I measured nitrate concentration and primary production (PP)  biweekly for 
23 months in one headwater stream.  I would like to use linear regression 
to determine if PP is related to nitrate concentration.  My dataframe is 
called "data" and consists of the vectors Rdate, PP, and nitrate.  Rdate 
is the observation date in class "date" and PP is primary production.

I first fit a linear model with uncorrelated residuals using gls:
M.lm <- gls (PP  ~ nitrate, data=data)

I then looked for evidence of auto-correlation using acf:
acf (residuals ( M.lm, type = 'normalized'))

The plot showed a clear pattern, so I reran the gls using an AR-1 
auto-correlation structure:
M.AR1 <-  gls (PP  ~ nitrate, correlation = corAR1 (form =~Rdate), 
data=data)

I then compared the AIC values of the M.lm and M.AR1:
AIC(M.lm, M.AR1)

M.AR1 had a much lower AIC score than M.lm, suggesting that it is the 
better model. 

I then inspected a plot of the standardized residuals against the fitted 
values for M.AR1:
plot (M.AR1)

This plot shows a pattern of decreasing residual spread with increasing 
nitrate concentration, a clear indication on non-homogeneity.

My questions are:
Is the M.AR1 model appropriate for the question I am trying to answer? All 
examples of AR-1 residual structure I could find are used in the context 
of testing for a time effect.  I am not interested in a time effect.  Is 
is still appropriate to use the AR1 auto-correlation structure?
If M.AR1 is OK, should I be worried about non-homogeneity in the 
residuals?  If so, how should I go about dealing with it?  Could I add a 
new variance structure to the model (e.g., varExp).
The approach outlined here is drawn from my interpretation of chapters 4 
and 6 of Zuur et al. 2009.  Mixed effects models and extensions in ecology 
with R.

Thanks,
Jake

Research Ecologist
US EPA
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