Network Trees: A Method for Recursively Partitioning Covariance Structures

Abstract

In many areas of psychology, correlation-based network approaches (i.e., psychometric networks) have become a popular tool. In this paper, we propose an approach that recursively splits the sample based on covariates in order to detect significant differences in the structure of the covariance or correlation matrix. Psychometric networks or other correlation-based models (e.g., factor models) can be subsequently estimated from the resultant splits. We adapt model-based recursive partitioning and conditional inference tree approaches for finding covariate splits in a recursive manner. The empirical power of these approaches is studied in several simulation conditions. Examples are given using real-life data from personality and clinical research.

Appendix: Score Functions of the Multivariate Gaussian Distribution

We derive the score functions of the multivariate Gaussian distribution for individual observations. In order to fit the networktree model described in the paper, one only requires the score functions of the correlation parameters. However, for the sake of completeness the score functions for all parameters, i.e., mean, standard deviation, and correlations, are derived. The first derivatives of the log-likelihood, as given in Eq. (2), w.r.t. the parameters $\mu _k$, $\sigma _k$, and $\rho _{kl}$ are given below. Note that we use the scalar parameter expressions rather than vector notation. The partial derivative w.r.t. $\mu _k$ is:

$$\begin{aligned} \frac{\partial \ell }{\partial \mu _k} = \sum _{l=1}^{p} \varsigma _{kl} (y_l - \mu _l), \end{aligned}$$

where $\varsigma _{kl}$ denotes the element in the k-th row and l-th column of the inverse of the covariance matrix $\varvec{\Sigma }^{-1}$. The partial derivative w.r.t. $\sigma _k$ is:

$$\begin{aligned} \frac{\partial \ell }{\partial \sigma _k} = -\frac{1}{\sigma _k} + \frac{1}{\sigma _k} \left( \frac{y_k - \mu _k}{\sigma _k}\right) \sum _{l=1}^{p} \omega _{kl} \left( \frac{y_l - \mu _l}{\sigma _l}\right) , \end{aligned}$$

where $\omega _{kl}$ denotes the element in the k-th row and l-th column of the inverse of the correlation matrix $\mathbf {R}^{-1}$. Finally, the partial derivative w.r.t. $\rho _{kl}$ is:

$$\begin{aligned} \frac{\partial \ell }{\partial \rho _{kl}} = -\frac{1}{2} \omega _{ij} + \frac{1}{2} \left( \sum _{m=1}^{p} \omega _{km}\left( \frac{y_m - \mu _m}{\sigma _m}\right) \right) \left( \sum _{m=1}^{p} \omega _{lm}\left( \frac{y_m - \mu _m}{\sigma _m}\right) \right) . \end{aligned}$$

Note that these are the ones used in our NT approach.