The role of odds ratios in joint species distribution modeling

Abstract

Joint species distribution modeling is attracting increasing attention these days, acknowledging the fact that individual level modeling fails to take into account expected dependence/interaction between species. These joint models capture species dependence through an associated correlation matrix arising from a set of latent multivariate normal variables. However, these associations offer limited insight into realized dependence behavior between species at sites. We focus on presence/absence data using joint species modeling, which, in addition, incorporates spatial dependence between sites. For pairs of species selected from a collection, we emphasize the induced odds ratios (along with the joint occurrence probabilities); they provide a better appreciation of the practical dependence between species that is implicit in these joint species distribution modeling specifications. For any pair of species, the spatial structure enables a spatial odds ratio surface to illuminate how dependence varies over the region of interest. We illustrate with a dataset from the Cape Floristic Region of South Africa consisting of more than 600 species at more than 600 sites. We present the spatial distribution of odds ratios for pairs of species that are positively correlated and pairs that are negatively correlated under the joint species distribution model.

Appendix

I: We consider, in detail, the connection between the correlation arising under the latent multivariate normal model for species pairs and the associated odds ratio. We draw on some older work relating bivariate normal probabilities to the associated bivariate correlation. There is a substantial literature, with multivariate extensions, and we only note two papers here: Gupta (1963) and Slepian (1962).

The basic result we need is the following:

Theorem: Suppose $\left( \begin{array}{c} Z_{1} \\ Z_{2} \\ \end{array} \right)$

$\sim$ BivN$\left( \left( \begin{array}{c} 0 \\ 0 \\ \end{array} \right) \right.$, $\left. \left( \begin{array}{cc} 1 &{} \rho \\ \rho &{} 1\\ \end{array} \right) \right)$. Then, $P(Z_{1} \le c_1, Z_{2} \le c_2)$ is non-decreasing in $\rho$.

We apply this result to (1). For fixed $\mu _{i}^{(j)}$ and $\mu _{i}^{(j')}$, by simple probability calculations, we have $p_{i,00}^{(j,j')}$ non-decreasing in $\rho ^{(j,j')}$, we have $p_{i,01}^{(j,j')}$ non-increasing in $\rho ^{(j,j')}$, we have $p_{i,10}^{(j,j')}$ non-increasing in $\rho ^{(j,j')}$, and we have $p_{i,11}^{(j,j')}$ non-decreasing in $\rho ^{(j,j')}$. As a result, the numerator in (1) is non-decreasing in $\rho ^{(j,j')}$ while the denominator in (1) is non-increasing in $\rho ^{(j,j')}$. So, altogether, we have $\theta _{i}^{(j,j')}$ non-decreasing in $\rho ^{(j,j')}$ for all i and $(j,j')$ pairs.

As a corollary, since $\theta _{i}^{(j,j')} =1$ when $\rho ^{(j,j')} = 0$, we must have $\theta _{i}^{(j,j')} \ge 1$ when $\rho ^{(j,j')} > 0$ and $\theta _{i}^{(j,j')} \le 1$ when $\rho ^{(j,j')} < 0$.

II: We offer some brief words regarding how calculation of probabilities is implemented. Under Markov chain Monte Carlo model fitting, we obtain posterior samples, say $\mu _{i,b}^{(j)}, \mu _{i,b}^{(j')}, \rho _{b}^{(j,j')}, b=1,2,\ldots ,B$. Each term in (1) is a double integral which is a function of $(\mu _{i}^{(j)}, \mu _{i}^{(j')}, \rho ^{(j,j')})$. So, each sample, $\mu _{i,b}^{(j)}, \mu _{i,b}^{(j')}, \rho _{b}^{(j,j')}$ produces a posterior realization of each of the four terms on the right side of (1), e.g., $p_{i,00,b}^{(j,j')}$, hence a posterior realization, $\theta _{i,b}^{(j,j')}$. Across $b=1,2,\ldots ,B$, we obtain a posterior sample of size B for each of the terms on the right side of (1) as well as the induced odds ratio. The double integrals that are needed can be computed using approximations or numerically. In fact, if we work with the functional JSDM and specification (ii) above, then $Z_{ij}$ and $Z_{ij'}$ are conditionally independent given $L^{F}_{ij}$, $L^{F}_{ij'}$, $L^{R}_{ij}$ and $L^{R}_{ij'}$. So, we only need to calculate univariate cumulative normal distribution functions.

A fully Monte Carlo alternative is to generate many $(Z_{ij}, Z_{ij'})$ pairs, hence many $(Y_{ij}, Y_{ij'})$ pairs for each $\mu _{i,b}^{(j)}, \mu _{i,b}^{(j')}, \rho _{b}^{(j,j')}$. The collection can be placed into a $2 \times 2$ table from which we can obtain a posterior realization of each of the probabilities on the right side of (1) as well the odds ratio.