Re: df with random effects

# Re: df with random effects

```
> Hello:
>
> In the example Ex11a, we have 3 lines and 9 sires,(actually the sires are
nested
> in the lines). There is 65 observations and 74 animals in total. The addition
of
> an animal effect or a sire effects does not change the degrees of freedom of
the
> error (62), which are 65-2-1. The test for lines will be with 2, 62 df with
any
> number of other random effects?.
>
> If we use Mixed of SAS with a sire model we got 56 df for the denominator
> for testing
> lines, which comes from deducting 2 df for lines, 9-3 = 6 df for sires/lines
> and 1 df for mu. All other things are identical.
>
> Could someone explain the difference in df's?.

For testing for line, one can argue for 6, 56, 62 or some intermediate value
as the appropriate error degrees of
freedom.

The value of 56  is obtained by treating sires as fixed in ASREML.
But if sirtes are then regarded as random, nested within lines, lines
should be tested against the 'sire' variance rather than against the
'residual' variance.

In general, it is not easy to work out the proper denominator
degrees of freedom for any test of fixed effects in a mixed model.

The EX11A example is sufficient to demonstrate the problem.

If we fit  AD ~ mu Line sire  we get the analysis of variance

SOurce    df     MS      F
Line       2    2227    16.81 [against Error], 6.8 against Sire
SIre       6     327.2   2.47
Error     56     132.5

If we equate the sire MS to its expection, we get a sire variance component
of about   (327.2 - 132.5)/7 = 27.8

From the mixed model fitting AD ~ mu Line !r sire

SOurce    df     MS      F
Line       2      ?    6.43   [ which agrees with the test against Sire above]

Error     62     132.4

The variance component for sires in this model is 27.2; similar to the ANOVA
estimate.  It is similar but not exact because the data is not fully
balanced and 7 is only approximately the average progeny per sire [65/9].

This is most obvious in a Split plot analysis where some components
would be tested against an 'Error A' and others against an 'Error B'.

I believe Kenward and ROger (1997) discuss this problem [Biometrics
53: 983-997].

I thought I had a discussion of this in the manual but apparantly not.

Essentially, only in a few well defined cases can we work out
precisely what the error degrees of freedom should be.  However,
we see in the example that ASREML does give the appropriate F statistic
[even if we do not know its proper distribution].

Some programs just quote a Wald statistic distributed as Chi-square but this
test
assumes the error variance is known [infinite df].  I therefore prefer
the F test where I can usually get some indication [ie use my knowledge
of the structure of the data to quess] of what the error
df should be.

I hope this helps.
>
> Thank you
>
> Hugo
>

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
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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