RE: BOV
From: "Liang Zhao"
Subject: RE: [NMusers] BOV
Date: Thu, September 23, 2004 1:03 pm
Hello Qi,
Please let me explain things in conjunction with Yaning's excellent
derivation shown at http://www.geocities.com/wangyaning2004/unequalbov.pdf.
Here (Omega_BSV, Omega_BOVj)=function(MSE, MSB, MSO, appropriate
degree of freedoms)=function(CLij, CLi.avg, CL..avg, d.f.s). When the
derivation makes expectations of MSE CLij, CLi., CL.j, Omega_BSV, Omega_BOVj,
it actually assuming that the average of CLi.avghat is 0 and CL.javghat is 0,
otherwise the estimation (biased or unbiased) for those values can never be
possible. You will NEVER EVER know or estimate what is the true CLhat because
it has been always confounded with BSV or BOV or whatever other random things
without making the above assumptions by whatever routes of theoretical proof
or derivation. This has not been stated explicitly in stat books, but please
see through this.
Since the original question is whether or not Omega_BSV and Omega_BOV are
estimable, I am just using the solvability of equation system as a shortcut
way to facilitate the thinking. Now I am fully convinced that even the design
is not a balanced design, Omega_BSV and Omega_BOV will still be estimable.
Since (1) the equation system will still be solvable and (2) you can use less
degree of freedom to estimate Omega_BSV and Omega_BOV in conjunction with
Yaning's derivation. A fractional balanced design to reduced the number of
runs is very possible.
As for the orthogonal projection, yep, the goal is to minimize the orthogonal
distances from the data points to the fitted line. However, the space dimensions
will be the number of subjects in the trial rather than the response variables to
the fitted line. It had taken me some time to establish that view. It is necessary
for the sum of weighted residual distance to be 0 under the assumption of normal
distribution. If I am making a statement of this, I am pretty sure it is the case
for the intraindividual variabilities. As far as interindividual variability is
concerned, my intuition tells me that is true but you can challenge me with
your proof.
It is a very stimulating discussion and I do have learned quite a lot. Please
forgive me if I am not strict statistical terms.
Liang Zhao PhD
Division of Pharmaceutics
The Ohio State Univ.