comparable pairwise and multivariate associations from presence-absence data

multivariate techniques for network construction

I adopted two multivariate approaches to compare to the pairwise approach: One, which uses partial correlation coefficients, has been introduced in psychology and was the subject of a 2017 tutorial by Epskamp and Fried. If \(m\) variables \(y_i\mid 1\leq i\leq m\) have standard deviations \(\sigma_i\), then the full partial correlation of \(y_i\) and \(y_j\) is the standardized regression coefficient \(\displaystyle r'_{ij}=\frac{\sigma_j}{\sigma_i}\beta_{ij}\) from the linear model predicting \(y_i\) from all other variables—or, equivalently, \(\displaystyle r'_{ji}=\frac{\sigma_i}{\sigma_j}\beta_{ji}\) from the model predicting \(y_j\). The partial correlation network \(G'\) is aggregated from the \(r'_{ij}\). Since these correlations are controlled for the effects of other variables, \(G'\) should include far fewer transitive correlations than a pairwise network.

Conveniently, partial correlations can be calculated from whatever pairwise correlations one begins with.

I also wanted to borrow from the current ecology literature. On my reading, ecology has always been at the cutting edge of multivariate statistical analysis, and lately there have been proposed several correlation and network models to overcome the well-documented limitations of pairwise techniques. These proposals have been motivated by a desire to capture a variety of interactions among species, geography, and environment. The reviews i’ve found aren’t recent enough to have surveyed the most interesting examples, which include Ulrich & Gotelli (2007), Ovaskainen, Hottola, & Siitonen (2010), Cazelles, Araújo, Mouquet, & Gravel (2016), and Morueta-Holme et al (2015).

Mostly because the authors included a tutorial for their method, implemented in R, in their supporting information, i adopted the joint distribution model (JDM) proposed by Pollock and colleagues.¹ Though the model was developed to handle both variable interactions (“endogenous” effects) and case-level covariates (“exogenous” effects), for the coming illustration i’m only concerned with endogenous information.

This simplified JDM assumes that each row of a \(0,1\)-matrix \(X\in\{0,1\}^{n\times m}\) is obtained from a latent multivariate normal distribution with center \(\vec\mu=[\,\mu_1\,\cdots\,\mu_m\,]\) and covariance matrix \(\Sigma\). The \(\mu_j\) encode the prevalences of the conditions while \(\Sigma\) encodes their correlations. If \(Z\sim N(\vec\mu,\Sigma)\), then the rows of \(X\) are assumed to have been generated from samples \(\vec z_i=[\,z_1\,\cdots\,z_m\,]\) as \[x_{ij}=\begin{cases} 0 & z_{ij}\leq 0 \\ 1 & z_{ij}>0 \end{cases}\text.\] The model is hierarchical with respect to the meta-parameters \(\mu\) and \(\sigma\) and is fit using Bayesian methods. I’ll denote the correlation matrix estimated this way as \(\hat{\mathrm{P}}=(\hat\rho_{ij})\), since Pollock and colleagues designate the underlying correlation matrix \(\mathrm{P}\) (capital \(\rho\)).

a comparable pairwise construction

Since partial correlations can be calculated from any sample correlation data, a grand comparison of these three approaches now requires only a (pairwise) correlation coefficient that meaningfully compares to \(\hat\rho_{ij}\). The JDM relies on a latent multivariate normal, so my natural choice—once i’d found it—was a correlation coefficient based on a latent bivariate normal: the tetrachoric correlation coefficient, often denoted \(r_t\).

The setup for \(r_t\) is a distribution \(N(\mu,\Sigma)\) with means \(\vec\mu=[\,\mu_1,\mu_2\,]\) and covariance \[\Sigma=\left(\begin{array}{cc} \!{\sigma_1}^2\! & \!\rho\sigma_1\sigma_2\! \\ \!\rho\sigma_1\sigma_2\! & \!{\sigma_2}^2\! \end{array}\right)\text.\] The upshot is that, as in the JDM, the value of the statistic is the maximum-likelihood estimate of the latent correlation coefficient \(\rho\). Several estimation techniques are discussion in Drasgow’s entry for the Encyclopedia of Statistical Sciences.

Implementations of all three statistics are available but require a bit of lead-in, so i’ll come back to this topic—and a grand comparison—in a future post.