[R-sig-ME] heteroscedastic non-linear model with crossed random effects

[Ben Bolker]

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On 02/18/2011 03:46 PM, LE Street wrote:
> Dear all
> 
> I am trying to model the relationship between the leaf area (lai) and
> total foliar nitrogen (tfn) in vegetation plots, in order to understand
> the sources of variation in tfn across sites and vegetation types.
> 
> lai and tfn was measured on each plot once, 300 plots in total, across 5
> sites
> and 4 vegetation types.
> 
> 'Site' is therefore a factor from 1 to 5 and 'veg_type' is a factor from
> 1 to 4.
> 
> The theoretical relationship between lai and tfn is non-linear, of the
> form:
> 
> tfn = (No/g)*(1- exp(-g*lai))
> 
> where No and g are biologically meaningful parameters.
> 
> The most appropriate random effects structure for the model (I think) is
> to have crossed factors (vegetation types 1 to 4 all occurring at sites
> 1 to 5). The data are heteroscedastic with the variance of residuals
> increasing with the fitted values of tfn (though not for all groups).
> 
> My question is:
> 
> Is it possible to incorporate crossed factors in nlme? If so how?
> 
> Or, is it possible to incorporate the heteroscedasticity in nlmer? If so
> how?
> 
> I hope I've explained the problem clearly. I can find similar questions
> in the archives, but struggling to find a solution to this particular
> problem.

   The short answers are (1) yes, but not very easily; you will have to
dig into section 4.2 of Pinheiro and Bates for the answers (esp. see p.
163, "Cell Culture Bioassay with Crossed Random Effects"); (2) no.

 My bigger question for you is: why are you treating veg_type as a
random effect?  It would seem dicey on numerical grounds (estimating a
variance from 4 points is difficult), on philosophical/inferential
grounds (do you really think you can extrapolate to the population of
all vegetation types by knowing the variance estimated from four of
them?), and on more general biological grounds (I would normally guess
that you'd be more interested in the behavior of particular vegetation
types than in just the variance in their parameters).

  I can appreciate that you may want to "quantify the sources of
variation", but it would seem to make more sense to me to do this in the
general sense by estimating parameters for each type than in the narrow
sense by estimating variance in parameters across veg types.

  *If* you treat veg type as fixed then you don't have to deal with
crossed random effects.

  Alternatively, if log-transforming the data made sense you might be
able to handle your heteroscedasticity that way.

  Ben Bolker



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