Hi

I would like some advice in selecting and interpreting a potentially complex random effects structure.

I have repeated estimates of growth (up to 41 per fish) for 1086 fish (ID). These fish were born at different times (YOB= 33 levels), and therefore their growth spans different years (year= 46, accounts for random year to year variation not accounted for by main effects). I would like to explore the importance of intrinsic (age) and extrinsic (temp, medianDF, P10,P90,annSOI) effects on growth variation.

The way I see it, my random effects structure could include ID, ID nested within YOB, and crossed with year. It is also possible (and biologically plausible) that random slopes for age could be fitted to each of these (i.e. effect of age varies by ID, YOB or even year). This results in potentially quite a lot of different random effects structures to test (18), and it just so happens that the most complex (age|ID)+(age|YOB)+(age|year) fits best by quite a way (see below)! However, when I look at the summary of model (r), the intercept and slope for both YOB and year are highly negatively correlated (below). This pattern remains even when I simplify the fixed effects structure using model selection. I suspect that model (r) is overfitting (are there other diagnostics or rules of thumb for random effects correlation?). If so, how can I reconcile this with the results of the random effect structure selection? I thought to fit interactions between age and each of my extrinsic effects to potentially account for any extra age related variance not captured by the original fixed effects, but end up with the same complex random effects structure

Also, as an aside- if a variable is transformed in the fixed effects side (e.g. log(age) in my model), does it need to be entered in its transformed state as a random slope (e.g. (log(age)|ID) vs. (age|ID))? I haven't been able to find a discussion of this (most examples use untransformed fixed effects) and when I fit the model with transformed and untransformed random slopes, I get different results for both random and fixed effect estimates.

Kind regards

John

> anova(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r)##best is r
Data: epgrowth
Models:
b: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID)
d: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (1 | year)
f: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (1 | YOB)
a: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID)
j: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (1 | year) + (1 | YOB)
c: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (1 | year)
e: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (1 | YOB)
h: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (log(age) | YOB)
n: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (log(age) | year)
i: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (1 | year) + (1 | YOB)
l: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (1 | year) + (log(age) | YOB)
p: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (log(age) | year) + (1 | YOB)
g: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (log(age) | YOB)
m: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (log(age) | year)
k: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (1 | year) + (log(age) | YOB)
o: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (log(age) | year) + (1 | YOB)
q: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (1 | ID) + (log(age) | year) + (log(age) | YOB)
r: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI + (log(age) | ID) + (log(age) | year) + (log(age) | YOB)
  Df      AIC      BIC logLik    Chisq Chi Df Pr(>Chisq)
b  9  -6259.4  -6190.8 3138.7
d 10 -10519.6 -10443.5 5269.8 4262.243      1     <2e-16 ***
f 10  -6263.4  -6187.3 3141.7    0.000      0          1
a 11  -7681.1  -7597.3 3851.5 1419.643      1     <2e-16 ***
j 11 -11013.0 -10929.2 5517.5 3331.924      0     <2e-16 ***
c 12 -11267.2 -11175.8 5645.6  256.146      1     <2e-16 ***
e 12  -7685.3  -7593.9 3854.6    0.000      0          1
h 12  -8168.8  -8077.4 4096.4  483.486      0     <2e-16 ***
n 12 -10838.0 -10746.6 5431.0 2669.233      0     <2e-16 ***
i 13 -11674.1 -11575.1 5850.0  838.092      1     <2e-16 ***
l 13 -11577.5 -11478.5 5801.7    0.000      0          1
p 13 -11601.1 -11502.1 5813.6   23.628      0     <2e-16 ***
g 14  -8351.8  -8245.1 4189.9    0.000      1          1
m 14 -11549.7 -11443.0 5788.8 3197.916      0     <2e-16 ***
k 15 -12000.6 -11886.3 6015.3  452.889      1     <2e-16 ***
o 15 -12186.8 -12072.5 6108.4  186.234      0     <2e-16 ***
q 15 -12173.9 -12059.6 6101.9    0.000      0          1
r 17 -12625.4 -12495.9 6329.7  455.509      2     <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1



> summary(r)
Linear mixed model fit by REML
Formula: log(growth) ~ log(age) + temp + medianDF + P10 + P90 + annSOI +      (log(age) | ID) + (log(age) | year) + (log(age) | YOB)
   Data: epgrowth
    AIC    BIC logLik deviance REMLdev
-12562 -12432   6298   -12659  -12596
Random effects:
Groups   Name        Variance  Std.Dev. Corr
 ID       (Intercept) 0.0574019 0.239587
          log(age)    0.0043767 0.066157 -0.412
 year     (Intercept) 0.4889119 0.699222
          log(age)    0.1043839 0.323085 -0.973
 YOB      (Intercept) 0.6581276 0.811251
          log(age)    0.1312008 0.362216 -0.975
 Residual             0.0177149 0.133097
Number of obs: 15014, groups: ID, 1086; year, 46; YOB, 33

Fixed effects:
              Estimate Std. Error t value
(Intercept) -4.0159786  1.2443690  -3.227
log(age)    -0.8895772  0.0809892 -10.984
temp         0.2248723  0.0805410   2.792
medianDF    -0.0002813  0.0007255  -0.388
P10          0.0001387  0.0001214   1.142
P90         -0.0004180  0.0009267  -0.451
annSOI      -0.0060916  0.0044218  -1.378

Correlation of Fixed Effects:
         (Intr) log(g) temp   mednDF P10    P90
log(age) -0.115
temp     -0.988 -0.026
medianDF  0.187  0.016 -0.210
P10      -0.179 -0.021  0.194 -0.871
P90      -0.551  0.008  0.539 -0.695  0.491
annSOI    0.408  0.011 -0.394  0.090 -0.211 -0.270




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