Time as both fixed and random term

Hi Lionel & List,

An easy-to-implement approach estimating overall time trends (i.e.,  
including Time as a fixed effect) while accounting for deviations from  
this trend for each plot could be to include random Time-by-Plot trends.  
This will result in the 'right' degrees of freedom for the overall time  
trend (have you thought about possible Treatment-by-Time interactions?) so  
statement ii) may be true if the Time trend varies among Plots or if you  
want to account for your study design in terms of Time. Possible meanings  
of including "Time" as both a fixed and a random term was just recently  
discussed by the list, but I think here you actually refer to having a  
random Time-by-Plot interaction term in your model.

Nevertheless, you can model this potential within-plot across-time data  
correlation by many other different ways, depending on your data. The  
above-mentioned random coefficient model (fitting random Time trends for  
each Plot) is only one way and your data may fit other covariance models  
better (e.g., when treating Time as a factor: ar1, us, ante, etc). Maybe  
it is best you check a book on time series models to get a better overview  
what is possible and how to decide on an adequate covariance structure for  
your data.
Not all of the many possible covariance structures can be fitted in lme4,  
nlme may be more flexible.

One of the most complicated covariance structures (that needs loads of  
data) to start with would be:
Biomass ~ Treatment + Time + (factor(Time)|Plot)

One of the least complicated would be:
Biomass ~ Treatment + Time + (Time|Plot)

Hope this helps,
Paul

Dear List,

In my work we usually deals with measures sampled repeatedly on  
experimental units over several time points and with specific  
treatments. For example we sampled plant biomass on 100 experimental  
plots at 5 different time point (say season or year). Some people argue  
that in this context we should model time as both a fixed effect term  
(as continuous variable) and random effect term in order to compute the  
correct numbers of degrees of freedom to test our treatment effects  
(usually considered as a continuous variables).

This is how such a model would look like:

Biomass ~ Treatment + Time + (1|Plot) + (1|Time)

In my experience having the same term has both fixed and random results  
in very low estimated standard deviation for the random term, which  
makes me skeptical about this approach. But having very little knowledge  
about how to correctly estimate the numbers of degrees of freedom I  
would like to ask you:

(i) if such a model makes sense,
(ii) if the argument "we need to have time as both fixed and random term  
to get the correct number of degrees of freedom" is valid
(iii) if such an alternative model: "Biomass ~ Treatment + Time +  
(1|Plot)" would be more appropriate.

Thanks for your input,
Lionel

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Paul Debes
DFG Research Fellow
University of Turku
Department of Biology
It?inen Pitk?katu 4
20520 Turku
Finland