Appendix 1. ‘jags’ model code corresponding to the simple linear regression model (no taxonomic terms), an ANCOVA with species-level coefficients, and a three level taxonomic hierarchical model (species, family and order) with uncorrelated slope and intercept variance components.
###Linear regression (no taxonomic coefficients)
model{
int~dnorm(0, 0.001);
slope~dnorm(0, 0.001);
resid.sigma~dunif(0,100)
resid.tau<- 1/(resid.sigma*resid.sigma)
for (i in 1:n){
deln.obs[i]~dnorm(deln.pred[i], resid.tau)
deln.pred[i]<-int+slope*logwt[i]
e.y[i]<- deln.obs[i]-deln.pred[i]
}
}
###Species-level ANCOVA model
model{
##priors
resid.tau~dgamma(0.001,0.001)
resid.sd<-1/sqrt(resid.tau)
for(i in 1:numsp){
sp.int[i]~dnorm(0, 0.001);
sp.slope[i]~dnorm(0, 0.001);
}
for (i in 1:n){
deln.obs[i]~dnorm(deln.pred[i], resid.tau)
deln.pred[i]<-sp.int[species[i]]+sp.slope[species[i]]*logwt[i]
e.y[i]<- deln.obs[i]-deln.pred[i]
}
}
###Three level hierarchical model with uncorrelated random slope and intercept variances at each taxonomic level
model{
##priors
resid.tau~dgamma(0.001,0.001)
resid.sd<-1/sqrt(resid.tau)
int.global~dnorm(0,0.001)
slope.global~dnorm(0,0.001)
sp.slope.sd~dunif(0,10)
sp.slope.tau<-1/(sp.slope.sd*sp.slope.sd)
ord.slope.sd~dunif(0,10)
ord.slope.tau<-1/(ord.slope.sd*ord.slope.sd)
fam.slope.sd~dunif(0,10);
fam.slope.tau<-1/(fam.slope.sd*fam.slope.sd);
##generate random effects for each level
for(j in 1:numsp){
sp.slope[j]~dnorm(0,sp.slope.tau);
}
for(q in 1:numfam){
fam.slope[q]~dnorm(0,fam.int.tau);
}
for(k in 1:numord){
ord.slope[k]~dnorm(0,ord.slope.tau);
}
##model
for (i in 1:n){
int[i]<- int.global + sp.int[sp[i]] + fam.int[fam[i]] + ord.int[ordr [i]]
slope[i]<- (slope.global + sp.slope[sp[i]] + fam.slope[fam[i]] + ord.slope[ordr[i]] )*logwt[i]
deln.pred[i]<-int[i] + slope[i]
deln.obs[i]~dnorm(deln.pred[i], resid.tau)
e.y[i]<- deln.obs[i]-deln.pred[i]
}
}
Appendix 2. Taxonomic composition, sample size (n), and mean, standard deviation (S.D.), minimum, and maximum body mass (g) of fish sampled for nitrogen stable isotope analysis from Puget Sound, Washington, USA.
Order Family Species n Mean S.D. Min Max
Scorpaeniformes Cottidae Chitonotus pugetensis 20 20.5 10.4 4.3 41.5
Pleuronectiformes Paralichthyidae Citharichthys sordidus 18 139.3 99.8 9.6 339.4
Pleuronectiformes Paralichthyidae Citharichthys stigmaeus 10 4.9 3.0 2.5 12.9
Clupeiformes Clupeidae Clupea pallasii 16 66.9 53.6 22.2 233.1
Perciformes Embiotocidae Cymatogaster aggregata 26 19.5 24.0 4.1 106.7
Gadiformes Gadidae Gadus macrocephalus 4 2567 2521 290.7 5328.7
Pleuronectiformes Pleuronectidae Glyptocephalus zachirus 15 136.5 225.9 13.7 941.7
Scorpaeniformes Hexagrammidae Hexagrammos stelleri 8 70.2 51.0 20.0 186.9
Chimaeriformes Chimaeridae Hydrolagus colliei 32 519.1 413.5 3.2 1358
Pleuronectiformes Pleuronectidae Lepidopsetta bilineata 20 261.2 136.0 28.7 555.2
Scorpaeniformes Cottidae Leptocottus armatus 13 196.1 136.8 68.5 469.9
Perciformes Zoarcidae Lycodes pacificus 10 51.3 51.1 12.9 164.9
Pleuronectiformes Pleuronectidae Lyopsetta exilis 11 37.6 16.3 14.4 64.4
Gadiformes Merlucciidae Merluccius productus 23 317.1 408.3 7.3 1469
Gadiformes Gadidae Microgadus proximus 18 18.1 17.1 2.1 68.9
Pleuronectiformes Pleuronectidae Microstomus pacificus 9 245.6 191.9 24.5 510.2
Scorpaeniformes Cottidae Myoxocephalus polyacanthocephalus 8 169.3 169.8 7.0 548.1
Scorpaeniformes Hemitripteridae Nautichthys oculofasciatus 4 54.8 16.1 30.9 65.8
Pleuronectiformes Pleuronectidae Parophrys vetulus 34 149.9 146.6 3.7 508.7
Pleuronectiformes Pleuronectidae Pleuronichthys coenosus 12 159.8 62.0 43.8 267.8
Scorpaeniformes Agonidae Podothecus accipenserinus 14 27.8 14.2 4.9 57.0
Batrachoidiformes Batrachoididae Porichthys notatus 14 71.4 58.4 19.1 239.3
Pleuronectiformes Pleuronectidae Psettichthys melanostictus 7 216.5 136.5 94.8 450.8
Rajiformes Rajidae Raja binoculata 9 810.1 1080 25.3 3227
Rajiformes Rajidae Raja rhina 19 596.3 428.8 51.4 1355
Perciformes Embiotocidae Rhacochilus vacca 10 28.8 26.7 12.8 90.1
Scorpaeniformes Cottidae Scorpaenichthys marmoratus 4 9206 1448 7991 10861
Scorpaeniformes Scorpanidae Sebastes auriculatus 17 116.0 222.1 8.5 946.1
Scorpaeniformes Scorpanidae Sebastes maliger 17 366.1 227.5 6.8 844.8
Scorpaeniformes Scorpanidae Sebastes proriger 4 177.7 77.8 108.1 277.0
Scorpaeniformes Scorpanidae Sebastes ruberrimus 4 1692 1692 353.7 4006
Squaliformes Squalidae Squalus suckleyi 23 1263 805.7 46.1 2327
Gadiformes Gadidae Theragra chalcogramma 16 215.7 244.7 3.7 761.0
Scorpaeniformes Agonidae Xeneretmus latifrons 7 11.7 2.3 8.5 14.6
Scorpaeniformes Agonidae Xeneretmus triacanthus 4 17.4 3.4 13.4 20.5
Appendix 3. Estimated slope and intercept coefficient and their respective 95% HPD credible intervals from a taxonomically nested (species, family, order) hierarchical model of δ15N values from the Puget Sound demersal fish community using log10-transformed body mass (g).
95% HDP credible interval 95% HDP credible interval
Species Slope Lower CI Upper CI Intercept Lower CI Upper CI
Chitonotus pugetensis 0.624 0.060 1.139 14.304 14.959 13.743
Citharichthys sordidus 0.699 0.226 1.226 13.532 14.154 12.789
Citharichthys stigmaeus 0.653 0.125 1.239 13.501 13.825 13.068
Clupea pallasii 0.727 -0.120 1.481 12.485 13.788 11.396
Cymatogaster aggregata 0.205 -0.362 0.757 13.615 14.199 13.061
Gadus macrocephalus 1.218 0.754 1.714 12.425 12.684 12.152
Glyptocephalus zachirus 0.715 0.218 1.168 12.868 13.526 12.293
Hexagrammos stelleri 0.463 -0.617 1.443 13.203 14.820 11.738
Hydrolagus colliei 1.024 0.667 1.390 12.660 13.089 12.258
Lepidopsetta bilineata 0.678 0.124 1.147 12.771 13.539 12.191
Leptocottus armatus 0.717 0.153 1.323 13.686 14.447 12.858
Lycodes pacificus 0.359 -0.510 1.208 13.787 15.018 12.605
Lyopsetta exilis 0.707 0.166 1.249 12.692 13.372 12.043
Merluccius productus 1.925 1.482 2.383 9.789 10.389 9.166
Microgadus proximus 1.155 0.736 1.646 12.522 12.886 11.986
Microstomus pacificus 0.812 0.373 1.333 12.732 13.200 12.094
Myoxocephalus polyacanthocephalus 0.736 0.230 1.274 13.628 14.196 12.974
Nautichthys oculofasciatus 0.931 -0.280 2.049 12.785 14.506 11.212
Parophrys vetulus 0.710 0.401 1.015 12.570 12.958 12.182
Pleuronichthys coenosus 0.807 0.271 1.429 12.180 12.863 11.331
Podothecus accipenserinus 0.805 -0.033 1.566 13.790 14.881 12.880
Porichthys notatus 0.746 -0.128 1.513 12.081 13.421 10.963
Psettichthys melanostictus 0.697 0.073 1.277 11.561 12.270 11.011
Raja binoculata 0.373 -0.166 0.867 13.823 14.396 13.343
Raja rhina 0.701 0.201 1.218 13.601 14.128 12.991
Rhacochilus vacca 0.298 -0.449 0.995 13.311 14.204 12.473
Scorpaenichthys marmoratus 0.637 0.261 1.124 13.736 13.656 13.539
Sebastes auriculatus 1.543 1.109 1.985 11.673 12.217 11.103
Sebastes maliger 1.532 1.117 1.947 11.709 12.183 11.223
Sebastes proriger 1.379 0.610 2.029 11.872 12.845 11.155
Sebastes ruberrimus 1.244 0.592 1.859 11.830 12.440 11.446
Squalus suckleyi 0.014 -0.446 0.490 14.325 14.698 13.950
Theragra chalcogramma 1.114 0.738 1.520 12.577 12.997 12.086
Xeneretmus latifrons 0.622 -0.251 1.402 14.346 15.240 13.611
Xeneretmus triacanthus 0.748 -0.128 1.634 13.753 14.705 12.789
Family
Agonidae 0.747 -0.013 1.466 13.761 14.566 13.063
Batrachoididae 0.748 -0.143 1.548 12.364 13.388 11.617
Chimaeridae 0.999 0.500 1.554 12.623 12.885 12.220
Clupeidae 0.716 -0.121 1.598 12.658 13.557 11.729
Cottidae 0.700 0.264 1.104 13.710 14.088 13.410
Embiotocidae 0.272 -0.329 0.892 13.534 14.069 12.921
Gadidae 1.163 0.718 1.560 12.418 12.780 12.186
Hemitripteridae 0.935 -0.088 2.057 12.784 14.036 11.359
Hexagrammidae 0.577 -0.497 1.437 13.195 14.555 12.266
Merlucciidae 1.729 0.962 2.356 10.515 11.245 10.023
Paralichthyidae 0.682 0.187 1.220 13.371 13.664 12.942
Pleuronectidae 0.736 0.405 1.109 12.529 12.833 12.139
Rajidae 0.557 0.052 1.059 13.601 13.994 13.226
Scorpanidae 1.364 0.791 1.825 11.901 12.540 11.460
Squalidae 0.114 -0.586 0.791 14.142 14.745 13.590
Zoarcidae 0.327 -0.567 1.135 13.786 14.734 12.931
Order
Batrachoidiformes 0.758 -0.079 1.540 12.742 13.573 11.901
Chimaeriformes 0.864 0.194 1.531 12.760 13.390 12.149
Clupeiformes 0.735 -0.112 1.560 12.862 13.733 11.894
Gadiformes 1.131 0.470 1.864 12.159 12.893 11.315
Perciformes 0.490 -0.213 1.179 13.377 14.143 12.628
Pleuronectiformes 0.724 0.145 1.259 12.955 13.553 12.493
Rajiformes 0.655 -0.026 1.329 13.208 13.905 12.637
Scorpaeniformes 0.842 0.339 1.316 12.964 13.456 12.521
Squaliformes 0.473 -0.372 1.246 13.461 14.344 12.621
Community mean 0.739 0.253 1.193 12.949 13.467 12.503
Appendix 4. Variance components estimated in a taxonomically nested (species, family, order) hierarchical model of δ15N values for the Puget Sound demersal fish community. Variance parameters correspond to random slope and intercept coefficients modeled at each level. Residual variance is also presented and relates individual-level variation in δ15N values. To aid interpretation, variance components are depicted in which the model was fit with the log10-transformed body mass centered on zero.
95% HDP interval 95% HDP interval
Slope Lower CI Upper CI Intercept Lower CI Upper CI
Species 0.0462 0.0002 0.2570 0.1624 0.0475 0.4007
Family 0.2016 0.0032 0.7762 0.2601 0.0296 0.8010
Order 0.1632 0.0005 1.0837 0.0876 0.0002 0.7957
Residual 0.4083 0.3588 0.4679