[R-sig-ME] longitudinal analysis of nested samples

[Giancarlo Sadoti]

Greetings list,

I'm helping a colleague with an analysis of spatially-clustered longitudinal data.  She is interested in testing for change in vigor (V) in a set of trees over time.

In each of 5 years (YR), 4 trees (TR) were measured per point, 10 points (PT) were measured along a transect. There may be two additional levels (transect and site), but let's just consider one site on one transect for now.  V is normally-distributed, fyi.

My (basic) understanding of longitudinal models suggests I use the following model formulation (using lme4):

>lmer(V ~ YR + (YR|TR), data=TREES)

Linear mixed model fit by REML 
Formula: V ~ YR + (YR | TR) 
   Data: TREES 
  AIC  BIC logLik deviance REMLdev
 1423 1449 -705.5     1415    1411
Random effects:
 Groups   Name        Variance   Std.Dev. Corr   
 TR       (Intercept) 0.00247886 0.049788        
          YR          0.00081403 0.028531 -0.065 
 Residual             0.71588684 0.846101        
Number of obs: 560, groups: TR, 1

Fixed effects:
             Estimate Std. Error t value
(Intercept) 206.17857   35.89736   5.744
YR           -0.10107    0.03367  -3.002

Correlation of Fixed Effects:
   (Intr)
YR -0.531

The model indicates a negative trend in vigor over time. However, this only addresses the first level in the hierarchy (TR), if I wanted to address within-point variance in vigor, would I simply add a random intercept for point (1|PT)? (see below) Likewise, if I had additional levels of sampling (e.g. transect [TR]), would adding (1|TR) be appropriate?

> lmer(V ~ YR + (YR|TR) + (1|PT), data=TREES)
Linear mixed model fit by REML 
Formula: V ~ YR + (YR | TR) + (1 | PT) 
   Data: TREES 
  AIC  BIC logLik deviance REMLdev
 1326 1356 -655.8     1309    1312
Random effects:
 Groups   Name        Variance   Std.Dev.  Corr  
 PT       (Intercept) 1.4817e-01 0.3849224       
 TR       (Intercept) 2.8743e-03 0.0536120       
          YR          1.2003e-06 0.0010956 0.271 
 Residual             5.7145e-01 0.7559429       
Number of obs: 560, groups: PT, 11; TR, 1

Fixed effects:
             Estimate Std. Error t value
(Intercept) 206.16368   32.07247   6.428
YR           -0.10107    0.01601  -6.313

Correlation of Fixed Effects:
   (Intr)
YR -0.998

It appears V varies between points (PT), and the model is much improved.  Is this the correct route?

Many thanks, and pardon me if this has been asked previously.

Giancarlo