, 2019-03-23 23:13 (920 d 03:46 ago) @ Helmut Posting: # 20073 Views: 11,601 ## Pooled vs IBD T-R in SAS from non-SASian Dear Helmut, » A request to the SASians of the forum: Please run Proc MIXED with DDFM=SATTERTHWAITE of the pooled data and both DDFM-options on the data where S is excluded. Here's my (naive) attempt <...> ODS output estimates = _pooled; proc mixed data=ex631 method=reml ITDETAILS maxiter=200; class sequence subject period formula carry; model ln_auc= sequence period formula/ddfm=SATTERTHWAITE outp=out; random subject(sequence); estimate 't T-R' formula -1 0 1/cl alpha=0.10;; run; Data _pooled; set _pooled; pe=round(100*exp(estimate), 0.00001); lowerCL = lower; upperCL = upper; lower=round(100*exp(lowerCL), 0.00001); upper=round(100*exp(upperCL), 0.00001); run; Data noS; set ex631; if formula = 'R' | formula = 'T'; run; ODS output estimates = _TRonly; proc mixed data=noS method=reml ITDETAILS maxiter=200; class sequence subject period formula carry; model ln_auc= sequence period formula/ddfm=SATTERTHWAITE outp=out; random subject(sequence); estimate 'T-R' formula -1 1/cl alpha=0.10;; run; Data _TRonly; set _TRonly; pe=round(100*exp(estimate), 0.00001); lowerCL = lower; upperCL = upper; lower=round(100*exp(lowerCL), 0.00001); upper=round(100*exp(upperCL), 0.00001); run; Results:  Option Label Estimate StdErr DF tValue Probt Alpha Lower Upper pe lowerCL upperCL Pooled "t T-R" 0.1497 0.03848 115 3.89 0.0002 0.1 108.97 123.81 116.153 0.0859308889 0.2135523619 IBD T-R 0.1488 0.03793 56.8 3.92 0.0002 0.1 108.92 123.65 116.047 0.0854035502 0.2122465728 Kind regards, Mittyri Helmut ★★★  Vienna, Austria, 2019-03-23 23:36 (920 d 03:23 ago) @ mittyri Posting: # 20074 Views: 11,582 ## Pooled vs IBD T-R in SAS from non-SASian Hi mittyri, » » A request to the SASians of the forum: Please run Proc MIXED with DDFM=SATTERTHWAITE of the pooled data and both DDFM-options on the data where S is excluded. » Here's my (naive) attempt THX a lot. I added your result to the comparison above. Dif-tor heh smusma 🖖 Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮 Science Quotes mittyri ★★ Russia, 2019-03-23 23:50 (920 d 03:09 ago) @ Helmut Posting: # 20076 Views: 11,448 ## mixed in R Dear Helmut, since you are very quick with answers even Saturday night, I'm not editing my previous post but commenting it here » Extracting the required stuff is beyond me. We had many discussions already in the forum and the verdict was “mixed effects models for BE are not possible in R”. Hmmm... If I remember correctly we were struggling with FDA model where sophisticated 'repeated' statement exists. We successfully crosschecked EMA method B (simple mixed effects model with a Subject as random effect). So linear mixed effects models are possible until FDA-style is required . Kind regards, Mittyri Helmut ★★★  Vienna, Austria, 2019-03-24 01:23 (920 d 01:36 ago) @ mittyri Posting: # 20077 Views: 11,589 ## mixed in R (EMA B ≠ FDA) Hi mittyri, » If I remember correctly we were struggling with FDA model where sophisticated 'repeated' statement exists. » We successfully crosschecked EMA method B (simple mixed effects model with a Subject as random effect). Are you reminding Detlew and me obout our TODO-list? » So linear mixed effects models are possible until FDA-style is required . Yes, but that’s the point. However, recycled our code: library(lmerTest) # (requires lme4, Matrix) dta <- read.table("exam45.dat", header=TRUE, na.strings="99999", colClasses=c(rep("factor", 4), rep("numeric", 2))) names(dta) <- "treatment" TR.only <- dta[dta$treatment != "S", ] ci            <- data.frame(rep(NA, 2), rep(NA, 4)) res           <- data.frame(method=c(rep("pooled", 2), rep("IBD", 2)),                             PE=NA, CL.lo=NA, CL.hi=NA, CV=NA,                             DFM=rep(c("Satterthwaite", "Kenward-Roger"), 2),                             DF=NA, stringsAsFactors=FALSE) for (j in 1:4) {   if (j == 1) { # pooled (all at once)     muddle <- lmer(log(AUC) ~ sequence + period + treatment + (1|subject),                               data=dta)   }   if (j == 3) { # IBD (S excluded)     muddle <- lmer(log(AUC) ~ sequence + period + treatment + (1|subject),                               data=TR.only)   }   sum.muddle  <- summary(muddle, ddf=res$DFM[j]) log.pe <- sum.muddle$coefficients["treatmentT", "Estimate"]   ci[j, 1:2]  <- round(100*exp(log.pe + c(-1, +1) *                            qt(1-0.05, sum.muddle$coef["treatmentT", "df"]) * sum.muddle$coef["treatmentT", "Std. Error"]), 2)   res$PE[j] <- round(100*exp(log.pe), 2) res$CL.lo[j] <- ci[j, 1]; res$CL.hi[j] <- ci[j, 2] res$CV[j]   <- round(100*sqrt(exp(sum.muddle$devcomp$cmp[["sigmaREML"]]^2)-1), 2)   res$DF[j] <- signif(sum.muddle$coefficients["treatmentT", "df"], 5) } print(res, row.names=FALSE)

Gives:

Method     PE  CL.lo  CL.hi    CV           DFM      DF pooled 116.15 108.97 123.81 21.20 Satterthwaite 115.040 pooled 116.15 108.97 123.81 21.20 Kenward-Roger 114.630    IBD 116.05 108.92 123.65 20.84 Satterthwaite  56.823    IBD 116.05 108.91 123.65 20.84 Kenward-Roger  56.468

Similar same.

Though the DFs are slightly different, the CIs look only identical due to rounding.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Brus
☆

Spain,
2021-06-09 16:48
(111 d 11:11 ago)

@ Helmut
Posting: # 22398
Views: 2,003

## Williams design 3-way

Dear Helmut,

I have a question on this topic that you mentioned a long time ago:

» A 6×3 design is needed in order to ‘extract’ two 2×2 tables, which are also balanced. Although the full 6×3 table will be used in the analysis of AUC and Cmax, you will need these 2×2s for the nonparametric analysis of tmax (un­fortunately there’s no confidence interval based nonparametric method available for more than two formulations/periods). The asterisks * denote pseudo-sequences and pseudo-periods, e.g. P1* means only that the treatment was given in a period prior to P2* – irrespective of the true study period:
» +----+------------+  -->  +----+--------+  and  +----+--------+» |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*|» +----+------------+       +----+--------+       +----+--------+» | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 |» | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  |» | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  |» | S4 | T   R2  R1 |       | S1*| T   R1 |       | S1*| T   R2 |» | S5 | R1  T   R2 |       | S2*| R1  T  |       | S1*| T   R2 |» | S6 | R2  R1  T  |       | S2*| R1  T  |       | S2*| R2  T  |» +----+------------+       +----+--------+       +----+--------+ »                             ^   balanced          ^   balanced
» A common mistake is to design the study as a set of 3×3 latin squares, which will lead (especially if the sample size is small and in the case of drop outs) to extremely imbalanced data sets:
» +----+------------+  -->  +----+--------+  and  +----+--------+» |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*|» +----+------------+       +----+--------+       +----+--------+» | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 |» | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  |» | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  |» +----+------------+       +----+--------+       +----+--------+ »                             ^ imbalanced          ^ imbalanced

According to what you said, 6 sequences had to be done with a 3-way study to get balanced sequences on the two 2x2 tablets. But, with following approach, balanced sequences would also be achieved:

ABC
BAC
CAB
CBA

It's possible? What problems would it have? It is not williams square design, right?

Best regards,
Helmut
★★★  Vienna, Austria,
2021-06-09 17:48
(111 d 10:12 ago)

@ Brus
Posting: # 22399
Views: 1,973

## Williams design 3-way

Hi Brus,

» I have a question on this topic that you mentioned a long time ago: […]

» According to what you said, 6 sequences had to be done with a 3-way study to get balanced sequences on the two 2x2 tablets. But, with following approach, balanced sequences would also be achieved:
»
» ABC
» BAC
» CAB
» CBA

That’s not balanced for carryover. However, since we cannot correct for unequal carryover anyway (see there), we have to avoid it by design, i.e., sufficiently long washouts between treatment periods.

» It's possible? What problems would it have?

What is your idea behind such a design? Actually a simple Latin Square (ABC|BCA|CAB) would do as well. If you want to make European assessors happy, opt for the 6-sequence Williams’ design.

Most important: Evaluate the study by the ‘Two at a Time’ approach.

» It is not williams square design, right?

Correct.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
Brus
☆

Spain,
2021-06-10 14:33
(110 d 13:26 ago)

@ Helmut
Posting: # 22402
Views: 1,877

## Williams design 3-way

Hi Helmut,

» That’s not balanced for carryover. However, since we cannot correct for unequal carryover anyway (see there), we have to avoid it by design, i.e., sufficiently long washouts between treatment periods.

Why?

IMHO, in my 3x4x6 design proposal,

ABC
BAC
CAB
CBA

Eeach formulation is administered once per subject, occurs the same number of times in each period, and any formulation is preceded equally often by each of the other formulations. In this way it is a balanced design. What benefit does williams design bring?

» What is your idea behind such a design?

Make a balanced design with as few sequences as possible.

» Actually a simple Latin Square (ABC|BCA|CAB) would do as well. If you want to make European assessors happy, opt for the 6-sequence Williams’ design.

Will Simple Latin Square be accepted by EMA assessors? Will my proposal design be accepted by EMA assessors?

Best regards,
Helmut
★★★  Vienna, Austria,
2021-06-10 15:03
(110 d 12:57 ago)

@ Brus
Posting: # 22403
Views: 1,983

## Williams design 3-way

Hi Brus,

» » That’s not balanced for carryover.
»
» Why?

Sorry (see below).

» Eeach formulation is administered once per subject, occurs the same number of times in each period, and any formulation is preceded equally often by each of the other formulations. In this way it is a balanced design.

So far, so good. But: In your design you have to assume equal period effects. Let’s count how many times the treatments are administered in each period.

   1 2 3 ──────── A  1 2 1 B  1 2 1 C  2 0 2

» What benefit does williams design bring?

It’s also balanced for period effects, which is really important.

    1 2 3 ───────── s1  A B C s2  A C B s3  B A C s4  B C A s5  C A B s6  C B A ───────── A   2 2 2 B   2 2 2 C   2 2 2

» » What is your idea behind such a design?
»
» Make a balanced design with as few sequences as possible.

Got it.

» » Actually a simple Latin Square (ABC|BCA|CAB) would do as well. If you want to make European assessors happy, opt for the 6-sequence Williams’ design.
»
» Will Simple Latin Square be accepted by EMA assessors?

Possible. Statistically nothing speaks against it. We don’t have carryover in the model and it’s balanced for period effects.

» Will my proposal design be accepted by EMA assessors?

Unlikely. If you have true period effects, your estimates will be biased and you have no means to correct it.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
vezz
☆

Erba (CO), Italy,
2021-06-10 17:23
(110 d 10:36 ago)

@ Helmut
Posting: # 22404
Views: 1,945

## Williams design 3-way

» Unlikely. If you have true period effects, your estimates will be biased and you have no means to correct it.

This is not to support the design proposed by Brus, but it seems to me that with those sequences we should still be able to obtain unbiased estimates of treatment effect also in presence of a period effect (based on the model suggested by the EMA Guideline on BE).

However, for some treatment comparisons the SE will be larger compared to a Latin square design with the same sample size.

I look forward for your thoughts!

Kind regards,

Stefano
Helmut
★★★  Vienna, Austria,
2021-06-10 20:14
(110 d 07:45 ago)

@ vezz
Posting: # 22405
Views: 1,911

## Williams design 3-way

Hi vezz,

» » Unlikely. If you have true period effects, your estimates will be biased and you have no means to correct it.
»
» This is not to support the design proposed by Brus, but it seems to me that with those sequences we should still be able to obtain unbiased estimates of treatment effect also in presence of a period effect (based on the model suggested by the EMA Guideline on BE).

Nope. Let’s consider a simple example. A = B = C = 1, no carryover, additive model.
• Latin Square
No period effects
     p1  p2  p3 s1  A=1 B=1 C=1 s2  B=1 C=1 A=1 s3  C=1 A=1 B=1 A/C = (s1p1+s2p3+s3p2)/(s1p3+s2p2+s3p1)=(1+1+1)/(1+1+1) = 1 B/C = (s1p2+s2p1+s3p3)/(s1p3+s2p2+s3p1)=(1+1+1)/(1+1+1) = 1
Period effects (p2 = p1–0.5, p3 = p2+1)
      p1   p2   p3 s1  A=1 B=0.5 C=1.5 s2  B=1 C=0.5 A=1.5 s3  C=1 A=0.5 B=1.5 A/C = (s1p1+s2p3+s3p2)/(s1p3+s2p2+s3p1)=(1+1.5+0.5)/(1.5+0.5+1) = 1 B/C = (s1p2+s2p1+s3p3)/(s1p3+s2p2+s3p1)=(0.5+1+1.5)/(1.5+0.5+1) = 1
• Brus’ design
No period effects
      p1  p2  p3 s1  A=1 B=1 C=1 s2  B=1 A=1 C=1 s3  C=1 A=1 B=1 s4  C=1 B=1 A=1 A/C = (s1p1+s2p2+s3p2+s4p3)/(s1p3+s2p3+s3p1+s4p1)=(1+1+1+1)/(1+1+1+1) = 1 B/C = (s1p2+s2p1+s3p3+s4p2)/(s1p3+s2p3+s3p1+s4p1)=(1+1+1+1)/(1+1+1+1) = 1
Period effects (p2 = p1–0.5, p3 = p2+1)
     p1   p2   p3 s1  A=1 B=0.5 C=1.5 s2  B=1 A=0.5 C=1.5 s3  C=1 A=0.5 B=1.5 s4  C=1 B=0.5 A=1.5 A/C = (s1p1+s2p2+s3p2+s4p3)/(s1p3+s2p3+s3p1+s4p1)=(1+0.5+0.5+1.5)/(1.5+1.5+1+1) = 0.7 B/C = (s1p2+s2p1+s3p3+s4p2)/(s1p3+s2p3+s3p1+s4p1)=(0.5+1+1.5+0.5)/(1.5+1.5+1+1) = 0.7

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
vezz
☆

Erba (CO), Italy,
2021-06-11 09:11
(109 d 18:48 ago)

@ Helmut
Posting: # 22406
Views: 1,807

## Williams design 3-way

Hi Helmut,

in your example you are assuming a model with no period effect, while I am considering the usual model for the analysis of BE studies, including the period effect.

With the period effect included in the model, the estimates of treatment effect are unbiased also using Brus' design.

Kind regards,

Stefano
Brus
☆

Spain,
2021-06-11 12:19
(109 d 15:41 ago)

@ vezz
Posting: # 22408
Views: 1,864

## Williams design 3-way

Dear Vezz and Helmut,

I think I understood Helmut. With my design (3x4x3), the carryover effect would be covered, since it generates balanced sequences for two-to-time comparisons, and any formulation is preceded equally often by each of the other formulations, but it does not cover the period effect, since that not all treatments are the same number of times in all periods (for example, in my design treatment "A" is administered 1 time in period 1, twice in period 2 and 1 time in period 3. In this way, in the event that there was a period effect, it would not be offset by the design and it would affect the results. Right?

As Vezz says, if you consider period effect in your model and the software itself already incorporates that correction in case there is period effect, you could have unbiased results, in fact, considering the period effect and the rest of possible effects (sequence, treatment,...) in the model, any design would be acceptable, right ?

But at this point, a question comes to mind, if our software is capable of calculating the period, sequence, group, and treatment effect, what need is there to make balanced designs?

In this way IMHO, and if I'm not mistaken, after kindly discussing it, I think the ideal option for a 3 treatment crossover study is 3x6x3 williams design.

Best regards,
Relaxation
★

Germany,
2021-06-11 12:42
(109 d 15:17 ago)

@ vezz
Posting: # 22409
Views: 1,831

## Williams design 3-way

Hi Stefano.

I was a little bit hesitant to answer, as I am and likely will remain a noob in the statistical background. Maybe I also misunderstand your question, which is quite possible.
But there are a lot of people around here to correct the answer, if necessary, and its SOP-Friday...

I think I get your point, but I also think that the period effect is implemented above.
The numbers already include both as mean=µ+π = treatment + period effect.

As an example:
the term s2p2 above is not the treatment mean for A in sequence 2 and period 2 (=1), it is the observed mean and includes the effect of period 2 (1-0.5=0.5).

If we would like to split that up into more detail, when sxpy is the treatment only (defined by combination of sequence and period), we can add Py as the period effect for more clarity.
Then:
A-C = ((s1p1 + P1) + (s2p2 + P2) + (s3p2 + P2) + (s4p3 + P3))
- ((s1p3 + P3) + (s2p3 + P3) + (s3p1 + P1) + (s4p1 + P1)).
See, as in Helmuts post, the period effects are implemented. But in the discussed design (P1+P2+P2+P3 = -0.5) and (P3+P3+P1+P1 = 1) do not completely eliminate each other as they would in a Latin Square. Thus treatment difference is biased (to +0.5 or 3.5/5=0.7) and not easily corrected.

Now, lets both wait and see, whether I got it right .

Best regards,

Relaxation.
Helmut
★★★  Vienna, Austria,
2021-06-11 15:41
(109 d 12:18 ago)

@ Relaxation
Posting: # 22410
Views: 1,799

## Period effects in general

Hi Relaxation and all,

» Now, lets both wait and see, whether I got it right .

You did. In simple terms: In – any – crossover period effects mean out if (and only if) the number of treatments in each period are equal because then they are affected to the same extent.
That’s one of the reasons why we design studies for balanced sequences. Due to dropouts sequences may be imbalanced, which biases the outcome. However,
• unless the degree of imbalancedness is extreme, the bias is negligible and
• theoretically it would be possible to exclude subjects in order to get balanced sequences. That would mean a loss in power and it is ethically questionable since available data is discarded. Never saw it in practice.

Dif-tor heh smusma 🖖
Helmut Schütz The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes
vezz
☆

Erba (CO), Italy,
2021-06-11 16:56
(109 d 11:03 ago)

@ Relaxation
Posting: # 22411
Views: 1,861

## Williams design 3-way

» I was a little bit hesitant to answer, as I am and likely will remain a noob in the statistical background. Maybe I also misunderstand your question, which is quite possible.
» But there are a lot of people around here to correct the answer, if necessary, and its SOP-Friday...
»
» I think I get your point, but I also think that the period effect is implemented above.
» The numbers already include both as mean=µ+π = treatment + period effect.

Hi Relaxation and Helmut,

the period effect was taken into account by Helmut when generating the data, but not when analysing them.

I am not able to explicitly describe each single step behind the estimation of the regression model (this would take some time!), but I will share with you the SAS code replicating the example.

Some notes:
- Treatments are coded as 1-2-3 instead of A-B-C.
- Without loss of generality, I am assuming 6 subjects per sequence and with the PARMS statement in the MIXED procedure I keep the residual variance fixed at 10.
- In the first MIXED procedure a model not including the period effect is estimated, while in the second one the model includes the period effect.

data a01 (drop=i t1-t3);    input t1 t2 t3;    seq+1;    do i=1 to 6;          subj+1;       period=1; tmt=t1; output;       period=2; tmt=t2; output;       period=3; tmt=t3; output;    end;    datalines;    1 2 3    2 1 3    3 1 2    3 2 1    ; run; data a02;    set a01;    y=1*(period=1)+.5*(period=2)+1.5*(period=3); run; proc mixed data=a02;    class seq subj tmt;    model y = seq subj(seq) tmt;    parms 10 / hold=1;    lsmeans tmt / diff; run; proc mixed data=a02;    class seq subj period tmt;    model y = seq subj(seq) period tmt;    parms 10 / hold=1;    lsmeans tmt / diff; run;

Estimated treatment differences (SE) by the first model (no period effect):
- A vs. B: 3.89E-16 (0.9129)
- A vs. C: -0.3750 (0.9129)
- B vs. C: -0.3750 (0.9129)

Estimated treatment differences (SE) by the second model (period effect included):
- A vs. B: 3.7E-16 (0.9129)
- A vs. C: -753E-17 (1.0206)
- B vs. C: -79E-16 (1.0206)

We may notice that in the second model:
- The estimate of the treatment effect is always practically 0, therefore unbiased.
- SEs are not identical as the design is not balanced for period.

Kind regards,

Stefano Ing. Helmut Schütz 