Bioequivalence and Bioavailability Forum • 2-stage maximum likelihood of CV/RMSE, given T/R

ElMaestro
★★★

Denmark,
2012-10-22 17:06
(4562 d 03:29 ago)

Posting: # 9442
Views: 7,854

2-stage maximum likelihood of CV/RMSE, given T/R [Two-Stage / GS Designs]

Dear all,

as usual I am fiddling with my compiler. And no, this is not a metaphore. At least not this time.

What I am thinking at present is this: For Potvin method B/C/D we use the CV from stage 1 in the sample size estimation for stage 2, but we do not use the observed T/R; in stead we use fixed T/R=0.9 or 0.95 etc. On the other hand we do use observed T/R when we test for BE at both stages.

So, let's be consequent and propose we will use the fixed T/R of 0.90 or 0.95 for evaluation of BE at stage 1 and dimensioning of st. 2. To do this we must then find the maximum likelihood of the anova RMSE (s, CV) given the model, the fixed T/R ratio and the observations. Can anyone suggest how to accomplish that?

Many thanks for any input.

—
Pass or fail!
ElMaestro

Helmut
★★★

Vienna, Austria,
2012-10-22 17:40
(4562 d 02:55 ago)

@ ElMaestro
Posting: # 9443
Views: 6,795

Not sure whether I understand this

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Gentleman!

❝ What I am thinking at present is this: For Potvin method B/C/D we use the CV from stage 1 in the sample size estimation for stage 2, but we do not use the observed T/R; in stead we use fixed T/R=0.9 or 0.95 etc.

Yes, since the framework is not fully adaptive (taking estimated variance + effect size into account) but only a sample size re-estimation. Full adaptive designs are nice if the sample size is really (!) large. If we have two estimates (T/R and CV) from the small sample sizes in BE the penalty might be terrible. The distribution of total sample sizes in B/C/D is already highly skewed to the right. In some preliminary sims I got awful results – especially if the first stage was relatively small (compared to a fixed design of the same CV) and the T/R was by chance >±10% from 1. IMHO, B/C/D (fixed α_adj.) no way. Would be interesting how the ol’ pirate’s framework performs in such a situation.

❝ On the other hand we do use observed T/R when we test for BE at both stages.

Sure.

❝ So, let's be consequent and propose we will use the fixed T/R of 0.90 or 0.95 for evaluation of BE at stage 1

What if a study passes (actual T/R) with 0.05 or 0.0294 (C), 0.0294 (B), or 0.0280 (D) in stage 1 and fails with T/R 0.95 (B/C) or 0.90 (D)? Following the same logic should we reject a conventional fixed sample study which was planned for T/R 0.95, CV somefink (T/R <0.95 – but passes due to a lower CV)?

❝ … and dimensioning of st. 2.

Well, that’s exactly what B/C/D does, IMHO.

❝ To do this we must then find the maximum likelihood of the anova RMSE (s, CV) given the model, the fixed T/R ratio and the observations.

– again.

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

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ElMaestro
★★★

Denmark,
2012-10-22 18:34
(4562 d 02:01 ago)

@ Helmut
Posting: # 9444
Views: 6,768

Not sure whether I understand this

Post reply

Dear Helmut,

❝ :confused: – again.

I am sorry for this gibberish.

Lemme try and explain with a simple made-up case to illustrate my point:
After dosing 4x500 mg of Schützomycin to two random subjects (let's call them Helmut and Detlew for simplicity), I measure how high they can jump. We are interested in knowing the average jump height of these two... erm.. shall we call them 'cases'.
For Helmut I measure (in cm):39,41,51,43,53
For Detlew I measure (in cm):50,39,56,40,42
...and I will now assume IID and blahdeeblahdeeblah and therefore I use a linear model even though jump height presumably cannot be lower than zero and blahblahblah. Let's just do this in R:

Height=c(c(39,41,51,43,53), c(50,39,56,40,42))

Case=as.factor( c(   rep("Helmut",5), rep("Detlew",5)))

Model=lm(Height~0+Case)

 

summary(Model)



Call:

lm(formula = Height ~ 0 + Case)



Residuals:

   Min     1Q Median     3Q    Max

 -6.40  -5.15  -2.90   5.35  10.60



Coefficients:

           Estimate Std. Error t value Pr(>|t|)

CaseDetlew   45.400      3.043   14.92 4.02e-07 ***

CaseHelmut   45.400      3.043   14.92 4.02e-07 ***

---

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



Residual standard error: 6.804 on 8 degrees of freedom

Multiple R-squared: 0.9823,     Adjusted R-squared: 0.9779

F-statistic: 222.6 on 2 and 8 DF,  p-value: 9.712e-08

(let me add: I will of course not fit with an intercept because then the model coefficients are not the LSMeans notch notch wink wink ;-)

).

OK, so the two cases seem to perform equally terribly miserably bad under these assumptions for the normal linear model. Lord have mercy. In addition, I see that the residual st error is 6.804 under these circumstances. Aha...
But now I will introduce a game-changer: Let's say I know that the actual Mean for Helmut is 45 and that the actual Mean for Detlew is 47. These are not estimates but true values. Given that I know this, and given our observations, what would then be the most likely residual st error?

—
Pass or fail!
ElMaestro

d_labes
★★★

Berlin, Germany,
2012-10-23 18:44
(4561 d 01:51 ago)

@ ElMaestro
Posting: # 9452
Views: 6,736

Not sure whether I understand this

Post reply

Dear ElMaestro,

❝ But now I will introduce a game-changer: Let's say I know that the actual Mean for Helmut is 45 and that the actual Mean for Detlew is 47. These are not estimates but true values. Given that I know this, and given our observations, what would then be the most likely residual st error?

What about this one: Subtract the 'known (true) means' from the measurement values and analyze them by fitting a model with all the other effects you are supposing to have to account for. The MSE of the fit is then the residual std error I think.

But I must confess that I'm totally unsure what you attempt to do in your original question :confused:

.
You never know the true T/R ratio. Even if you have done your study (first stage or pivotal). The best you have is an estimate.
And assuming a true T/R ratio of say 0.95 within the 2-stage design evaluation is not for fitting a model to the data but only for powering the second stage enough under the assumption that you get an estimate equal or better than this true ratio.

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2012-10-23 18:55
(4561 d 01:40 ago)

@ d_labes
Posting: # 9453
Views: 6,645

Not sure whether I understand this

Post reply

Dear d_labes,

❝ What about this one: Subtract the 'known (true) means' from the measurement values and analyze them by fitting a model with all the other effects you are supposing to have to account for. The MSE of the fit is then the residual std error I think.

thanks for your input. I will try and see if I can make it work.

❝ But I must confess that I'm totally unsure what you attempt to do in your original question :confused: .

❝ You never know the true T/R ratio. Even if you have done your study (first stage or pivotal). The best you have is an estimate.

❝ And assuming a true T/R ratio of say 0.95 within the 2-stage design evaluation is not for fitting a model to the data but only for powering the second stage enough under the assumption that you get an estimate equal or better than this true ratio.

When we calculate type I errors and power in Potvin's scenarios we assume a T/R. This is a 'known', in contrast to the estimates we get from the sampled data. My point is this:
When we calculate the sample size for the second stage we apply a known (=assumed) T/R and a measured variability, but the measured variability reflects the sampled T/R. I am therefore thinking we could empirically try to calculate a maximum likelihood estimate of the variability given the known (=assumed T/R) rather than the observed T/R and given the observations.

So, in essence one suggestion is to somehow fixed the two first parameters of the effects vector and allow the rest to be fit, and see what variability comes out at the other end. If it works the same to subtract known effects from the sampled data then I am happy. Ís the resulting s biased?

—
Pass or fail!
ElMaestro

Helmut
★★★

Vienna, Austria,
2012-10-23 19:17
(4561 d 01:18 ago)

@ ElMaestro
Posting: # 9454
Views: 6,696

Not sure whether I understand this

Post reply

Hi ElMaestro!

❝ When we calculate type I errors and power in Potvin's scenarios we assume a T/R. This is a 'known', in contrast to the estimates we get from the sampled data.

Not quite. You are mixing two things up, IMHO. T/R is an assumption – not “known”. Hard-core statisticians told me that (even for a fixed design) we never should use term “sample size calculation”, but “sample size estimation” instead because T/R is assumed and CV an estimate at the best (or also only an assumption).

❝ When we calculate the sample size for the second stage we apply a known (=assumed) T/R and a measured variability,

Known ≠ assumed. We never know.

❝ but the measured variability reflects the sampled T/R.

Why? Variance is independent from a shift in location.

Too stupid for the rest of your post.

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

The quality of responses received is directly proportional to the quality of the question asked. 🚮
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ElMaestro
★★★

Denmark,
2012-10-23 22:51
(4560 d 21:44 ago)

@ Helmut
Posting: # 9455
Views: 6,747

Ach du lieber

Post reply

Hi Helmut,

❝ Not quite. You are mixing two things up, IMHO. T/R is an assumption – not “known”.

Not quite. You are mixing two things up, IMHO :-D

When we do simulations, we know the exact T/R we are simulating. This is the situation I am talking about here. My question is not related to the situation of an applicant conducting a two-stage trial when T/R is estimated and never known.
My thought was prolly a bad idea anyway. A shift in T or R (location?!) will affect the minimised SS, albeit if we tamper with it that way I no longer know if the resulting s derived from minimisation of SS will be unbiased. In principle the latter issue is of less importance for me if it turns out the type I error or power is improved.

—
Pass or fail!
ElMaestro