The purpose of this vignette is to learn how to estimate trophic position for multiple species or groups using stable isotope data (δ13C and δ15N). We can estimate trophic position using a one source model based on equations from Post (2002).
The equation for a one source model consists of the following:
$$ \text{Trophic Position} = \lambda + \frac{(\delta^{15}N_c - \delta^{15}N_b)}{\Delta N} $$
Where λ is the trophic
position of the baseline (e.g., 2
), δ15Nc
is the δ15N of the
consumer, δ15Nb
is the mean δ15N of the
baseline, and ΔN is
the trophic enrichment factor (e.g., 3.4).
To use this model with a Bayesian framework, we need to rearrange this equation to the following:
δ15Nc = δ15Nb + ΔN × (Trophic Position − λ)
The function one_source_model()
uses this rearranged
equation.
First we need to organize the data prior to running the model. To do this work we will use {dplyr} and {tidyr} but we could also use {data.table}.
When running the model we will use {trps} and {brms} and iterative processes provided by {purrr}.
Once we have run the model we will use {bayesplot} to assess models and then extract posterior draws using {tidybayes}. Posterior distributions will be plotted using {ggplot2} and {ggdist} with colours provided by {viridis}.
First we load all the packages needed to carry out the analysis.
In {trps} we have
a data set that has consumer and baseline data already joined for two
ecoregions (combined_iso
) using the same methods in getting started with trps. Let’s look at this data
frame.
combined_iso
#> # A tibble: 117 × 13
#> id common_name ecoregion d13c d15n d13c_b1 d15n_b1 d13c_b2 d15n_b2 c1
#> <int> <fct> <fct> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Lake Trout Anthropo… -22.3 17.6 -20.3 8.85 -26.4 7.72 -21.3
#> 2 2 Lake Trout Anthropo… -23.0 17.7 -20.1 8.77 -24.4 8.69 -21.3
#> 3 3 Lake Trout Anthropo… -21.2 16.7 -20.3 8.85 -24.8 7.99 -21.3
#> 4 4 Lake Trout Anthropo… -20.9 18.7 -20.1 8.77 -24.4 8.69 -21.3
#> 5 5 Lake Trout Anthropo… -20.7 18.0 -20.5 8.38 -24.8 7.99 -21.3
#> 6 6 Lake Trout Anthropo… -20.7 18.0 -20.1 8.34 -24.4 8.05 -21.3
#> 7 7 Lake Trout Anthropo… -22.8 17.8 -19.7 8.04 -24.1 8.79 -21.3
#> 8 8 Lake Trout Anthropo… -22.4 17.9 -20.1 8.56 -24.6 10.7 -21.3
#> 9 9 Lake Trout Anthropo… -20.9 18.4 -18.7 8.95 -24.3 10.6 -21.3
#> 10 10 Lake Trout Anthropo… -21.7 17.7 -20.8 9.28 -24.6 10.7 -21.3
#> # ℹ 107 more rows
#> # ℹ 3 more variables: n1 <dbl>, c2 <dbl>, n2 <dbl>
We can see that this data frame has isotope data for a second
baseline (dreissenids; d13c_b2
and d15n_b2
) as
well as the mean values for both baselines
(c1
-n2
). These columns for the second baseline
are useful when estimating trophic position using a two source model but
we do not need them for this analysis and they can be removed.
We can also confirm that this data set has one species, lake trout.
collected from two ecoregions in Lake Ontario.
Let’s remove the columns we don’t need, d13c_b2
,
d15n_b2
, c2
, n2
, and add λ to the data frame
(l1
). To do so we make a name
column that will
be the two groups we have, common_name
and
ecoregion
pasted together. We are doing this to make the
iterative processes easier.
combined_iso_update <- combined_iso %>%
dplyr::select(-c(d13c_b2, d15n_b2, c2, n2)) %>%
mutate(
l1 = 2,
name = paste(ecoregion, common_name, sep = "_")
) %>%
dplyr::select(id, common_name, ecoregion, name, d13c:l1)
Let’s view our completed data set.
combined_iso_update
#> # A tibble: 117 × 11
#> id common_name ecoregion name d13c d15n d13c_b1 d15n_b1 c1 n1
#> <int> <fct> <fct> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Lake Trout Anthropogenic Anth… -22.3 17.6 -20.3 8.85 -21.3 8.14
#> 2 2 Lake Trout Anthropogenic Anth… -23.0 17.7 -20.1 8.77 -21.3 8.14
#> 3 3 Lake Trout Anthropogenic Anth… -21.2 16.7 -20.3 8.85 -21.3 8.14
#> 4 4 Lake Trout Anthropogenic Anth… -20.9 18.7 -20.1 8.77 -21.3 8.14
#> 5 5 Lake Trout Anthropogenic Anth… -20.7 18.0 -20.5 8.38 -21.3 8.14
#> 6 6 Lake Trout Anthropogenic Anth… -20.7 18.0 -20.1 8.34 -21.3 8.14
#> 7 7 Lake Trout Anthropogenic Anth… -22.8 17.8 -19.7 8.04 -21.3 8.14
#> 8 8 Lake Trout Anthropogenic Anth… -22.4 17.9 -20.1 8.56 -21.3 8.14
#> 9 9 Lake Trout Anthropogenic Anth… -20.9 18.4 -18.7 8.95 -21.3 8.14
#> 10 10 Lake Trout Anthropogenic Anth… -21.7 17.7 -20.8 9.28 -21.3 8.14
#> # ℹ 107 more rows
#> # ℹ 1 more variable: l1 <dbl>
This example data is now ready to be analyzed.
We will use similar structure used in getting
started with trps to model trophic position, however, we first
split()
the data into a list for all groups and then use
map()
from {purrr} to run the model for
each group.
You will notice that the brm()
call is exactly the same
as when we ran the model for one group. The only difference here is when
using map()
, the data
argument in
brm()
needs to be replaced with .x
to tell
brm()
where to get the data.
Let’s run the model!
m1 <- combined_iso_update %>%
split(.$name) %>%
map( ~ brm(
formula = one_source_model(),
prior = one_source_priors(),
stanvars = one_source_priors_params(),
data = .x,
family = gaussian(),
chains = 2,
iter = 4000,
warmup = 1000,
cores = 4,
seed = 4,
control = list(adapt_delta = 0.95)
),
.progress = TRUE
)
#> Compiling Stan program...
#> Start sampling
#> ■■■■■■■■■■■■■■■■ 50% | ETA: 45s
#> Compiling Stan program...
#> Start sampling
Let’s look at the summary of both models.
m1
#> $`Anthropogenic_Lake Trout`
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: d15n ~ n1 + dn * (tp - l1)
#> dn ~ 1
#> tp ~ 1
#> Data: .x (Number of observations: 87)
#> Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1;
#> total post-warmup draws = 6000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> dn_Intercept 3.38 0.26 2.88 3.87 1.00 1645 1794
#> tp_Intercept 4.82 0.22 4.44 5.29 1.00 1644 1818
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.48 0.04 0.42 0.57 1.00 2124 2125
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).
#>
#> $`Embayment_Lake Trout`
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: d15n ~ n1 + dn * (tp - l1)
#> dn ~ 1
#> tp ~ 1
#> Data: .x (Number of observations: 30)
#> Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1;
#> total post-warmup draws = 6000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> dn_Intercept 3.37 0.25 2.89 3.86 1.00 1506 1878
#> tp_Intercept 4.54 0.20 4.21 4.96 1.00 1531 1891
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.61 0.09 0.47 0.81 1.00 2145 1903
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).
We can see that R̂ is 1, meaning that the variance among and within chains are equal (see {rstan} docmentation on R̂) and that ESS is quite large for both groups. Overall, this means that both models are converging and fitting accordingly.
Let’s look at the trace plots and distributions. We use
iwalk()
instead of map()
, as
iwalk()
invisibly returns .x
which is handy
when you want to call a function (e.g., plot()
) for its
side effects rather than its returned value. I have also added
grid.text()
from {grid}
to add the group names
to each plot.
We can see that the trace plots look “grassy” meaning the model is converging!
Let’s again look at the summary output from the model.
m1
#> $`Anthropogenic_Lake Trout`
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: d15n ~ n1 + dn * (tp - l1)
#> dn ~ 1
#> tp ~ 1
#> Data: .x (Number of observations: 87)
#> Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1;
#> total post-warmup draws = 6000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> dn_Intercept 3.38 0.26 2.88 3.87 1.00 1645 1794
#> tp_Intercept 4.82 0.22 4.44 5.29 1.00 1644 1818
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.48 0.04 0.42 0.57 1.00 2124 2125
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).
#>
#> $`Embayment_Lake Trout`
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: d15n ~ n1 + dn * (tp - l1)
#> dn ~ 1
#> tp ~ 1
#> Data: .x (Number of observations: 30)
#> Draws: 2 chains, each with iter = 4000; warmup = 1000; thin = 1;
#> total post-warmup draws = 6000
#>
#> Regression Coefficients:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> dn_Intercept 3.37 0.25 2.89 3.86 1.00 1506 1878
#> tp_Intercept 4.54 0.20 4.21 4.96 1.00 1531 1891
#>
#> Further Distributional Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.61 0.09 0.47 0.81 1.00 2145 1903
#>
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).
We can see that, for lake trout from the Anthropogenic
ecoregion, ΔN is
estimated to be 3.38
with l-95% CI
of
2.88
, and u-95% CI
of 3.87
. If we
move down to trophic position (tp
) we see trophic position
is estimated to be 4.82
with l-95% CI
of
4.44
, and u-95% CI
of 5.29
.
We can see that, for lake trout from the Embayment
ecoregion, ΔN is
estimated to be 3.37
with l-95% CI
of
2.89
, and u-95% CI
of 3.86
. If we
move down to trophic position (tp
) we see trophic position
is estimated to be 4.54
with l-95% CI
of
4.21
, and u-95% CI
of 4.96
.
We can check how well the model is predicting the δ15N of the
consumer using pp_check()
from {bayesplot}
. We
have to use map()
from {purrr}
to iterate over
the list that has our model objects.
m1 %>%
map(~ .x %>%
pp_check()
)
#> Using 10 posterior draws for ppc type 'dens_overlay' by default.
#> Using 10 posterior draws for ppc type 'dens_overlay' by default.
#> $`Anthropogenic_Lake Trout`
#>
#> $`Embayment_Lake Trout`
We can see that posteriors draws (yrep; light lines) for both groups are are effectively modeling δ15N of the consumer (y; dark line).
We use functions from {tidybayes} to do this
work. First we look at the the names of the variables we want to extract
using get_variables()
. Considering we have multiple models
in m1
that all have the same structure, we can just look at
the names of the first model object in m1
.
get_variables(m1[[1]])
#> [1] "b_dn_Intercept" "b_tp_Intercept" "sigma" "lprior"
#> [5] "lp__" "accept_stat__" "stepsize__" "treedepth__"
#> [9] "n_leapfrog__" "divergent__" "energy__"
You will notice that "b_tp_Intercept"
is the name of the
variable that we are wanting to extract. Next we extract posterior draws
using gather_draws()
, and rename
"b_tp_Intercept"
to tp
.
Again, considering we have multiple models in m1
we need
to use map()
to iterate over m1
to get the
posterior draws. Once we have iterated over m1
to extract
draws we can combine the results using bind_rows()
from {dplyr}. The variable
name
will have the name of the ecoregion and common name of
the species pasted to together by a _
. We need to separate
this string into the two variables we want, being ecoregion
and common_name
. We can do this by using
separate_wider_delim()
from {tidyr}. When using this
function it will separate the columns and keep them as
characters
, hence why the last step is to convert
ecoregion
into a factor
.
For your data you will likely have category names other than
ecoregion
and common_name
. Please replace with
the columns that fit your data structure.
post_draws_mg <- m1 %>%
map(~ .x %>%
gather_draws(b_tp_Intercept) %>%
mutate(
.variable = "tp"
) %>%
ungroup()
) %>%
bind_rows(.id = "name") %>%
separate_wider_delim(name, names = c("ecoregion", "common_name"),
delim = "_", cols_remove = FALSE) %>%
mutate(
ecoregion = factor(ecoregion,
levels = c("Anthropogenic", "Embayment")),
)
Let’s view the post_draws_mg
post_draws_mg
#> # A tibble: 12,000 × 8
#> ecoregion common_name name .chain .iteration .draw .variable .value
#> <fct> <chr> <chr> <int> <int> <int> <chr> <dbl>
#> 1 Anthropogenic Lake Trout Anthropog… 1 1 1 tp 4.49
#> 2 Anthropogenic Lake Trout Anthropog… 1 2 2 tp 4.90
#> 3 Anthropogenic Lake Trout Anthropog… 1 3 3 tp 4.74
#> 4 Anthropogenic Lake Trout Anthropog… 1 4 4 tp 4.87
#> 5 Anthropogenic Lake Trout Anthropog… 1 5 5 tp 4.86
#> 6 Anthropogenic Lake Trout Anthropog… 1 6 6 tp 5.17
#> 7 Anthropogenic Lake Trout Anthropog… 1 7 7 tp 5.16
#> 8 Anthropogenic Lake Trout Anthropog… 1 8 8 tp 5.11
#> 9 Anthropogenic Lake Trout Anthropog… 1 9 9 tp 4.94
#> 10 Anthropogenic Lake Trout Anthropog… 1 10 10 tp 4.95
#> # ℹ 11,990 more rows
We can see that the posterior draws data frame consists of seven variables:
ecoregion
common_name
.chain
.iteration
(number of samples after burn-in).draw
(number of samples from iter
).variable
(this will have different variables depending
on what is supplied to gather_draws()
).value
(estimated value)Note - the names of and items in the first two columns will vary depending on the names you split your data into.
Considering we are likely using this information for a paper or
presentation, it is nice to be able to report the median and credible
intervals (e.g., equal-tailed intervals; ETI). We can extract and export
these values using spread_draws()
and
median_qi
from {tidybayes}.
Again, because m1
is a list
of our model
objects, we need to map()
over the list to calculate these
values. Then we do the same procedures we have done before to combine
and restructure the outputs. Lastly, we use mutate_if()
to
round all columns that are numeric to two decimal points.
post_medians_ci <- m1 %>%
map(~ .x %>%
spread_draws(b_tp_Intercept) %>%
median_qi() %>%
rename(
tp = b_tp_Intercept
)
) %>%
bind_rows(.id = "name") %>%
separate_wider_delim(name, names = c("ecoregion", "common_name"),
delim = "_", cols_remove = FALSE) %>%
mutate(
ecoregion = factor(ecoregion,
levels = c("Anthropogenic", "Embayment")),
) %>%
mutate_if(is.numeric, round, digits = 2)
Let’s view the output.
post_medians_ci
#> # A tibble: 2 × 9
#> ecoregion common_name name tp .lower .upper .width .point .interval
#> <fct> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <chr> <chr>
#> 1 Anthropogenic Lake Trout Anthrop… 4.81 4.44 5.29 0.95 median qi
#> 2 Embayment Lake Trout Embayme… 4.53 4.21 4.96 0.95 median qi
I like to use {openxlsx} to
export these values into a table that I can use for presentations and
papers. For the vignette I am not going to demonstrate how to do this
but please check out {openxlsx}
.
Now that we have our posterior draws extracted we can plot them. For comparing trophic position among species or groups, I like using either violin plots, interval points, or slab plots for posteriors. We can access violins through {ggplot2} with the later being available in {ggdist}.
Let’s first look at the violin plot.
ggplot(data = post_draws_mg, aes(x = common_name,
y = .value,
fill = ecoregion)) +
geom_violin() +
stat_summary(fun = median, geom = "point",
size = 3,
position = position_dodge(0.9)
) +
scale_fill_viridis_d(name = "Ecoregion",
option = "G",
begin = 0.35,
end = 0.75, alpha = 0.65) +
theme_bw(base_size = 15) +
theme(
panel.grid = element_blank(),
legend.position = "inside",
legend.position.inside = c(0.85, 0.86)
) +
labs(
x = "Species",
y = "P(Trophic Position | X)"
)
Next, we’ll look at the point interval plot – but first we need to create our colour palette.
Now let’s plot the point intervals.
ggplot(data = post_draws_mg, aes(x = common_name,
y = .value,
group = ecoregion)) +
stat_pointinterval(
aes(point_fill = ecoregion),
point_size = 4,
interval_colour = "grey60",
position = position_dodge(0.4),
shape = 21,
) +
scale_fill_manual(aesthetics = "point_fill",
values = viridis_colours,
name = "Ecoregion") +
theme_bw(base_size = 15) +
theme(
panel.grid = element_blank(),
legend.position = "inside",
legend.position.inside = c(0.85, 0.86)
) +
labs(
x = "Species",
y = "P(Trophic Position | X)"
)
Congratulations we have successfully run a Bayesian one source trophic position model for one species in two ecoregions of Lake Ontario!