Bioequivalence and Bioavailability Forum • Inter and Total Variability

Ben
★

2011-09-02 15:43
(4848 d 07:07 ago)

Posting: # 7327
Views: 14,647

Inter and Total Variability [Power / Sample Size]

Dear all,

I looked at the presentation slides about sample size calculations from HS, see here. There's a formula for the inter CV (slide #8) and I do not quite understand why the exponent from the exponential is the difference of MSE_B and MSE_W divided by 2 (and not just MSE_B). Would be great if someone could give me some more details on that.
Also I read that one only gets the total variance out from a parallel study. The reason for that is what? Is it just the fact that the intra subject variability always exists, but cannot be measured by this kind of design?

Thank you in advance for clarification.

Best regards,
Benjamin

Edit: Moved to a category which fits the topic better, IMHO. [Helmut]

Helmut
★★★

Vienna, Austria,
2011-09-02 16:21
(4848 d 06:29 ago)

@ Ben
Posting: # 7328
Views: 13,017

Inter and Total Variability

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Dear Benjamin!

❝ […] Would be great if someone could give me some more details on that.

For references see this post and Hauschke et al.^1,2

The between-subject coefficient of variation \(CV_b\) is calculated from the between-subject variance \(\sigma_b^2\):
\begin{equation}
CV_b=\sqrt{e^{\sigma_b^2-1}}
\end{equation}
Since we don’t know the between-subject variance of the population, \(CV_b\) has to be estimated from the model’s \(MSE_b\) and \(MSE_w\):
\begin{equation}
CV_b\approx\sqrt{e^{(MSE_b-MSE_w)/2}-1}
\end{equation}

❝ Also I read that one only gets the total variance out from a parallel study. The reason for that is what? Is it just the fact that the intra subject variability always exists, but cannot be measured by this kind of design?

Exactly! The fact that you observed just one occasion, doesn’t mean that variability between occasions (in the same subjects – therefore ‘within’ or ‘intra’) does not exist.

Hauschke D, Steinijans VW, Pigeot I. Bioequivalence Studies in Drug Development: Methods and Applications. New York: Wiley; 2007. p 88.
A note for users of Phoenix/WinNonlin: The output of a 2×2 cross-over study erroneously gives \(CV_b\) based on (1)…
Edit: Removed from the output in Phoenix 7.0; now it is up to you to apply the correct formula in a ‘custom transformation’ from the variance(s). BTW, if the EMA’s ‘all fixed effects’ model is used, only the within-subject variance is available.

PS: THX for bringing the problems with the contact form and the registration to my attention!

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

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Ben ★ 2011-09-03 13:28 (4847 d 09:22 ago) @ Helmut Posting: # 7333 Views: 12,880	Inter and Total Variability Post reply
	Thank you very much for this quick reply. I will check out all the references (haven't done all of them so far). Best regards, Benjamin

Ben
★

2011-09-08 21:22
(4842 d 01:28 ago)

@ Ben
Posting: # 7342
Views: 12,759

Inter and Total Variability

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Dear all,

I will just reply to this post instead of opening a new one (since one issue I want to talk about is again about variabilities).

I read in other posts (here) that one cannot get inter (or intra) variability from the total variability and one should avoid rule of thumbs like CV_intra = 60% of CV_total. I agree (I saw the data from Lansoprazole) - at least when talking about CVs. But what about just taking intra variance to be half of the total variance (when only total variance is given)? This approach seems to be a conservative estimate. Usually intra variance is less than or equal to inter variance. It also makes sense coming from the correlation between two responses on the same subject, which is equal to 1/2 if and only if the within variance equals the between variance. Based on this one can calculate CVs. Or not? (the Lansoprazole example should also fulfill this)
I'm trying to get the degrees of freedom for the following design: one single group, fixed sequence, uncontrolled with respect to time effects (intra-subject design), say n subjects receive k days treatment A, then another k days they receive treatment A and B. More mathematically
log(response) = overall mean + subject + trt effect + error.

Now, consider the following approach. We have 2n values, we lose n-1 df because of the subjects, 2-1 df because of treatment effects and the "usual" -1 because of the overall mean, hence the degrees of freedom is n-1.
I'm wondering whether this is correct or not. Let's say we use the same approach for a 2x2 cross-over design, that is,
log(response) = overall mean + sequence + subject + period + trt effect + error,

then we end up with (assume n₁ = n₂ = n/2) 2n - (n-1) - (2-1) - (2-1) - (2-1) - 1 = n-3.
But we should have n-2 (don't we?) and hence the approach may not be correct at all...?

Thank you in advance for your thoughts and help on that.

Best regards,
Benjamin

Helmut
★★★

Vienna, Austria,
2011-09-11 13:55
(4839 d 08:55 ago)

@ Ben
Posting: # 7346
Views: 12,936

Inter and Total Variability

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Dear Benjamin,

answering only your first question (now) …

❝ […] But what about just taking intra variance to be half of the total variance (when only total variance is given)? This approach seems to be a conservative estimate.

Not always (see below).

❝ Usually intra variance is less than or equal to inter variance.

What it your drug/formulation is ‘unusual’? If you have only a parallel study you simply don’t know.

❝ It also makes sense coming from the correlation between two responses on the same subject, which is equal to 1/2 if and only if the within variance equals the between variance. Based on this one can calculate CVs. Or not? (the Lansoprazole example should also fulfill this)

OK, here are the complete data of the previous post (only the two extremes):*


Study           CV_intra  CV_inter  CV_total intra/total   MSE_w     MSE_B     MSE_t 

Methylphenidate  7.00%  19.13%  20.41%    34.27%     0.004882 0.076746 0.039225

Lansoprazole    47.01%  25.14%  54.60%    86.09%     0.199667 0.322229 0.262708

If you base the sample size estimation in a X-over on \(\sigma_{intra}^2\approx\sigma_{total}^2/2\) you would end up with CV_intra of 14.07% (↑) for methyphenidate and 37.47% (↓) for lansoprazole. In the former case you waste money – in the latter you blew the study.

See this post for corrections.

—
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

Ben
★

2011-09-14 00:03
(4836 d 22:47 ago)

@ Helmut
Posting: # 7350
Views: 12,566

Inter and Total Variability

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Dear Helmut,

thank you for your post. So just for me to get this straight: MSE_t is not the sum of MSE_w and MSE_B, it's MSE_w + (MSE_B - MSE_W)/2. And σ²_inter is estimated by (MSE_B - MSE_W)/2 and not MSE_B (in the references you posted these values are plugged in for calculating the corresponding CV, so I guess these values themselves represent direct estimates for the variances). Then of course it makes sense. A conservative estimator for the intra variability would then be MSE_t.

Best regards

Helmut
★★★

Vienna, Austria,
2011-09-14 02:43
(4836 d 20:07 ago)

@ Ben
Posting: # 7351
Views: 12,832

Inter and Total Variability

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Dear Benjamin,

THX for thinking it over again and forcing me to dig out old references. My output (given by a software I have written in the mid-1980s) didn’t contain MSE_t (only SSE_t and df). In this table I thoughtlessly used MSE_t = SSE_t/df…

❝ […] MSE_t is not the sum of MSE_w and MSE_B, it's MSE_w + (MSE_B - MSE_W)/2.

Which reduces to MSE_t = (MSE_B + MSE_W)/2. q.e.d.
I should have paid more attention to my own slide. :angry:

Therefore:


Study           CV_intra  CV_inter  CV_total   MSE_w     MSE_B     MSE_t 

Methylphenidate  7.00%  19.13%  20.41%   0.004882 0.076746 0.040814

Lansoprazole    47.01%  25.14%  54.60%   0.199667 0.322229 0.260948

The percentage CV_total intra/total is nonsense (apples and oranges).

❝ And σ²_inter is estimated by (MSE_B - MSE_W)/2 and not MSE_B (in the references you posted these values are plugged in for calculating the corresponding CV, so I guess these values themselves represent direct estimates for the variances).

Right.

❝ Then of course it makes sense. A conservative estimator for the intra variability would then be MSE_t.

Also correct. But then we are relying on:
[image]

In the methylphenidate example conservatism will be very expensive.

—
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

Ben ★ 2011-09-18 14:36 (4832 d 08:15 ago) @ Helmut Posting: # 7367 Views: 12,425	Inter and Total Variability Post reply
	Dear Helmut, thanks for confirming and clarifying again. ❝ In the methylphenidate example conservatism will be very expensive. I agree. Any thoughts on the degrees of freedom problem? Best regards

Ben
★

2011-10-06 20:51
(4814 d 02:00 ago)

@ Helmut
Posting: # 7438
Views: 12,219

Inter and Total Variability

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Dear Helmut/all,

I'd like to pick up the topic on how to estimate σ²_inter again.

❝ ❝ And σ²_inter is estimated by (MSE_B - MSE_W)/2 and not MSE_B (in the references you posted these values are plugged in for calculating the corresponding CV, so I guess these values themselves represent direct estimates for the variances).

❝

❝ Right.

When using SAS's PROC MIXED then the variance of the subject effect and the variance of the error term will be printed in the table "Covariance Parameter Estimates". Does SAS internally calculate this estimate by (somehow) using the fact that (MSE_B - MSE_W)/2 ? I found this website which says that SAS calculates (MSE_B - MSE_W)/3. (Example 2; here we have balanced data). I'm confused. Which one is the "correct" estimate? Do I have to multiply the value from SAS by 3 and then divide by 2 to get the "right one"? Does this calculation depend on whether someone uses GLM or MIXED (but the formula (MSE_B - MSE_W)/2 seems to be an general one...)?
Thank you in advance!

Best,
Ben

d_labes
★★★

Berlin, Germany,
2011-10-07 15:33
(4813 d 07:17 ago)

@ Ben
Posting: # 7440
Views: 12,492

Inter-subject variance sasophylistic (GLM vs. MIXED)

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Dear Ben,

❝ When using SAS's PROC MIXED then the variance of the subject effect and the variance of the error term will be printed in the table "Covariance Parameter Estimates". Does SAS internally calculate this estimate by (somehow) using the fact that (MSE_B - MSE_W)/2 ?

No. The variance of the random subject effect is a direct parameter of the underlying model and is estimated within Proc MIXED by REML method - a non-linear optimisation method. No sum-of-squares decomposition / mean squares are used within that method. That's the reason why you don't get an ANOVA like table within that SAS procedure.

❝ I found this website which says that SAS calculates (MSE_B - MSE_W)/3. (Example 2; here we have balanced data).

This example you cite is a '3 period' study, measurements at 3 time points for each subject, therefore the formula differs from that one for the 2x2 crossover design.
Note that this formula is given for the calculation of the inter-subject variance within ANOVA (Proc GLM).

—
Regards,

Detlew

ElMaestro
★★★

Denmark,
2011-09-18 16:14
(4832 d 06:37 ago)

@ Ben
Posting: # 7368
Views: 12,531

Inter and Total Variability

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Hi Ben,

❝ 2. I'm trying to get the degrees of freedom for the following design: one single group, fixed sequence, uncontrolled with respect to time effects (intra-subject design), say n subjects receive k days treatment A, then another k days they receive treatment A and B. More mathematically

❝ log(response) = overall mean + subject + trt effect + error.

❝

❝ Now, consider the following approach. We have 2n values, we lose n-1 df because of the subjects, 2-1 df because of treatment effects and the "usual" -1 because of the overall mean, hence the degrees of freedom is n-1.

❝ I'm wondering whether this is correct or not. Let's say we use the same approach for a 2x2 cross-over design, that is,

❝ log(response) = overall mean + sequence + subject + period + trt effect + error,

❝

❝ then we end up with (assume n₁ = n₂ = n/2) 2n - (n-1) - (2-1) - (2-1) - (2-1) - 1 = n-3.

❝ But we should have n-2 (don't we?) and hence the approach may not be correct at all...?

I think....
In the 2,2,2-BE example you know the sequence if you know the subject's period and treatment coding of the model matrix (or vice versa). Thus you need to add one df in the equation above to get n-2.
For your model, assuming you code A and A+B as two individual factor levels, you might say if there's a tick for A then there's no tick for A+B and vice versa, so loss of 2-1 df here. Then df=n-1 looks right to me.

—
Pass or fail!
ElMaestro

Ben ★ 2011-09-19 21:43 (4831 d 01:07 ago) @ ElMaestro Posting: # 7371 Views: 12,362	Inter and Total Variability Post reply
	Thanks ElMaestro, that does sound reasonable!