Bioequivalence and Bioavailability Forum

randombadger
☆

UK,
2011-04-28 15:02
(4738 d 13:49 ago)

Posting: # 6963
Views: 11,726

Replicate Designs & IntraSubj CV [General Statistics]

Hello all,

I have a 3 period, 2 treatment XO design and would like to calculate the intra-subject CV for the two treatments.

According to FDA guidance, I should use the following SAS code:

PROC MIXED DATA= cmax ;

  CLASSES armcd usubjid period treat ;

  MODEL LOG_PARM = armcd period treat / DDFM=KR;

  RANDOM treat/TYPE=FA0(2) SUB=usubjid G;

  REPEATED/GRP=treat SUB=usubjid;

  ESTIMATE 'B vs A' treat -1 1/CL ALPHA=0.1;

  LSMEANS treat/ cl alpha=0.1;

RUN;

however I see my colleagues are using the following:

PROC MIXED DATA= cmax ; 

   CLASS USUBJID period treat ARMCD ; 

   MODEL LOG_PARM = treat period ARMCD  /ddfm=KR;  

   RANDOM USUBJID(ARMCD) ;

   REPEATED/GRP=treat SUB=usubjid ;

   ESTIMATE "B vs A"  TREAT  -1 1 /cl alpha=0.1; 

   LSMEANS treat/ cl alpha=0.1;

RUN;

and I am finding the intra-subject CV's are quite different. I am a little confused regarding which RANDOM statement to use. Can anyone explain the differences and which statement is best to use?

My colleagues also included a carryover in the model but I believe the washout periods are long enough hence not required.

Thanks,
RB

Helmut
★★★

Vienna, Austria,
2011-04-28 17:00
(4738 d 11:51 ago)

@ randombadger
Posting: # 6965
Views: 10,440

Replicate Designs & IntraSubj CV

Post reply

Dear randombadger!

❝ I have a 3 period, 2 treatment XO design and would like to calculate the intra-subject CV for the two treatments.

Can you be a little be more specific? In some designs it’s possible to calculate CVs for both treatments, and in others not. FDA’s model for the commonly used partial replicate [TRR|RTR|RRT] is overspecified – you may get a number for CV_WT, but only with a SAS-warning and being nonsense (since T is not replicated). Examples of 3-period replicate designs allowing estimation of both CVs are [TRT|RTR] or [TRT|RTR|TRR|RTT]. So which design do you have?

❝ According to FDA guidance, […]

Almost. You should use ddfm=satterth (Satterthwaite’s degrees of freedom) – not ddfm=kr (Kenward-Rogers); though I guess there shouldn’t be too much of a difference in results.

❝ however I see my colleagues are using the following: […]

Can you ask them for a reference for their code?

❝ My colleagues also included a carryover in the model but I believe the washout periods are long enough hence not required.

Carryover should go the statistical waste-bin. Stephen Senn devoted (almost) an entire book to this topic.

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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

randombadger
☆

UK,
2011-04-28 17:25
(4738 d 11:26 ago)

@ Helmut
Posting: # 6967
Views: 10,264

Replicate Designs & IntraSubj CV

Post reply

Hi,

❝ An example of a 3-period replicate design which allows estimation of both CVs is [TRT|RTR]. So which design do you have?

We're using TRR, RTT design.

❝ Can you ask them for a reference for their code?

I suspect they're adding the carryover based on the SC Chow & JP Liu section on 2 sequence dual designs but yes, I plan to discuss with them their thinking behind the RANDOM statement as it's same as you'd use for a 2x2 XO design.

Thanks!

Helmut
★★★

Vienna, Austria,
2011-04-28 19:09
(4738 d 09:41 ago)

@ randombadger
Posting: # 6968
Views: 10,324

Replicate Designs & IntraSubj CV

Post reply

Dear randombadger!

❝ We're using TRR, RTT design.

I see. The “extra-reference design”. Not optimal because biased in the presence of period effects.

❝ I suspect they're adding the carryover based on the SC Chow & JP Liu section on 2 sequence dual designs but yes, I plan to discuss with them their thinking behind the RANDOM statement as it's same as you'd use for a 2x2 XO design.

With your code and Chow/Liu’s Table 9.3.3 untransformed data, I can reproduce their CI of (-1.48, 2.83), whilst FDA’s code gets hickups (Newton's algorithm converged with modified Hessian. Output is suspect.) and gives a much wider CI (-5.62, 6.97). BTW, I’m using Phoenix6.2; last post from my side, I guess we need a SAS-guru here (Detlew, are you here?).

—
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz

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

d_labes
★★★

Berlin, Germany,
2011-04-29 13:33
(4737 d 15:17 ago)

@ randombadger
Posting: # 6972
Views: 10,675

Inter-subject dance with the wolves

Post reply

Dear randombadger!

The badger [zool.] or to badger :-D

?

Dear Helmut!

Let me snap-in.
RB's question is a wide field. It is tilled with many matrices some beyond the horizon of a normal mortal mind.

The difference of your two codes lies in the specification of the inter-subject covariances.
The FDA model specifies different variabilities of the treatment groups plus a correlation of T vs. R within a subject.
Let us write it down in the CSH notation which is another possibility in the FDA Statistical guidance:

(   s²_bT     rho*s_bT*s_bR )

( rho*s_bT*s_bR   s²_bR     )

This notation is easier to interpret than the FA0(2) parametrisation which is only for assuring that the G-matrix is positive definite.

From that specification the so-called subject-by-treatment interaction variance is defined as

s²_D = s²_bT + s²_bR - 2*rho*s_bT*s_bR

This term is also known under the name subject-by-formulation interaction.
It had became famous in the days of IBE - individual bioequivalence.

If you are convinced that there doesn't exists such a horrible thing ;-)

like subject-by-treatment-interaction set it to zero!
This is only possible under the two conditions

1: s_bT = s_bR = s_between i.e. equal between-subject variabilities of T and R

2: rho = 1, i.e. perfect correlation

as you can easily verify.
Then the G-matrix above reduces to

( s²_between   s²_between )

( s²_between   s²_between )

Unfortunately there is no TYPE=blabla in the SAS Proc MIXED code to specify exactly this structure in the FDA like syntax.
But your second code with the different RANDOM statement is fitting exactly this model!

You can eventually try the TYPE=CS which specifies a matrix according to

( s²+s₁   s²   )

(  s²    s²+s₁ )

and start with the parameter s₁=0 and hold it at this value during the REML fit. Or cross your fingers and hope that the data are best fit with s₁=0 (called CS in the SAS output).

Hope this all makes sense to you. To me only after some beer :cool:

.
The key message: Different models of the inter-subject covariance -> different intra-subject CVs.
But I'm convinced if the subject-by-treatment interaction is approximately negligible there shouldn't be great differences in the intra-subject CVs.
RB, can you eventually post your values?

BTW: Concerning carry-over in the model I fully coincide with Helmut.
Concerning the DDFM=Satterth or DDFM=KR its a matter of taste. The FDA code arose from times where the Kenward-Roger method was not implemented or experimental in SAS. But its the more modern method and commonly seen as more appropriate in case of small sample sizes.
DDFM=KR is used through out in the book B. Jones and M.G. Kenward "Design and Analysis of Cross-over Trials" Chapman & Hall/CRC, Boca Raton (2nd ed. 2003). Not so astonishing if you look at the second author :-D

—
Regards,

Detlew

jdetlor
☆

2011-05-02 20:21
(4734 d 08:29 ago)

@ d_labes
Posting: # 6977
Views: 10,085

How many subjects?

Post reply

Dear d_labes, Dear All!

d_labes this is an excellent reply IMHO. :clap:

The only addition I would make would be to point out that the SUBJ= option in the RANDOM statement is absent from the second set of code. This is critical to the proper allocation of the between and within variabilities. If we ignore this option in the RANDOM statement we are assuming the data is from 1 subject only, and thus the G matrix is not in a block-diagonal format. The data from each subject in our actual dataset is not treated as independent, and our between-subject variabilities are reduced, while our within-subject variabilities are increased.

I would guess this is the case with randombadger's results.

J. Detlor

d_labes
★★★

Berlin, Germany,
2011-05-03 10:26
(4733 d 18:24 ago)

@ jdetlor
Posting: # 6978
Views: 10,254

G matrix

Post reply

Dear J. Detlor!

❝ d_labes this is an excellent reply IMHO.

Thanks for the flowers.

❝ The only addition I would make would be to point out that the SUBJ= option in the RANDOM statement is absent from the second set of code.

❝ ...

❝ If we ignore this option in the RANDOM statement we are assuming the data is from 1 subject only, and thus the G matrix is not in a block-diagonal format.

Really?

I got the G matrix for the second code set (armcd renamed to sequence) for the above mentioned Example 9.3.3 in Chow/Liu - a 2-sequence-3-period replicate study as:

                                Estimated G Matrix



 Row  Effect             sequence  subject    Col1      Col2     Col3    ...



   1  subject(sequence)  RTT        1       75.6238

   2  subject(sequence)  RTT        3                 75.6238

   3  subject(sequence)  RTT        4                           75.6238

... 

                                        ...  Col10     Col11     Col12   ...

  10  subject(sequence)  TRR        2       75.6238 

  11  subject(sequence)  TRR        5                 75.6238

  12  subject(sequence)  TRR        6                           75.6238

...

For me this looks block diagonal at its best :cool:

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2011-05-03 11:15
(4733 d 17:36 ago)

@ d_labes
Posting: # 6979
Views: 10,138

G matrix

Post reply

Dear d_labes,

I am just the lowest of amateur scum, so please forgive me for interfering with your divine elaborations.

❝ For me this looks block diagonal at its best :cool:

It is diagonal, yes, but unblocked, which I think was what was meant.
To a person with a walnut-sized brain that kind of matrix creates headache since the first three lines tell the values correspond to different subjects.
Output of G perhaps could be a QC tool for the aficionados?

—
Pass or fail!
ElMaestro

d_labes ★★★ Berlin, Germany, 2011-05-03 11:51 (4733 d 16:59 ago) @ ElMaestro Posting: # 6980 Views: 10,064	G matrix Post reply
	Dear ElMaestro, ❝ I am just the lowest of amateur scum, ... Bescheidenheit ist eine Zier, doch weiter kommt man ohne ihr . (pretended to Wilhelm Busch) (Sorry can't translate that. Google translate: Humility is a virtue, but further you get without her) ❝ It is diagonal, yes, but unblocked, ... Sorry. What was the question? — Regards, Detlew

randombadger
☆

UK,
2011-05-03 14:03
(4733 d 14:47 ago)

@ d_labes
Posting: # 6982
Views: 10,000

Results

Post reply

Dear all,

Thank you for your responses so far, they have been very supportive. Please see below for results from my analyses:

Model 1

PROC MIXED DATA= cmax ; 

   CLASS USUBJID period treat ARMCD ; 

   MODEL LOG_PARM = treat period ARMCD  /ddfm=KR;  

   RANDOM USUBJID(ARMCD) /G;

   REPEATED/GRP=treat SUB=usubjid ;

   ESTIMATE "B vs A"  TREAT  -1 1 /cl alpha=0.1; 

   LSMEANS treat/ cl alpha=0.1;

RUN;



Estimated G Matrix



                                                Planned

                        Unique Subject          Arm

   Row  Effect          Identifier              Code    Col1     Col2



     1  USUBJID(ARMCD)  XXX XXX-X-XXX-00011001  ABB     0.07056      

     2  USUBJID(ARMCD)  XXX XXX-X-XXX-00011003  ABB             0.07056

(block diagonal, apologies for the lack of formatting)
etc

Intra-Subject

CV A (%)    CV B (%)

 31.98       10.02

Model 2

PROC MIXED DATA= cmax ;

  CLASSES armcd usubjid period treat ;

  MODEL LOG_PARM = armcd period treat / DDFM=KR;

  RANDOM treat/TYPE=FA0(2) SUB=usubjid G;

  REPEATED/GRP=treat SUB=usubjid;

  ESTIMATE 'B vs A' treat -1 1/CL ALPHA=0.1;

  LSMEANS treat/ cl alpha=0.1;

RUN;





Estimated G Matrix

                 Unique Subject

Row    Effect    Identifier            TREAT    Col1     Col2



1    TREAT     XXX XXX-X-XXX-00011001    A     0.1036   0.05780

2    TREAT     XXX XXX-X-XXX-00011001    B     0.05780  0.07110

Intra-Subject

CV A (%)    CV B (%)

 22.88        9.78

As you can see, the differences in CV's are quite large hence my concern on the most appropriate code to apply. Note: the treatments were nowhere near bioequivalent.

Edit: Formatted. You may use BBCodes (see here). [Helmut]

d_labes
★★★

Berlin, Germany,
2011-05-03 16:32
(4733 d 12:18 ago)

@ randombadger
Posting: # 6984
Views: 9,996

Results - comments

Post reply

Dear randombadger,

that results show for me:

The FDA model (model 2), not making any assumptions about equal variances, gives you somewhat different inter-subject variabilities (CV_bA=33%, CV_bB=27%) plus a non-negligible subject-by-formulation interaction
s²_D = g₁₁+g₂₂-2*g₁₂ = 0.0591 s_D = 0.2431
Remember s_D>0.15 for log-transformed values was considered as big in the old days of IBE.
The constraint model 1 (assuming s²_D = 0 which lead to equal inter-subject variabilities) must absorb these inter-subject variance-covariance terms into the other random terms, namely the intra-subject (residual) variances attributed to the treatments A or B. Therefore the intra-subject CV especially for treatment A rises.
I bet some EURO cent that the model 2 fits your data better. Have a look at the Akaike information criterion (AIC) or Bayesian information criterion (BIC) under Fit statistics to verify this. These information criteria (there are some other in the SAS output if you run your Proc MIXED code with the option IC) you can use to choose your model.
If we interpret treatment A for a moment as Reference the differences in the intra-subject CVs between both models can make the step from not allowed to allowed scaled, widened acceptance ranges (sABE). Since the constraint model is a step from the FDA model to the EMA Q&A recommended evaluation we could be happy for our sponsors .

—
Regards,

Detlew

randombadger
☆

UK,
2011-05-03 18:53
(4733 d 09:57 ago)

@ d_labes
Posting: # 6986
Views: 10,007

Results - comments

Post reply

Dear d_labes,

Thank you for your feedback.

Here are the fit statistics for the 2 models:

Model 1

-2 Res Log Likelihood        24.0

AIC (smaller is better)      30.0

AICC (smaller is better)     30.5

BIC (smaller is better)      33.0

Model 2

-2 Res Log Likelihood        22.0

AIC (smaller is better)      32.0

AICC (smaller is better)     33.3

BIC (smaller is better)      37.0

Interestingly, the 1st model is a better fit.

RB

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

d_labes ★★★ Berlin, Germany, 2011-05-04 10:07 (4732 d 18:44 ago) @ randombadger Posting: # 6989 Views: 9,955	Lost cents Post reply
	Dear randombadger, ❝ Interestingly, the 1st model is a better fit. Give me your bank account details to obtain my cents . — Regards, Detlew

ElMaestro ★★★ Denmark, 2011-05-03 16:16 (4733 d 12:34 ago) @ d_labes Posting: # 6983 Views: 10,075	G matrix Post reply
	Dear d_labes, ❝ Sorry. What was the question? The question was "Really?" — Pass or fail! ElMaestro

jdetlor
☆

2011-05-03 21:37
(4733 d 07:13 ago)

@ d_labes
Posting: # 6987
Views: 9,982

G matrix

Post reply

Dear d_labes,

❝ Thanks for the flowers.

You're welcome — I thought your write-up laying out the matrices deserved something. ;-)

❝ Really?

❝

❝ I got the G matrix for the second code set (armcd renamed to sequence) for the above mentioned Example 9.3.3 in Chow/Liu - a 2-sequence-3-period replicate study as:

❝

❝ Estimated G Matrix

❝

❝ Row Effect sequence subject Col1 Col2 Col3

❝ ...

❝

❝ 1 subject(sequence) RTT 1 75.6238

❝ 2 subject(sequence) RTT 3 75.6238

❝ 3 subject(sequence) RTT 4 75.6238

❝ ...

❝ ... Col10 Col11 Col12

❝ ...

❝ 10 subject(sequence) TRR 2 75.6238

❝ 11 subject(sequence) TRR 5 75.6238

❝ 12 subject(sequence) TRR 6 75.6238

❝ ...

❝

❝ For me this looks block diagonal at its best :cool: .

This is not a block-diagonal G-matrix. This is the default covariance structure for PROC MIXED in SAS (VC – variance components) when no TYPE= option is specified. It is true it does have values along the diagonal, but this demonstrates the VC structure.

The 'Dimensions' output from SAS will indicate how your Z-matrix is constructed, and thus your G-matrix. A blocked-diagonal G-matrix will be indicated by the number of columns in the Z-matrix defined per subject, with the number of max observations per subject matching correctly (max 3 observations for this design), for a total of max observations x the number of subjects.

A standard VC structure will list the number of columns in the Z-matrix equal to the subjects, with the number of subjects equal 1, and the max observations per subject equal to the total number of observations.

For the output above (Chow/Liu data) the SAS output dimensions will specify there are 18 (subject) columns (with 18x3 rows for this design) in the Z-matrix, with an 18x18 G-matrix (the full matrix of your excerpt above).

Replicate Designs & IntraSubj CV [General Sta­tis­tics]

Replicate Designs & IntraSubj CV

Replicate Designs & IntraSubj CV

Replicate Designs & IntraSubj CV

Inter-subject dance with the wolves

How many subjects?

G matrix

G matrix

G matrix

Results

Results - comments

Results - comments

Lost cents

G matrix

G matrix

Replicate Designs & IntraSubj CV [General Statistics]