Re: Variance components analysis in random effects ANOVA with one factor

Carlo F. wrote: 
On Wed, 11 Jan 2006 21:22:21 +0200, Anon. wrote:


Thanks Bobfor your two messages.

My concern is exactly that: how to test that the assumptions holds ?
Particularly, I should somehow test that all the covariance between
the random effects ("tau"s) and between random "tau"s and
errors("epsilon"s) are zero. But how can I do it if I don't have any
experimental value for tau_i and epsilon_i,j ?

You can't. By the assumptions, E(epsilon_i.)=0, so tau_i is estmated assuming epsilon_i._bar=0 (this induces a covariance between the estimates, though).



Hi, bob.

So to recap, One way could be: 

1) by the moment method applied to the model I get y.._bar=mu;

Yes. This is also the ML solution, if ou have balanced data and assume normality.

2) On the ground of the weak law of large numbers I assume that the items
in each levels , call it n, are big enough that the difference between
E(epsilon) and epsilon_i._bar negligeable/ converge in probability. Or
again I could invoke the method of the moments not to estimate E(epsilon)
which I assumed equal zero, but "to force" epsilon_i._bar to zero. In this
way, I get, for each of the i random effects tau(i=1..a), n realisations
of the random effect tau_i, for each tau_i. tau_i_hat=y_i._bar -
y_.._bar_bar  (i=1..a);

Yes. It makes sense that a point estimate forces epsilon_i._bar to zero: of course it is unlikely to be precisely that, but you can calulate confidence limits to see how far out you could be.

3) At this point I can compute the residuals: e_ij=y_ij- tau_i_hat;

Yes.

4) Now I can :

4a) Am I correct saying the following? " At the i-th level of tau, It is
impossibile to test cov (tau_i, e_ij) =0. in fact, sample correlation
between tau_i_hat and e_ij does not mean anything because e_ij are a set
of single realisations of differrent random variables epsilon_ij
and not n realisations of a single random variable. " So, this assumption
will be not verified nor supported by any experimental evidence or
reasoning. Correct ?

Well, almost. The e_ij's are realisations of the same random variable, but the estimates depend on the estimate of tau_i. Alas, there's nothing you can do to change it.

4b) plot of the residuals versus run order and autocorrelogram to support
idea of independence of the N=n*a random variables epsilon_ij 

Yes, assuming run order makes some sort of sense scientifically.

4c) plot of the residuals versus levels and/or y_i_., that now, based on
1)+2) I can accept as "fitted" value. To see omoschedaschity and/or
patterns

Plot against levels. You will also be able to spot outliers. One trick is to "jitter" the Levels, i.e. add/subtract a little bit, so that you move the ponts apart but it is still clear which level they are in. Or use boxplots.

4d) some Test on equal sigmas of the residuals. Here, now I envisage
further doubts. but never mind.

Yes.  The plots might be enough to tell you what's going on.

4e) Normality: via Anderson Darling I rejected: so for further analyses
(confidence interval, F-test), I need to thing something different or, at
least for F-test invoke the robustness.

I would draw normal probability plots: they can show you what is going on, e.g. outliers or skewness. With enough data, you can get any test to be significant, but it may mean nothing in practice.

In box-hunter-hunter, there is a quite different approach: I'll try to mix
the two.

Have fun! There is no one algorithm for this sort of thing: there are a bunch of tools that you can use to help learn about the data. I then to favour the graphical ones because then I can see what the data is doing.

And what about the following:

Straight away from the model definition, it seems that the y_ij must be
with cov !=0 for same i or same j (see also Evgenii's paper). Thus, what
I've got at a certain level, say i', is that y_i'j with j=1..n are not
independant. 

True:it's called the intra-class correlation.

Therefore all the y_ij are not independant. And what if the
autocorrelogram via acf() in R seems to say autocorrelations all close to
zero ? How would you interpret it ? 

It would suggest that the tau_i's are all almost equal...

You should examine the autocorrelation of the residuals. The assumptions of independence are assumptions about the tau_i's and epsilon_ij's, not the y_ij's.

Bob

--
Bob O'Hara

Dept. of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax:  +358-9-191 51400
WWW:  http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: http://www.jnr-eeb.org

.

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