From: Rolf Turner <r.turner_at_auckland.ac.nz>

Date: Tue, 13 May 2008 10:39:36 +1200

R-help_at_r-project.org mailing list

https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Mon 12 May 2008 - 22:44:56 GMT

Date: Tue, 13 May 2008 10:39:36 +1200

On 13/05/2008, at 4:09 AM, Douglas Bates wrote:

> I'm entering this discussion late so I may be discussing issues that

*> have already been addressed.
**>
**> As I understand it, Federico, you began by describing a model for data
**> in which two factors have a fixed set of levels and one factor has an
**> extensible, or "random", set of levels and you wanted to fit a model
**> that you described as
**>
**> y ~ effect1 * effect2 * effect3
**>
**> The problem is that this specification is not complete.
*

At *last* (as Owl said to Rabbit) we're getting somewhere!!!

I always knew that there was some basic fundamental point about this business that I (and I believe many others) were simply missing. But I could not for the life of me get anyone to explain to me what that point was. Or to put it another way, I was never able to frame a question that would illuminate just what it was that I wasn't getting. I now may be at a stage where I can start asking the right questions.

> An interaction of factors with fixed levels and a factor with random

*> levels can mean, in the lmer specification,
**>
**> lmer(y ~ effect1 * effect2 + (1| effect3) + (1|
**> effect1:effect2:effect3), ...)
**>
**> or
**>
**> lmer(y ~ effect1 * effect2 + (effect1*effect2 | effect3), ...)
**>
**> or other variations. When you specify a random effect or an random
**> interaction term you must, either explicitly or implicitly, specify
**> the form of the variance-covariance matrix associated with those
**> random effects.
**>
**> The "advantage" that other software may provide for you is that it
**> chooses the model for you but that, of course, means that you only
**> have the one choice.
*

Now may I start asking what I hope are questions that will lift the fog a bit? Let us for specificity consider a three-way model with two fixed effects and one random effect from the good old Rothamstead style agricultural experiment context: Suppose we have a number of species/breeds of wheat (say) and a number of fertilizers. These are fixed effects. And we have a number of fields (blocks) --- a random effect. Each breed-fertilizer combination is applied a number of times in each field. We ***assume*** that that the field or block effect is homogeneous throughout. This may or may not be a ``good'' assumption, but it's not completely ridiculous and would often be made in practice. And probably *was* made at Rothamstead. The response would be something like yield in bushels per acre. The way that I would write the ``full'' model for this setting, in mathematical style is: Y_ijkl = mu + alpha_i + beta_j + (alpha.beta)_ij + C_k + (alpha.C)_ik + (beta.C)_jk + (alpha.beta.C)_ijk + E_ijkl The alpha_i and beta_j are parameters corresponding to breed and fertilizer respectively; the C_k are random effects corresponding to fields or blocks. Any effect ``involving'' C is also random. The assumptions made by the Package-Which-Must-Not-Be-Named are (I think) that C_k ~ N(0,sigma_C^2) (alpha.C)_ik ~ N(0,sigma_aC^2) (beta.C)jk ~ N(0,sigma_bC^2) (alpha.beta.C)_ijk ~ N(0,sigma_abC^2) E_ijkl ~ N(0,sigma^2) and these random variables are *all independent*. Ahhhhhhhh ... perhaps I'm on the way to answering my own question. Is it this assumption of ``all independent'' which is questionable? It seemed innocent enough when I first learned about this stuff, lo these many years ago. But .... mmmmmaybe not! To start with: What would be the lmer syntax to fit the foregoing (possibly naive) model? I am sorry, but I really cannot get my head around the syntax of lmer model specification, and I've tried. I really have. Hard. I know I must be starting from the wrong place, but I haven't a clue as to what the *right* place to start from is. And if I'm in that boat, I will wager Euros to pretzels that there are others in it. I know that I'm not the brightest bulb in the chandelier, but I'm not the dullest either. Having got there: Presuming that I'm more-or-less on the right track in my foregoing conjecture that it's the over-simple dependence structure that is the problem with what's delivered by the Package-Which-Must- Not-Be-Named, how might one go about being less simple-minded? I.e. what might be some more realistic dependence structures, and how would one specify these in lmer? And how would one assess whether the assumed dependence structure gives a reasonable fit to the data?

> If you can describe how many variance components you think should be

*> estimated in your model and what they would represent then I think it
**> will be easier to describe how to fit the model.
*

How does this fit in with my conjecture (above) about what I've been missing all these years? Does it fit? How many variance components are there in the ``naive'' model? It looks like 5 to me ... but maybe I'm totally out to lunch in what I think I'm understanding at this stage. (And besides --- there are three sorts of statistician; those who can count, and those who can't.) Thank you for your indulgence. cheers, Rolf Turner ######################################################################Attention:\ This e-mail message is privileged and confid...{{dropped:9}}

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