> Date: Thu, 16 Jul 1998 19:08:03 +1000 (GMT+1000)
> From: Vincenzo Matassa <s185152@student.uq.edu.au>
> To: asreml@ram.chiswick.anprod.csiro.au
> Subject: Hmmm!!!
> MIME-Version: 1.0
>
> Dear Arthur
> Sorry but I need to ask.....
>
> Okay, say I fit a simple Mixed Model
>
> Yield ~ mu !r genotypes
>
> Now shouldn't the estimate of the ERROR Variance comp. be the same as the
> variance of the Residuals. (in the .sln file)
>
> i.e var(Residuals).
NO (except in the fixed model case)
The residuals are e=y-XB-Zu
But Error SS is y'Py = y'(y-XB)
so the SS of residuals e'e is only the same thing
if there are NO random effects (No u) and independent uniform errors
[I use ^ to represent the inverse i.e. ^{-1} in latex
when e'e = y'(I - X(X'X)^X) (I-X(X'X)^X) y
= y'(I - X(X'X)^X)y since the middle term is idempotent in
this case
= y'(y-XB)
In the more general case,
e'e = y'(V^ - V^X(X'V^X)^X'V^)(V^ - V^X(X'V^X)^X'V^)y
It needs to be e'Ve to reduce to y'(y-XB)
COnsider an example. The midsow data has 3 reps of 23 varities.
Fitting y ~ var gives
LogL=-53.2460 S2= 2.151 46 df 1.00000
with
S> sum(MIDBLUE$Res ^2)
[1] 98.929
Fitting y ~ mu !r var gives
LogL=-89.1171 S2= 4.366 68 df 0.10000 1.00000
LogL=-83.4606 S2= 3.032 68 df 0.46659 1.00000
LogL=-81.0803 S2= 2.374 68 df 1.03988 1.00000
LogL=-80.7901 S2= 2.179 68 df 1.40776 1.00000
LogL=-80.7839 S2= 2.151 68 df 1.47809 1.00000
S> sum(MIDBLUP$Res ^2)
[1] 107.64
The BLUE variety effects are mu + (Yi.-3mu) / 3
The BLUP variety effects are mu + (Yi.-3mu)/(3+1/1.478)
So the devisor is changed from 3 to 3.6766
Thus, the residuals are increased hence their increased sum of squares.
Can we see what this increase is.
The reduction in variety effects is .6766/3.6766 = .184
Squaring to the SS scale gives .033867
So the Variety SS will decrease by a factor of .966133
The FIXED variety SS can be calculated as
22 * EMS (1 + k gamma) = 22 * 2.151 * (1 + 3x1.4781)
= 119.55 * 2.151
So the e'e will increase 0.033867 * 119.55 * 2.151 = 8.709
which is the difference in SS of residuals 107.64 - 98.93)
I hope this is enough to convince you that
you cannot estimate the error variance easily from the
sum of squares of residuals except in the fixed model with IID errors
Arthur
>
> Many thanks again Arthur.
>
> Kind Regards
>
> Vince
>
> Vince Matassa
> Department Of Agriculture
> BIOMETRICS SECTION
> University Of Queensland
> Brisbane 4072
> Australia
>
>
<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
Arthur Gilmour PhD email: Arthur.Gilmour@agric.nsw.gov.au
Senior Research Scientist (Biometrics) fax: <61> 2 6391 3899
NSW Agriculture <61> 2 6391 3922
Orange Agricultural Institute telephone work: <61> 2 6391 3815
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