Hello all,

I am trying to use lme to examine how a response variable (Chla) changes 
over time in different treatments (2 Temp & 2 Light levels).  Within each 
treatment combination, there are two replicate tanks (each with unique 
TankID) with coral fragments in them.  All tanks are subject to the same 
environment until Time=0, when treatments are imposed, and Chla is measured 
for each tank at six times, including Time 0 just as the experiment 
commences. The model is:

Chla.1 <- lme(Chla ~ Temp*Light* Time - Temp*Light, random = ~1 | TankID, 
method="ML")

The reasoning here is that each tank’s intercept (Chla at Time 0) is a 
random draw from a common distribution regardless of treatment, but that 
the trend in Chla over time may vary among treatment combinations.  Based 
on the help files, two separate threads from the archives, and the Pinheiro 
and Bates nlme 3.0 manual, I became confused about which of two ways to 
check for a first-order temporal autocorrelation:

Chla.1b <- lme(Chla ~ Temp*Light* Time - Temp*Light, random = ~1 | TankID, 
corr = corCAR1(form = ~Time | TankID), method="ML")

Chla.1c <- lme(Chla ~ Temp*Light* Time - Temp*Light, random = ~1 | TankID, 
corr = corCAR1(form = ~1 | TankID), method="ML")

Comparing these fits with inspection of plot(ACF(chla.model1),alpha=0.05) 
suggests to me that there are problems with both of my attempts.  For the 
ACF plot, the correlation at lag 1 is about –0.3, and sticks out beyond the 
confidence limits.  By contrast, the two models' correlation parameters are 
not negative (phi = +0.13 and ~0 respectively), and the log-likelihood 
values are identical to the original model, suggesting no evidence of 
autocorrelation.  Our times are not equally spaced (they vary from 5-8 days 
apart), and I gather than ACF assumes they are, but my troubleshooting 
(summarized below) suggests to me that my problem is bigger than this.  I 
think I have not used corCAR1 properly, and am hoping someone can point me 
in the right direction.

Attempted troubleshooting:

1.  To check whether the discrepancy between ACF and the lme fits was due 
entirely to the unequal spacing of measurements, I created a bogus time 
variable (Time2) that was equally spaced (running from 0 to 5 in steps of 
1).  I then re-fit all of the above models with Time2 replacing Time in the 
function calls, and get the same kinds of problems (phi ~ 0 in the model 
fits, while ACF plot suggests a negative correlation at lag 1).

2.  Still using the bogus equally-spaced time variable, I replaced corCAR1 
with corAR1.  Now, the two different specifications of “form” yield 
identical parameter estimates and MLLs; the estimates of phi agree with 
those from the ACF plot; and the models actually do fit better than the 
equivalent model without autocorrelation.

Any advice would be greatly appreciated.

Sincerely,
Sean Connolly 
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