Accurate Estimation of Structural Equation Models with Remote Partitioned Data

Abstract

This paper focuses on a privacy paradigm centered around providing access to researchers to remotely carry out analyses on sensitive data stored behind firewalls. We develop and demonstrate a method for accurate estimation of structural equation models (SEMs) for arbitrarily partitioned data. We show that under a certain set of assumptions our method for estimation across these partitions achieves identical results as estimation with the full data. We consider two situations: (i) a standard setting with a trusted central server and (ii) a round-robin setting in which none of the parties are fully trusted, and extend them in two specific ways. First, we formulate our methods specifically for SEMs, which have become increasingly common models in psychology, human development, and the behavioral sciences. Secondly, our methods work for horizontal, vertical, and complex partitions without needing different routines. In application, this method will serve to increase opportunities for research by allowing SEM estimation without transfer or combination of data. We demonstrate our methods with both simulated and real data examples.

A Appendix

1.1 A.1 RAM Algebra

We briefly exhibit here the method we use for defining SEMs and transforming the model parameters to model implied means and covariance matrices. These model implied matrices are then used to calculate log likelihoods iteratively given the data. Optimizing over these matrices is equivalent to optimizing over the model parameters, giving us our estimates.

The SEM path diagram has a one-to-one relationship with the Multivariate normal mean and covariance matrices for the manifest variables. We construct this relationship through the use of RAM matrix algebra. For this we define five matrices denoted A, S, F, M, and I. These matrices contain both fixed and free model parameters. The free parameters are to be estimated and will be changed during optimization, while the fixed parameters do not change. In these matrices, free parameters are denoted with a greek symbol and the fixed parameters are designated by a constant number.

Recall the path diagram shown in Fig. 4. For this example model, the RAM algebra proceeds as follows. The A (“asymmetric”) matrix defines all regression parameters or one-headed arrows in the path diagram. It has number of rows and columns equal to the number of combined latent and manifest variables, with the column designating the path origin and the row designation the destination.

The S (“symmetric”) matrix defines are variance parameters or two-headed arrows in the path diagram in the same way as the A matrix.

The F (“filter”) matrix acts a filter for the manifest variables. It has columns equal to the combined number of latent and manifest variables but rows equal only the number of manifest variables. For each manifest variable it has a one on the diagonal.

The M (“mean”) matrix defines the mean parameters if any for the latent and manifest variables. These are not always included in the path diagrams.

Finally an I (“identity”) matrix is included, with columns and rows equal to the number of combined latent and manifest variables.

Using these matrices, we obtain the corresponding model implied mean ($\mu$) and covariance matrices ($\varSigma$) of the manifest variables based on the chosen parameters. The following equations give this crucial relationship.

$$\begin{aligned} \varSigma = F * (I - A)^{-1} * S * ((I - A)^{-1})^T * F^T \end{aligned}$$

(10)

$$\begin{aligned} \mu = F * (I - A)^{-1} * M \end{aligned}$$

(11)