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*To*: s185152@student.uq.edu.au*Subject*: Re: e'e does not estimate error variance in mixed model*From*: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>*Date*: Fri, 17 Jul 1998 09:39:51 +1000 (EST)*Cc*: asreml@ram.chiswick.anprod.csiro.au*Reply-To*: Arthur Gilmour <gilmoua@ornsun.agric.nsw.gov.au>*Sender*: owner-asreml@chiswick.anprod.csiro.au

> 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 Forest Rd, ORANGE, 2800, AUSTRALIA home: <61> 2 6362 0046 ASREML is currently free by anonymous ftp from pub/aar on ftp.res.bbsrc.ac.uk Point your web browser at ftp://ftp.res.bbsrc.ac.uk/pub/aar/ in the IACR-Rothamsted information system http://www.res.bbsrc.ac.uk/ To join the asreml discussion list, send the message subscribe to asreml-request@chiswick.anprod.CSIRO.au The address for messages to the list is asreml@chiswick.anprod.CSIRO.au <> <> <> <> <> <> <> "The men marveled, saying 'Who can this be, that even the winds and the sea obey Him?'" Matthew 8:27 <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>