In this vignette we explain how to fit, evaluate and interpret a joint trait distribution model (JTDM). We refer to Poggiato et al., in prep. “Joint models and predictions of community traits” for the full description of JTDMs, and we rather focus here on presenting the R package by applying it to the dataset of the case study presented in the publication.

JTDMs relate a vector of average community traits (e.g. community weighted mean, CWM) to a vector of environmental covariates. The matrix Y contains the CWM traits of specific leaf area (SLA), leaf nitrogen content (LNC) and plant height of 116 plant community sites situated along 21 elevation gradients (www.orchamp.osug.fr) in the French Alps.

```
data(Y)
summary(Y)
#> SLA LNC Height
#> Min. : 3.857 Min. :14.86 Min. :10.80
#> 1st Qu.:13.225 1st Qu.:19.71 1st Qu.:23.49
#> Median :16.349 Median :21.42 Median :35.39
#> Mean :20.500 Mean :22.06 Mean :34.87
#> 3rd Qu.:21.923 3rd Qu.:23.88 3rd Qu.:44.17
#> Max. :66.664 Max. :34.30 Max. :76.38
```

We consider two environmental covariates: Growing Degree Days (GDD, the sum of temperature of days with positive temperature in the growing season) to represent the average length and intensity of the growing season and intensity of freezing events (Freezing Degree Days, the sum of temperature of days with negative temperature in the growing season, FDD). To account for habitat type, we included a variable that was set to 1 when the site was in a forest, 0 otherwise (hereafter habitat). These data are stored in the matrix X.

```
data(X)
summary(X)
#> GDD FDD forest
#> Min. : 467.4 Min. :-36.838 Min. :0.0000
#> 1st Qu.:1037.3 1st Qu.:-14.768 1st Qu.:0.0000
#> Median :1320.7 Median :-11.520 Median :0.0000
#> Mean :1357.6 Mean :-12.758 Mean :0.3621
#> 3rd Qu.:1684.9 3rd Qu.: -9.461 3rd Qu.:1.0000
#> Max. :2861.5 Max. : -2.522 Max. :1.0000
```

JTDMs infer a linear regression for each CWM trait as a function of
the environmental covariates, together with an inter-traits residual
covariance matrix. Therefore, the parameters of a JTDM are the
regression coefficients \(B\) and the
residual covariance \(\Sigma\). Notice
that inferring a generalized linear model is possible, even thus not yet
implemented in our package. The inference of JTDM is implemented in the
Bayesian framework with the function `jtdm_fit`

. This
functions samples from the posterior distribution, which has been
analytically determined. Therefore, there is no need for classical MCMC
convergence checks. The syntax of the function is similar to the popular
function `lm`

. This function requires the response matrix Y,
the predictor matrix X and a (right-hand only) formula to specify how
CWM traits depends on environmental variables. We choose here to include
in the model linear and quadratic terms (using orthogonal polynomials)
of GDD and FDD in interaction with habitat to enable non linear
trait-environment relationships and to allow for different
trait-environment relationships in forest and open habitats. The is very
fast, however, to ensure that the whole vignette runs quickly fast, we
only sample a relative low number of samples from the posterior
distribution (100 here). To obtain reliable results, the number of
samples should be increased (around 1000).

```
# Short MCMC to obtain a fast example: results are unreliable !
m = jtdm_fit(Y = Y, X = X,
formula = as.formula("~poly(GDD,2)+poly(FDD,2)+poly(GDD,2):forest+poly(FDD,2):forest"),
sample = 100)
summary(m)
```

Since we ran the JTDM with very short chains, the MCMC has not converged. You can try to run the model with longer chains and check its convergence.

We can now look at the inferred the regression coefficients. The
`getB`

function provides the MCMC samples of the regression
coefficients matrix \(B\), together
with its mean, and \(95\%\) credible
interval.

`get_sigma`

provides instead the MCMC samples of the
residual covariance matrix \(\Sigma\)
together with its mean and \(95\%\)
credible interval.

We can plot these parameters with the function `plot`

We can predict CWM traits on a (new) set of sites using
`jtdm_predict`

. Here, we predict on the training dataset and
compute goodness of fit measures of these predictions like \(R^2\) and RMSE, by setting
`validation=TRUE`

. We can choose whether to obtain the full
posterior predictive distribution (`FullPost=TRUE`

which is
more time consuming) or just its posterior mean
(`FullPost=FALSE`

, which we advice if the aim is to only
compute goodness of fit measures).

```
predictions = jtdm_predict(m, Xnew = X, Ynew = Y,validation = T)
predictions$R2
#> SLA LNC Height
#> 0.5879573 0.3521668 0.5212710
predictions$RMSE
#> SLA LNC Height
#> 7.597665 2.799570 9.129341
```

We can evaluate the performances of the model using a K-fold
cross-validation using the function `jtdmCV`

We can now analyze the inferred trait-environment relationships using
the function `partial_response`

, that computes and plots the
partial response curve of a focal trait along a focal environmental
gradient. This function takes as input the model `m`

, the
focal environmental variable `indexGradient`

and the focal
trait `indexTrait`

, using the names of environmental
variables and traits as specified in the column names of X and Y
respectively. The function builts a dataset of environmental variables
by building the gradient of the focal environmental variable and keeping
all other variables fixed to their mean. Then, it predicts the marginal
distribution of the focal trait for the so-built environmental dataset.
The user can choose whether to obtain the full predictive distribution
(`FullPost=TRUE`

), or the predictive distribution of the mean
term (`FullPost="mean"`

, which we advice here), or just the
posterior mean (`FullPost="mean"`

). The function outputs are
the plot of the inferred trait-environment relationship and the
predictions of the model (used to produce the plot). For example, we can
plot the partial response curve relationship between SLA and GDD

The user can eventually provide a given gradient for the focal
environmental variable `XFocal`

(which is otherwise built on
regular grid with length given by `grid.length`

). The user
can also choose to fix the non focal environmental variables to another
value. For example, we can obtain the partial response curve of SLA and
GDD in forest

We can then compute the partial response curves of pairwise CWM trait
combinations together with their \(95\%\) credible region, what we define in
the publication as the most likely CWM combination and envelop of
possible CWM combinations. This is done by the function
`ellipse_plot`

, that takes as input the model `m`

,
the focal environmental variable `indexGradient`

and the two
focal traits `indexTrait`

, using the names of environmental
variables and traits as specified in the column names of X and Y
respectively. The function builds a gradient of the focal environmental
variable while keeping all other variables fixed to their mean, and then
predict and plots the joint distribution of the focal traits. The user
can choose whether to obtain the full predictive distribution
(`FullPost=TRUE`

), or the predictive distribution of the mean
term (`FullPost=F`

, which we advice here to obtain smoother
curves).

```
# plot the pairwise SLA-LNC partial response curve along the GDD gradient
ellipse_plot(m, indexTrait = c("SLA","LNC"), indexGradient = "GDD")
```

The user can also choose to fix the non-focal environmental variables to another value. For example, we can obtain the partial response curves of the most likely CWM combination and envelop of possible CWM combinations of SLA and GDD in forest.

The `jtdm`

package allows to define a region in the
community-trait space and compute their joint probabilities for a given
set of environmental conditions. This is done by the function
`joint_trait_prob`

. To define a given region in the
community-trait space, the user has to define the focal traits
`indexTrait`

(any number of traits is accepted) and
trait-specific thresholds through the parameter `bounds`

.
`bounds`

is a list of the length of `indexTrait`

,
where each element of the list is a vector of length two. The vector
represents the inferior and superior bounds of the region for the
specified trait. For example, if we consider two traits (e.g. SLA and
LNC), `bounds=list(c(20,Inf),c(20,Inf))`

corresponds to the
region in the community-trait space where both SLA and LNC both take
values greater than 20. We can then define the sites (i.e. the set of
environmental conditions) in which to compute joint probabilities. For
example we can compute joint probabilities of both SLA and LNC to be
greater than 20 in a high altitude site. This measures the relative
suitability of communities where both SLA and LNC are higher than 20 in
a high altitude site.

```
joint_trait_prob(m, indexTrait = c("SLA","LNC"), Xnew = X["VCHA_2940",],
bounds = list(c(20,Inf),c(20,Inf)),
FullPost = TRUE)$PROBmean
#> 1
#> 0.1099954
```

Unsurprisingly, the probability is low. Then, we compute how this
probability vary along the GDD gradient using the function
`joint_trait_prob_gradient`

. The function builts a dataframe
where the focal variable varies along a gradient and the other
(non-focal) variables are fixed to their mean (but see FixX parameter
for fixing non-focal variables to user-defined values) and predict the
joint probabilities along the gradient.

```
joint = joint_trait_prob_gradient(m, indexTrait = c("SLA","LNC"),
indexGradient = "GDD",
bounds = list(c(mean(Y[,"SLA"]),Inf),c(mean(Y[,"SLA"]),Inf)),
FullPost = TRUE)
```

We can then plot such predictions.

```
#> Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
#> ℹ Please use `linewidth` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
```

As climatic conditions become more favorable (i.e. GDD increases), the probability of having high values of both traits increases.