## total (pooled) variance: examples [Power / Sample Size]

Dear zizou,

thanks for the answers.

I have checked it now

I made up (Cmax) concentrations for 3x6x3 Williams design (with treatments A,B,C) with 18 subjects (complete dataset, all subjects completed all periods) and run it with Phoenix WNL.

Here are the results:

a)

Inter-subject CV given directly with mixed model approach = 13.4 %

Inter-subject CV calculated from ANOVA table with denominator 3 = 13.4 % (right)

Inter-subject CV calculated from ANOVA table with denominator 2 = 16.5 % (wrong)

So here when using (wrong) All at once approach, the number in the denominator should be the same as the total number of treatments (the number of all treatments we are evaluating)

b)

Inter-subject CV given directly with mixed model approach = 6.6 %

Inter-subject CV calculated from ANOVA table with denominator 3 = 5.4 % (wrong)

Inter-subject CV calculated from ANOVA table with denominator 2 = 6.6 % (right)

But here when using Two at a Time approach (where we are evaluating just 2 treatments), it seems that we have to use number 2 in the denominator. So it seems that the denominator is affected by whether the model is complete or incomplete.

c)

Here we should of course as above also probably use number 2 in the denominator but here interestingly something else happens:

Inter-subject CV given directly with mixed model approach = ERROR

Inter-subject CV calculated from ANOVA table with denominator 3 = ERROR

Inter-subject CV calculated from ANOVA table with denominator 2 = ERROR

Even when using Subject as fixed effect (which is the solution for such cases according to Certara) MSE (0.22973) is larger than MSS (0.20599), therefore inter-subject variance is negative (-0.0119; the same number is given directly with mixed model). Therefore the calculation of inter-subject CV is impossible (error).

Why is this so and what should be done in this case? This probably happens because intra-subject CV is so much higher than inter-subject CV? I know these data that I used is made up but nevertheless it could happen with real-world data as well?

thanks for the answers.

❝ No, for both approaches the denominator is 3. It's not affected by whether the model is complete or incomplete.** Nevertheless someone else could check it.** I read about it many years ago (mentioned here and statistical background somewhere else, I don't remember where). And "a long time ago in a galaxy far, far away" I also recalculated the estimate (obtained from ANOVA - GLM and equation with 3 as denominator) using the mixed model where the inter-subject variability is obtained more directly.

I have checked it now

I made up (Cmax) concentrations for 3x6x3 Williams design (with treatments A,B,C) with 18 subjects (complete dataset, all subjects completed all periods) and run it with Phoenix WNL.

Here are the results:

a)

**All at once approach**Inter-subject CV given directly with mixed model approach = 13.4 %

Inter-subject CV calculated from ANOVA table with denominator 3 = 13.4 % (right)

Inter-subject CV calculated from ANOVA table with denominator 2 = 16.5 % (wrong)

So here when using (wrong) All at once approach, the number in the denominator should be the same as the total number of treatments (the number of all treatments we are evaluating)

b)

**Two at a Time approach (incomplete block design); B vs. C**Inter-subject CV given directly with mixed model approach = 6.6 %

Inter-subject CV calculated from ANOVA table with denominator 3 = 5.4 % (wrong)

Inter-subject CV calculated from ANOVA table with denominator 2 = 6.6 % (right)

But here when using Two at a Time approach (where we are evaluating just 2 treatments), it seems that we have to use number 2 in the denominator. So it seems that the denominator is affected by whether the model is complete or incomplete.

c)

**Two at a Time approach (incomplete block design); A vs. C**Here we should of course as above also probably use number 2 in the denominator but here interestingly something else happens:

Inter-subject CV given directly with mixed model approach = ERROR

Inter-subject CV calculated from ANOVA table with denominator 3 = ERROR

Inter-subject CV calculated from ANOVA table with denominator 2 = ERROR

Even when using Subject as fixed effect (which is the solution for such cases according to Certara) MSE (0.22973) is larger than MSS (0.20599), therefore inter-subject variance is negative (-0.0119; the same number is given directly with mixed model). Therefore the calculation of inter-subject CV is impossible (error).

Why is this so and what should be done in this case? This probably happens because intra-subject CV is so much higher than inter-subject CV? I know these data that I used is made up but nevertheless it could happen with real-world data as well?

### Complete thread:

- Sample size estimation for parallel design Nav Coelho 2007-07-20 16:14 [Power / Sample Size]
- Parallel designs (total variance) Helmut 2007-07-20 16:29
- Parallel designs (total variance) Nav Coelho 2007-07-20 18:38
- CV-intra # 60% CV-inter Helmut 2007-07-22 01:17
- total (pooled) variance: examples Helmut 2007-07-23 17:44
- total (pooled) variance: examples Nav Coelho 2007-07-23 19:58
- total (pooled) variance: examples BEQool 2023-04-18 09:59
- Higher-Order Crossovers Helmut 2023-04-20 10:19
- Higher-Order Crossovers BEQool 2023-04-21 07:53

- total (pooled) variance: examples zizou 2023-04-21 21:23
- total (pooled) variance: examples BEQool 2023-04-22 18:20
- total (pooled) variance: examples zizou 2023-04-22 21:06
- total (pooled) variance: examplesBEQool 2023-04-25 10:57
- total (pooled) variance: examples zizou 2023-04-25 22:43
- total (pooled) variance: examples BEQool 2023-05-04 08:17

- total (pooled) variance: examples zizou 2023-04-25 22:43

- total (pooled) variance: examplesBEQool 2023-04-25 10:57

- total (pooled) variance: examples zizou 2023-04-22 21:06

- total (pooled) variance: examples BEQool 2023-04-22 18:20

- Higher-Order Crossovers Helmut 2023-04-20 10:19

- Parallel designs (total variance) Nav Coelho 2007-07-20 18:38
- Sample size estimation for parallel design Dipesh Jayswal 2007-07-24 08:52

- Parallel designs (total variance) Helmut 2007-07-20 16:29