[R] df pseudoreplication in lme model

[Lauren Meyer]

Hello, I am trying to assess weather or not my df are pseudoreplicated in my
lme model.

my study was undertaken on five fish (labeled PC) each tested in two
replicates(REP), across each combination of three treatments HOM, C18 and
CU, each of which had two levels; HOM(SON, BLD),C18 SML, BIG), CU (YES, NO).
The variable we are assessing is the amount of toxin extracted (TOX1). Also,
some data is missing, and has already been removed.

Here is the model I am using and output:
model<- lme(TOX1~HOM*C18*CU, random=~1|PC/REP, data=Data4, method="ML")
Linear mixed-effects model fit by maximum likelihood
 Data: Data4
      AIC      BIC    logLik
  220.603 244.5213 -99.30151

Random effects:
 Formula: ~1 | PC
        (Intercept)
StdDev:    1.574392

 Formula: ~1 | REP %in% PC
         (Intercept)  Residual
StdDev: 0.0001356862 0.9724221

Fixed effects: TOX1 ~ HOM * C18 * CU
                        Value Std.Error DF   t-value p-value
(Intercept)          3.729044 0.8204586 48  4.545073  0.0000
HOMSON               0.423330 0.5175211 48  0.817995  0.4174
C18SML              -1.160060 0.5475120 48 -2.118784  0.0393
CUYES                0.419067 0.4643966 48  0.902391  0.3714
HOMSON:C18SML       -0.645514 1.0385203 48 -0.621571  0.5372
HOMSON:CUYES        -0.436996 0.6953361 48 -0.628467  0.5327
C18SML:CUYES        -0.137128 0.7179371 48 -0.191003  0.8493
HOMSON:C18SML:CUYES  0.313720 1.2287607 48  0.255314  0.7996
 Correlation:
                    (Intr) HOMSON C18SML CUYES  HOMSON:C18SML HOMSON:CU
C18SML:
HOMSON              -0.254
C18SML              -0.240  0.361
CUYES               -0.283  0.449  0.424
HOMSON:C18SML        0.127 -0.472 -0.550 -0.224
HOMSON:CUYES         0.189 -0.744 -0.268 -0.668  0.351
C18SML:CUYES         0.183 -0.275 -0.763 -0.647  0.419         0.421
HOMSON:C18SML:CUYES -0.107  0.399  0.464  0.378 -0.845        -0.549
-0.599

Standardized Within-Group Residuals:
         Min           Q1          Med           Q3          Max
-4.090875567 -0.433368736 -0.007582723  0.498944076  2.603341469

Number of Observations: 65
Number of Groups:
         PC REP %in% PC
          5          10

As the three way interaction as well as all of the two way interactions were
deemed non-significant, I simplified the model, removing first the three way
interaction, then each two way interaction in turn, comparing each
subsequent model with the previous one using an ANOVA as per the example in
the R book on pg. 632. I have a final model of:

> model5<- lme(TOX1~HOM+C18+CU, random=~1|PC/REP, data=Data4, method="ML")
> summary(model5)
Linear mixed-effects model fit by maximum likelihood
 Data: Data4
       AIC      BIC   logLik
  214.0699 229.2906 -100.035

Random effects:
 Formula: ~1 | PC
        (Intercept)
StdDev:    1.567082

 Formula: ~1 | REP %in% PC
         (Intercept)  Residual
StdDev: 0.0005730032 0.9847228

Fixed effects: TOX1 ~ HOM + C18 + CU
                Value Std.Error DF   t-value p-value
(Intercept)  3.927801 0.7623505 52  5.152225  0.0000
HOMSON      -0.028203 0.2603204 52 -0.108341  0.9141
C18SML      -1.437095 0.2605651 52 -5.515302  0.0000
CUYES        0.214583 0.2675196 52  0.802122  0.4261
 Correlation:
       (Intr) HOMSON C18SML
HOMSON -0.125
C18SML -0.114  0.047
CUYES  -0.167 -0.152 -0.184

Standardized Within-Group Residuals:
         Min           Q1          Med           Q3          Max
-4.212407492 -0.433128656  0.003244622  0.618291014  2.578288257

Number of Observations: 65
Number of Groups:
         PC REP %in% PC
          5          10

However, I am unsure if these Df are pseudoreplicated and would like some
help in how to determine if this is the case. Thank you

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