Calculation of TP for one species using multiple Bayesian models
To demonstrate how tRophicPosition works, we will simulate some data. For example, imagine we have a fish species of interest that has mean (\(\pm\) SD) values of \(\delta^{15}\)N of 21‰ (\(\pm\) 1‰), and \(\delta^{13}\)C of -15‰ (\(\pm\) 1‰). Also, as in many aquatic ecosystems, this fish lives in a system with two sources of C and N, meaning that we have to deal with two different baselines. One of the baselines has mean (\(\pm\) SD) values of \(\delta^{15}\)N of 14.2‰ (\(\pm\) 1‰) and \(\delta^{13}\)C -17.3‰ (\(\pm\) 1‰), while the other baseline has mean values of 15.4‰ (\(\pm\) 1‰) and -12.7‰ (\(\pm\) 1‰) (\(\delta^{15}\)N and \(\delta^{13}\)C respectively). We will use the function generateTPData() to simulate this simple ecosystem.
consumer1 <- generateTPData(dCb1 = -17.3, dNb1 = 14.2,
dCc = -15, dNc = 21,
dCb2 = -12.7, dNb2 = 15.4,
DeltaN = 3.4, sd.DeltaN = 0.98,
DeltaC = 0.39, sd.DeltaC = 1.3,
consumer = "Consumer 1")In this function we have also indicated mean and sd values for trophic discrimination factors for DeltaN (\(\Delta\)N) and DeltaC (\(\Delta\)C), and also have indicated a name for the consumer. Remember that all the functions have a help page where you will find other options (write for example ?generateTPData in the console).
Now that we have our simple ecosystem simulated, we will plot it:
plot(consumer1)
In this figure we can see that sample of simulated fish consumers is exactly located in-between both baselines on the \(\delta^{13}\)C axis, and also, our fish is 6.8‰ above the baseline 1 in terms of its \(\delta^{15}\)N value. If we use a one baseline model, and if the trophic level of baseline 1 (lambda) is 2, we would expect a mean trophic position of 4 for our simulated fish species. If we use a two baselines model without fractionation for carbon, we would expect an alpha value of 0.5, which means that each baseline is contributing in 50% of the energy inputs to our fish. Finally, if we use a two baselines full model (including fractionation for \(\delta^{13}\)C), the trophic position of our fish would depend on both baselines, and the relative contribution of each baseline would depend on the trophic discrimination factors (\(\Delta\)N and \(\Delta\)C). Clearly, this is worth exploring more, so we will calculate (and compare) estimates for TP using these models with the function multiModelTP().
consumer1_models <- multiModelTP(consumer1)## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 106
## Unobserved stochastic nodes: 31
## Total graph size: 149
##
## Initializing model
##
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 206
## Unobserved stochastic nodes: 14
## Total graph size: 243
##
## Initializing model
##
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 313
## Unobserved stochastic nodes: 66
## Total graph size: 404
##
## Initializing modelmultiModelsTP() requires an object of the class isotopeData, which was created previously by the function generateTPData(). By default multiModelsTP() defines a lambda = 2 for the baselines, uses 2 chains (n.chains = 2) to do the Bayesian calculation with 20,000 adaptive iterations (n.adapt = 20000), 20,000 actual iterations (n.iter = 20000), 20,000 iterations as burnin (burnin = 20000) and a thinning of 10 (thin = 10). Furthermore, it estimates TP using three different Bayesian models: one baseline, two baselines and two baselines full (models = c("oneBaseline", "twoBaselines", "twoBaselinesFull"). The default setting is to limit the amount of output for each model run (print = FALSE). NB: The user can change any of these options, simply adding them as arguments to multiModelTP(), see ?multiModelTP() for details.
Also, by default this function uses Post’s (2002) assumptions on TDF (i.e. 56 values with mean 3.4 \(\pm\) 0.98 SD for \(\Delta\)N and 107 values with mean 0.39 \(\pm\) 1.3 SD for \(\Delta\)C).
For example, if we have used the option print = TRUE as argument (i.e. multiModelTP(print = TRUE)), we would have a trace plot for every model (to check graphically if our parameters of interest TP, muDeltaN and alpha have converged), and if more than 1 chain were used, also would have printed two statistics from the coda package: Gelman and Rubin’s convergence diagnostic statistics (see ?gelman.diag for a detailed explanation). Basically, we expect that potential scale reduction factors for each parameter is close to 1, which means that our model has converged and it is reliable.
These estimates have been generated using Post’s (2002) mean estimates for TDFs, which are commonly used across a wide range of ecological studies. Now that we have calculated the trophic position for our simulated fish in a simple ecosystem, we will do the same calculation, but this time, we will change the trophic discrimination factor from Post’s (2002) values to McCutchan’s et al. (2003) values in order to examine the sensitivity of the models to TDF estimates. To do this, we will we will generate consumer2 with the same isotope values as consumer1 and simply change the variables deltaN and deltaC within the object consumer2.
consumer2 <- generateTPData(dCb1 = -17.3, dNb1 = 14.2,
dCc = -15, dNc = 21,
dCb2 = -12.7, dNb2 = 15.4,
consumer = "Consumer 2")
consumer2$deltaN <- TDF(author = "McCutchan", element = "N")## You selected McCutchan's et al (2003)
## All d15N: 73 values with mean 2.3 +- 0.18 seconsumer2$deltaC <- TDF(author = "McCutchan", element = "C")## You selected McCutchan's et al (2003)
## All d13C: 102 values with mean 0.5 +- 0.13 seAnd now we will calculate the three Bayesian models of trophic position again, with the only difference being we have changed their TDF values.
consumer2_models <- multiModelTP(consumer2)## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 123
## Unobserved stochastic nodes: 31
## Total graph size: 166
##
## Initializing model
##
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 223
## Unobserved stochastic nodes: 14
## Total graph size: 260
##
## Initializing model
##
## Compiling model graph
## Resolving undeclared variables
## Allocating nodes
## Graph information:
## Observed stochastic nodes: 325
## Unobserved stochastic nodes: 66
## Total graph size: 416
##
## Initializing modelIn both cases we have calculated three Bayesian models (one baseline, two baselines without fractionation for C, and two baselines full model with C fractionation) for two simulated simple ecosystems consisting of one fish and 2 baselines, and the only difference between consumer 1 and consumer 2 is the trophic discrimination factor. In the first case we included Post’s (2002) values, and in the second case we included McCutchan’s et al. (2003) values for both N and C. We have saved the output of multiModelTP() into two objects consumer1_models and consumer2_models, and now we will analyze the results.
If you are using RStudio, in the environment tab you will see that both consumer1_models and consumer2_models are lists of 4 elements each. The first element is a data frame named TP. This dataframe has the posterior samples of TP for each model. The second element is also a data frame, named alpha in this case, which includes the posterior samples of alpha for 2 of the 3 models (only the two baselines models calculate alpha, i.e. the relative contribution of the baseline 1). The third element is a data frame as well, named gg (to indicate us that has the structure required to use the ggplot2 package). This data frame has summary values of both posterior samples of trophic position and alpha: median and 95% credibility interval, grouped by model, community and species. The fourth and last element is a list of 3 elements named samples, that has raw posterior mcmc samples if you want to summarise the information, or analyze it in a different way. The first three elements (data frames TP, alpha and gg) are based in these raw posterior samples of every Bayesian model, summarised by multiModelsTP() for convenience.
We can produce a detailed summary of the structure described above using the function str():
str(consumer1_models)In order to plot the mode and 95% credibility interval for both posterior trophic position and alpha, we can use the function credibilityIntervals():
# For consumer 1 (based on Post's (2002) TDF values)
credibilityIntervals(consumer1_models$gg, x = "model")
This figure provides mode (\(\pm\) 95% credibility interval) for posterior estimates trophic position from the three different models (one baseline two baselines and two baselines full), as well as alpha (only given for two baselines models).
# For consumer 2 (based on McCutchan's (2003) TDF values)
credibilityIntervals(consumer2_models$gg, x = "model")
The arguments df = consumer1_models$gg and x = model above, tells the function credibilityIntervals() to plot the data frame consumer1_models$gg using the model column within that data frame as the independent variable (x), and by default using the mode as the central tendency descriptor (see credibilityIntervals() for details). If you want to plot the median instead of the mode add y1 = "median and y2 = "alpha.median" as arguments within credibilityIntervals(). When we have multiple species or several communities we will change this (e.g. see the vignette multiple species calculation of trophic position).
Why we need of accurate trophic discrimination factors (TDF)?
Probably, the single and most important assumption to get accurate and reliable estimations of trophic position is the trophic discrimination factor, either for nitrogen \(\Delta^{15}\)N (one and two baselines models) and both \(\Delta\)N and \(\Delta\)C for the two baselines full model. Obviously, the use of a non-representative TDF estimate for either \(\Delta\)N or \(\Delta\)C can affect both trophic position estimations and the alpha parameter.
Now we will examine the impact of using the different TDFs for our simulated consumers. We have estimated trophic position and alpha for both consumers groups (differing only in TDF) and will test for statistical differences between posterior samples of these parameters. First, we must change the names of both posterior samples, because they have the same name (as they were calculated with multiModelsTP()):
# Here we see that we have 4002 posterior samples of 3 parameters
# (one for each Bayesian model) for consumer1
str(consumer1_models$TP)## List of 3
## $ 1b : num [1:4002] 4.09 3.91 3.94 3.73 3.93 ...
## $ 2b : num [1:4002] 3.74 3.77 3.8 3.99 3.7 ...
## $ 2bf: num [1:4002] 4.17 3.79 3.85 3.83 3.71 ...# And also 4002 posterior samples (for each 3 Bayesian models) for consumer2
str(consumer2_models$TP)## List of 3
## $ 1b : num [1:4002] 4.82 4.76 4.65 4.78 4.94 ...
## $ 2b : num [1:4002] 4.71 4.31 4.94 5.22 4.44 ...
## $ 2bf: num [1:4002] 5.28 4.88 5.23 5.61 5.03 ...# But the names of each variables are the same for both consumers
# For consumer1
names(consumer1_models$TP)## [1] "1b" "2b" "2bf"# For consumer2
names(consumer2_models$TP)## [1] "1b" "2b" "2bf"# So, we change them in order to compare them.
# To make things clear, consumer 1 will be "Post" and consumer 2 will be
# "McCutchan". # Also, one baseline Bayesian model will be model1, two
# baselines model will be model2 and two baselines full model will be model2F
names(consumer1_models$TP) <- c("Post-model1",
"Post-model2",
"Post-model2F")
names(consumer2_models$TP) <- c("McCutchan-model1",
"McCutchan-model2",
"McCutchan-model2F")A useful first step is to summarize the posterior distributions to get a quantitative idea of how different they are. For this, we will use the base function (i.e. that comes with R) sapply(). This function is part of the lapply functions that apply a function (in this case summary) over a list. As we have two lists (consumer1_models$TP and consumer2_models$TP) first we need to use the c operator, that combines them, then we will calculate a summary for the combined element:
# Here we combine posterior estimates of trophic position for both consumers
combined_models <- c(consumer1_models$TP, consumer2_models$TP)
# Then we calculate a summary of posterior trophic position
sapply(combined_models, summary)## Post-model1 Post-model2 Post-model2F McCutchan-model1
## Min. 3.608028 3.415638 3.493050 4.085558
## 1st Qu. 3.921280 3.756448 3.806252 4.808365
## Median 4.002738 3.827421 3.885791 4.986985
## Mean 4.006179 3.830818 3.888840 5.006483
## 3rd Qu. 4.084321 3.901013 3.962907 5.173656
## Max. 4.435879 4.246032 4.453742 6.580966
## McCutchan-model2 McCutchan-model2F
## Min. 4.003042 4.093534
## 1st Qu. 4.550024 4.685633
## Median 4.712765 4.856066
## Mean 4.733087 4.875364
## 3rd Qu. 4.899018 5.045211
## Max. 6.433009 6.257308# And we calculate the modes
getPosteriorMode(combined_models)## Post.model1 Post.model2 Post.model2F McCutchan.model1
## Posterior mode 4.003 3.824 3.895 4.957
## McCutchan.model2 McCutchan.model2F
## Posterior mode 4.671 4.825Here we have a rudimentary summaries of posterior trophic position estimates associated with each model for both consumers. For example, the median TP for consumer 1 based on the one baseline model (Post-model1) is higher than consumer 1 calculated with both 2 baselines (Post-model2) and 2 baselines full (Post-model2F) models. It appears that the same is true for consumer 2, where median TP estimated for the one baseline model (McCutchan-model1) is higher than that for both two baselines model (McCutchan-model2 and McCutchan-model2F).
As we are operating in a Bayesian universe, there is little sense to state a threshold under or above which both distributions will be considered different (as used in a frequentist universe e.g. p > 0.05). Instead, we think rather of comparing posterior distributions. For example, we can say that two posterior trophic position estimates will be considered different if their samples of posterior distributions are 100% different. In this stringent case and considering Post-model1 and McCutchan-model1 posterior estimates of trophic position, we would expect then that all the posterior estimates of Post-model1 are lesser than posterior estimates of McCutchan-model1.
compareTwoDistributions(combined_models$"Post-model1",
combined_models$"McCutchan-model1",
test = "<=")## [1] 1Above, we can see that trophic position posterior estimate of one baseline model using Post’s (2002) TDF assumptions has a proportion of 4000 observations over the 4000 we sampled (i.e. 4000 / 4000 = 1) that are lesser or equal than trophic position posterior estimate of the same one baseline model but using McCutchan’s (2003) TDF assumptions. Thus, we are pretty confident that posterior estimation of trophic position calculated with one baseline Bayesian model with Post’s (2000) assumptions is higher than posterior estimation of trophic position calculated with McCutchan’s TDF assumptions. Remember that this is the same simulated fish consumer, in the same simple ecosystem, only considering differents TDF assumptions.
By default compareTwoDistributions() evaluates how many observations from combined_models$"Post-model1" are \(\leq\) (less than or equal) than combined_models$"McCutchan-model1", randomly drawn from each of the models posterior distribution. We can change to \(\geq\), \(<\) or \(>\) stating the argument within pairwiseComparisons function (add test = ">=", for example).
We can make a more detailed comparison of the different models and TDFs using the pairwiseComparisons() function, which as its name suggests, produces a matrix of of pairwise comparisons:
pairwiseComparisons(combined_models, test = "<=")## [1] [2] [3] [4] [5] [6]
## [1] Post-model1 0.000 0.135 0.241 1.000 0.998 1.000
## [2] Post-model2 0.865 0.000 0.639 1.000 1.000 1.000
## [3] Post-model2F 0.759 0.361 0.000 1.000 1.000 1.000
## [4] McCutchan-model1 0.000 0.000 0.000 0.000 0.231 0.371
## [5] McCutchan-model2 0.002 0.000 0.000 0.769 0.000 0.647
## [6] McCutchan-model2F 0.000 0.000 0.000 0.629 0.353 0.000The matrix generated by this function is symmetrical, below the diagonal we have the results of quatifying the proportion of posterior samples of the model in row are less or equal than posterior samples of the model in the column, while above the diagonal we have the results of quantifying the proportion of posterior samples of the model in the column are higher than posterior samples of the model in the row.