Re: Spatially adjusted residuals
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Re: Spatially adjusted residuals

> Date: Fri, 18 Dec 1998 12:08:44 +0800

> From: "N.W. Galwey" <>
> Subject: Spatially adjusted residuals
> Dear Arthur,
>         I think I now understand what the nugget variance is and what a
> units term in the linear model does, and it is not what I want.   What I'm
> after, expressed in the terms of Equation 4.1 on p. 20 of "Spatial Analysis
> of Field Experiments" (Cullis, Gilmour, Frensham and Verbyla, 1998), is to
> obtain, for each observation, the components zeta and eta (I hope my Greek
> is correct) of the residual value e.

In ASREML, Zeta (the spatially dependent random error vector)
                 is the Residual (in the .sln file) and
                 is estimated with AR.AR autocorrelation.
           Eta is the 'units' term in the .sln file - independent errors

There is a possible TRAP here.  After writing the .sln file
but before doing the residual statistics graphics, 
  ASREML SOMETIMES does the residual statistics graphics on Zeta+Eta
  rather than on Zeta.  
  It does this when  'units' is in the model 
                     the data is presented in sorted order and
                                data sorting is not specified 
                                in the structure lines
                                e.g.  row 0 AR .1  rather than  row row AR .1
                     one or two dimensions.
  A message appears in the .asr file of this has happenened.
  If it has not happened, the variogram of Zeta (when Eta also fitted)
  will look particularly smooth
  (i.e.  Zeta and Eta are calculated separately and the variogram is 
  only of Zeta)
  ASREML does the addition of Eta + Zeta when it can for the purposes of
  the residual graphics because  the error is then comparable to the
  Zeta calculated when Eta (= units) is not fitted.

>         I think you may have told me at some point that this can't be done
> in ASREML, but I'm not sure whether I've explained the reason why I want to
> do it.   My intention can be most simply illustrated in the case of a
> randomised block design with one treatment factor.   The model is
> x(ij) = mu + t(i) + r(j) + e(ij) .
> I can present the treatment means in a bar chart with a standard error bar.
> (Not a display that I approve of, but one that scientists are familiar
> with.)   This is analogous to presenting a confidence band around the
> regression line for the response to a quantitative variable.   But the
> standard error of the treatment means will reflect only the variance of
> e(ij), not that of r(j).   
>         Now, the sceptical scientist might like me to add the individual
> data points to my display.   But if I plot the actual values x(ij), these
> will seem absurdly widely 
   [ONLY if there are HUGE r(j)]
                            scattered relative to the standard error bar, due
> to the contribution of r(j).   It seems to me that the values I should plot 
> mu + t(i) + e(ij)
> that is,
> interesting part + component of error that interferes with comparisons between
>                    interesting parts.
>         Returning to Equation 4.1, if all fixed effects are interesting and
> if the vector of parameters u is partitioned into interesting random effects
> (e.g. variety effects) and uninteresting block effects, then the equation
> becomes
> y = X*tau + Z'u(interesting) + Z"u(block) + zeta + eta
> and it seems to me that the values I should plot around my fitted spline are
> y = X*tau + Z'u(interesting) + eta
>         Do you agree, and if so can you help?

Unfortunately, I do not think it is as easy as that.  
Are we interested in the confidence about a mean, or th SE of a contrast!

The actual data values show the real spread of values and I take it
that they are therefore of interest.  It is only when comparing means
that you can subtract off  common effects.

I can however partially accept your argument about subracting off BLOCK
effects [especially if largely orthogonal to treatment] and fixed covariates 
(including say lin(row) say)
effects but in typical cases, cannot see that subracting out Zeta is
helpful.  The main problem I have is that the partition of the error into
Zeta and Eta is really based on extrapolation and modelling.
  What we have as base statistics is
the Variance and the Lagx Covariances.  Based on the smoothed relationship
of the covariances to  p  p^2  p^3   p^4  ...  we estimate  the scale
of Zeta,  and calculate the scale of Eta by difference.  [ using REML as
the optimimization method].  Consequently, since the partition of the
error into Zeta and Eta is so model dependent, I would not be
comfortable using just one component for adjustment.  

I am more comfortable using Zeta to plot the trend surface because it
can be regarded as a smoothed residual.

My recommendation would be to plot the actual data.  

I'll circulate this to the list becasue I would be interesed in hearing
other points of view.

> Merry Christmas,

Thanks.  and a Happy 1999.
> Nick
> _____________________________________________________________________
> N.W. Galwey,
> Faculty of Agriculture,
> University of Western Australia,
> Nedlands, WA 6709, Australia.
> Tel.: +61 9 380 1959 (direct line)
>       +61 9 380 2554 (switchboard)
> Fax:  +61 9 380 1108

Arthur Gilmour PhD                    email:
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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