[R-sig-ME] How to get estimates of time (Age) as well as Lots in a mixed model

[Robert A. LaBudde]

I have a small dataset that represents measurements across Lots of a 
processed food, where each lot has a particular Age since manufacturer.

The object of the study is twofold: 1) Determine the Lot-Lot random 
effect size, and 2) determine the degradation slope with Age.

Unfortunately, there is only one unique Lot for each particular 
unique Age. (A Lot is all manufacture from a particular date.) So the 
Lot effect is confounded to some extent with the Age effect.

My question is: What is the best way to get estimate of the two 
effects from this dataset? Should I use the Lot effect from a random 
effects model with no Age variable (so the Lot effect will include 
the Age degradation effect), or should I fit a mixed model anyway, 
and let the software sort them out (which it apparently does by 
dropping a level of Lots)?

I happen to be using the "nlme" package, so here is the example (I 
get a little larger Lot standard deviation when I include Age in the model):

 > blots<- read.table('pseudolots.txt', header=TRUE)
 > blots$ID<- factor(blots$ID)
 > blots$Unit<- factor(blots$Unit)
 > blots
    ID Lot Unit Age Conc
1   1   A    1   3 1.44
2   2   A    2   3 1.56
3   3   B    3  41 1.03
4   4   B    4  41 1.57
5   5   C    5 229 1.49
6   6   C    6 229 1.66
7   7   D    7 238 0.88
8   8   D    8 238 0.93
9   9   E    9 349 1.43
10 10   E   10 349 1.22
11 11   F   11 391 1.42
12 12   F   12 391 1.46
 > #Modeling
 > require('nlme')
Loading required package: nlme
 > fit1<- lme(Conc ~ Age, data=blots, random=~1|Lot)
 > summary(fit1)
Linear mixed-effects model fit by REML
  Data: blots
        AIC      BIC    logLik
   22.74858 23.95893 -7.374292

Random effects:
  Formula: ~1 | Lot
         (Intercept)  Residual
StdDev:   0.2320592 0.1786757

Fixed effects: Conc ~ Age
                  Value  Std.Error DF   t-value p-value
(Intercept)  1.3710359 0.18970706  6  7.227121  0.0004
Age         -0.0001449 0.00074846  4 -0.193538  0.8560
  Correlation:
     (Intr)
Age -0.823

Standardized Within-Group Residuals:
         Min          Q1         Med          Q3         Max
-1.59441923 -0.45471612 -0.06071212  0.52440223  1.42781630

Number of Observations: 12
Number of Groups: 6
 > anova(fit1)
             numDF denDF   F-value p-value
(Intercept)     1     6 154.51072  <.0001
Age             1     4   0.03746   0.856
 >
 > fit2<- lme(Conc ~ 1, data=blots, random=~1|Lot)
 > summary(fit2)
Linear mixed-effects model fit by REML
  Data: blots
        AIC     BIC    logLik
   8.122374 9.31606 -1.061187

Random effects:
  Formula: ~1 | Lot
         (Intercept)  Residual
StdDev:   0.2010265 0.1786757

Fixed effects: Conc ~ 1
                Value  Std.Error DF  t-value p-value
(Intercept) 1.340833 0.09693139  6 13.83281       0

Standardized Within-Group Residuals:
         Min          Q1         Med          Q3         Max
-1.57582754 -0.56625710 -0.01917496  0.56893404  1.44640790

Number of Observations: 12
Number of Groups: 6
 > anova(fit2)
             numDF denDF  F-value p-value
(Intercept)     1     6 191.3466  <.0001
================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

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