Re: e'e does not estimate error variance in mixed model

# Re: e'e does not estimate error variance in mixed model

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> 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)
 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)
 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
Forest Rd, ORANGE, 2800, AUSTRALIA                    home: <61> 2 6362 0046

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