N(μ, σ²) [Software]
Hi Yong,
No it isn’t! Back to the basics. The model of a 2×2×2 crossover assumes [sic] IID. For log-transformed data:
\(\log{(\mu_T / \mu_R)} = \mu_T - \mu_R\), which is estimated by the difference of the LSMs of log-transformed data \(\bar{x}_T - \bar{x}_R\). Now it gets interesting (i.e., the assumption!): In the balanced case for simplicity (where n1 = n2 and n = n1 + n2), \(\bar{x}_T - \bar{x}_R\) follows a normal distribution \(N\left ( \log{(\mu_T / \mu_R)}, 2\sigma^2 / n \right )\).
Since σ2 is unknown, it is estimated by the MSE from ANOVA. Then we can estimate \(CV = \sqrt{e^{MSE} - 1}\). Do you see \(\bar{x}_R\) in this derivation? I don’t. Remember that the normal distribution is described by two parameters, μ and σ2, which are independent. If you are interested in the variance component, please leave the mean(s) completely out of it (as it is correctly done in PHX/WNL for log-transformed data).
I think that for untransformed data dividing MSE by LSMR goes back to Kem Phillips, who wrote*
❝ ❝ ❝ I prefer the following formula :
❝ ❝ ❝ intersubject CV = sqrt(Var(Sequence*Subject)) / RefLSM
❝ ❝ Why?
❝ Because this is also used for ln_transformed data.
No it isn’t! Back to the basics. The model of a 2×2×2 crossover assumes [sic] IID. For log-transformed data:
\(\log{(\mu_T / \mu_R)} = \mu_T - \mu_R\), which is estimated by the difference of the LSMs of log-transformed data \(\bar{x}_T - \bar{x}_R\). Now it gets interesting (i.e., the assumption!): In the balanced case for simplicity (where n1 = n2 and n = n1 + n2), \(\bar{x}_T - \bar{x}_R\) follows a normal distribution \(N\left ( \log{(\mu_T / \mu_R)}, 2\sigma^2 / n \right )\).
Since σ2 is unknown, it is estimated by the MSE from ANOVA. Then we can estimate \(CV = \sqrt{e^{MSE} - 1}\). Do you see \(\bar{x}_R\) in this derivation? I don’t. Remember that the normal distribution is described by two parameters, μ and σ2, which are independent. If you are interested in the variance component, please leave the mean(s) completely out of it (as it is correctly done in PHX/WNL for log-transformed data).
❝ Like one formula for intrasubject CV for both transformed and none transformed data, I think this formula is generally accepted both for transformed and none transformed data.
I think that for untransformed data dividing MSE by LSMR goes back to Kem Phillips, who wrote*
“… σ being expressed as a percentage of a reference mean, that is, as a coefficient of variation; values of the difference in means are expressed as percentages of the same reference mean.”
That’s unfortunate and IMHO, not correct at all (Wolfgang Pauli would say: “That is not only not right; it is not even wrong!”). Again: The mean and variance are independent. So, the more I think about it: Even my idea of using the weighted global mean does not make sense. Paraphrasing Stephen Senn: Proving that apples are oranges by comparing the weight.- Phillips KM. Power of the Two One-Sides Tests Procedure in Bioequivalence. J Pharmacokinet Biopharm. 1990; 18(2): 137–44. doi:10.1007/BF01063556.
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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
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
Complete thread:
- Lack of IntersubjectCV in PHX yicaoting 2017-09-01 11:09
- CVs for untransformed data mittyri 2017-09-02 15:46
- CVs for untransformed data Helmut 2017-09-02 15:57
- CVs for untransformed data yicaoting 2017-09-02 17:50
- CVs for untransformed data Helmut 2017-09-02 19:20
- CVs for untransformed data yicaoting 2017-09-02 23:26
- N(μ, σ²)Helmut 2017-09-04 13:32
- N(μ, σ²) yicaoting 2017-09-05 02:46
- N(μ, σ²)Helmut 2017-09-04 13:32
- CVs for untransformed data yicaoting 2017-09-02 23:26
- CVs for untransformed data Helmut 2017-09-02 19:20
- CVs for untransformed data yicaoting 2017-09-02 17:50
- CVs for untransformed data yicaoting 2017-09-03 08:19
- CVs for untransformed data mittyri 2017-09-03 08:41
- CVs for untransformed data yicaoting 2017-09-04 11:35
- CVs for untransformed data mittyri 2017-09-03 08:41
- CVs for untransformed data Helmut 2017-09-02 15:57
- CVs for untransformed data mittyri 2017-09-02 15:46