In latent space item response models (LSIRMs), subjects and items are embedded in a low-dimensional Euclidean latent space. As such, interactions among persons and/or items can be revealed that are unmodeled in conventional item response theory models. Current estimation approach for LSIRMs is a fully Bayesian procedure with Markov Chain Monte Carlo, which is, while practical, computationally challenging, hampering applied researchers to use the models in a wide range of settings. Therefore, we propose an LSIRM based on two variants of regularized joint maximum likelihood (JML) estimation: penalized JML and constrained JML. Owing to the absence of integrals in the likelihood, the JML methods allow for various models to be fit in limited amount of time. This computational speed facilitates a practical extension of LSIRMs to ordinal data, and the possibility to select the dimensionality of the latent space using cross-validation. In this study, we derive the two JML approaches and address different issues that arise when using maximum likelihood to estimate the LSIRM. We present a simulation study demonstrating acceptable parameter recovery and adequate performance of the cross-validation procedure. In addition, we estimate different binary and ordinal LSIRMs on real datasets pertaining to deductive reasoning and personality. All methods are implemented in R package ‘LSMjml’ which is available from CRAN.

A social network comprises both actors and the social connections among them. Such connections reflect the dependence among social actors, which is essential for individuals’ mental health and social development. In this article, we propose a mediation model with a social network as a mediator to investigate the potential mediation role of a social network. In the model, the dependence among actors is accounted for by a few mutually orthogonal latent dimensions which form a social space. The individuals’ positions in such a latent social space are directly involved in the mediation process between an independent and dependent variable. After showing that all the latent dimensions are equivalent in terms of their relationship to the social network and the meaning of each dimension is arbitrary, we propose to measure the whole mediation effect of a network. Although individuals’ positions in the latent space are not unique, we rigorously articulate that the proposed network mediation effect is still well defined. We use a Bayesian estimation method to estimate the model and evaluate its performance through an extensive simulation study under representative conditions. The usefulness of the network mediation model is demonstrated through an application to a college friendship network.