Contents

- 1 Networks and Latent Variable Models: Equivalences, Distinctions and Combinations
- 1.1 Schedule
- 1.2 Description
- 1.3 Abstracts
- 1.3.1 Joost Kruis — How theoretically distinct mechanisms can generate identical (binary) observations.
- 1.3.2 Sacha Epskamp — Generalized Network Psychometrics: Combining Network and Latent Variable Models
- 1.3.3 Epskamp, S., Rhemtulla, M.T., & Borsboom, D. (in press). Generalized Network Psychometrics: Combining Network and Latent Variable Models. Psychometrika. Pre-print available at arxiv.org/abs/1605.09288.
- 1.3.4 Riet van Bork — How to think of model complexity?
- 1.3.5 Abe Hoffman — A comparison of latent variable vs network models using longitudinal data

# Networks and Latent Variable Models: Equivalences, Distinctions and Combinations

## Schedule

13:30 – 13:35: Introduction and walk-in

13:36 – 13:56: Joost Kruis — How theoretically distinct mechanisms can generate identical (binary) observations.

13:57 – 14:17: Sacha Epskamp — Generalized Network Psychometrics: Combining Network and Latent Variable Models

14:18 – 14:38: Riet van Bork — How to think of model complexity?

14:39 – 14:59: Abe Hofman — A comparison of latent variable vs network models using longitudinal data

## Description

In recent literature, network modeling has been suggested as an alternative to latent variable modeling of psychological phenomena. In this framework, covariation is assumed to arise as a result of direct interactions between observed variables. Probabilistic graphical models have been proposed that focus on the unique variance between observed variables, and have since been worked out as psychometric modeling frameworks. In addition, state of the art estimation methods and software packages have made this methodology available to empirical researchers have been developed. These advances have lead to a surge of publications in high-impact journals that utilized this methodology in diverse fields of psychological science.

While network modeling was proposed to contrast latent variable modeling, recent psychometric research has instead shown that network modeling and latent variable modeling are closely related, if not equivalent. The first speaker, **Joost Kruis**, will discuss these equivalences between psychometric common-cause models, common effect models, and network models. Due to the equivalent nature, network modeling allows for a different conceptualization and characterization of latent variable models. Consequently, the network models that have been proposed have direct utility to latent variable modeling and vise versa. The second speaker, **Hudson Golino, **makes use of this in retrieving both the number of underlying factors and their indicators using clustering detection on an estimated network model. The third speaker, **Sacha Epskamp**, continues discussing how network modeling and latent variable modeling complement each other and shows that by introducing a network model as a formal psychometric model one can generalize structural equation modeling to include network models on its residual (allowing for estimating a factor model where local independence is violated) or latent level (allowing for exploratory estimation of structural relationships).

If network modeling and latent variable modeling are so closely related, then it is questionable how one can assess the nature of the possible underlying causal mechanisms. That is, are the data generated by an unobserved common cause or by direct interactions between the observed variables? The earlier described equivalences show that fitting a latent variable or network model alone does not support either of these causal mechanisms. The last two speakers will therefore tackle the question on how to separate these models. The fourth speaker, **Riet van Bork**, will investigate how the network model and latent variable model differ in terms of model complexity. Finally, **Abe Hofmann**, will show that when longitudinal data is analyzed, network models and latent variable models can imply different testable hypotheses; the network approach best describes the development of learning mathematics.

In sum, this symposium will discuss the relationships between network modeling and latent variable modeling, benefits that can be obtained from combining these frameworks and methods on how to assess the underlying causal mechanisms in light of these model equivalences.

Unfortunately, the second speaker could not make it to IMPS 2017. His work on this topic can be read in recent publications in PlosOne and Intelligence.

## Abstracts

### Joost Kruis — How theoretically distinct mechanisms can generate identical (binary) observations.

Examining the structure of observed associations between measured variables is an integral part of psychometrics. At face value, associations inform about a possible relation between two variables, yet contain no information about the nature and directions of these relations. Making causal inferences from these associations requires the specification of a mechanism that explains the emergence of the associations.

With the arrival of the network perspective, as such a mechanism, in psychometrics we have a promising new contender a field that has been historically dominated by latent variable modelling. However, with the ever-increasing popularity of applying network models to data, it is important to subject this approach to some scrutiny. In this talk we therefore add a footnote to the application of network models to (binary) data. Specifically, we discuss three topics that can have an effect on the substantive interpretation of an obtained network structure.

First, we discuss a recent paper in which we describe the common cause (latent variable model), reciprocal affect (network model), and common effect (collider model), frameworks as, theoretically very distinct, mechanisms from which associations between variables can emerge. However, while theoretically distinct, we demonstrated in the paper that their associated statistical models for binary data are mathematically equivalent. Next, we discuss how the sparsity assumptions made by the lasso estimation method influence the network structure resulting from this procedure. Finally, we discuss how the view on specific parameters in the Ising model, affects the interpretation of networks in general.

### Sacha Epskamp — Generalized Network Psychometrics: Combining Network and Latent Variable Models

The formalization of the Gaussian graphical model (GGM), a popular undirected network model of partial correlation coefficients, as a formal psychometric model allows for its combination with the general framework of Structural Equation Modeling (SEM; Epskamp, Rhemtulla & Borsboom, in press). The GGM conceptualizes the covariance between psychometric indicators as resulting from pairwise interactions between observable variables in a network structure. This contrasts with standard psychometric models, in which the covariance between test items arises from the influence of one or more common latent variables. Here, we present two generalizations of the network model that encompass latent variable structures. In the first generalization, we model the covariance structure of latent variables as a network. We term this framework Latent Network Modeling (LNM) and show that, with LNM, a unique structure of conditional independence relationships between latent variables can be obtained in an explorative manner. In the second generalization, the residual variance-covariance structure of indicators is modeled as a network. We term this generalization Residual Network Modeling (RNM) and show that, within this framework, identifiable models can be obtained in which local independence is structurally violated. These generalizations allow for a general modeling framework that can be used to fit, and compare, SEM models, network models, and the RNM and LNM generalizations. This methodology has been implemented in the free-to-use software package, lvnet, which contains confirmatory model testing as well as two exploratory search algorithms: stepwise search algorithms for low-dimensional datasets and penalized maximum likelihood estimation for larger datasets.

References

### Epskamp, S., Rhemtulla, M.T., & Borsboom, D. (in press). Generalized Network Psychometrics: Combining Network and Latent Variable Models. *Psychometrika. *Pre-print available at arxiv.org/abs/1605.09288.

### Riet van Bork — How to think of model complexity?

This talk explores how to think of model complexity of regularized partial correlation network models compared to latent variable models. Within the latent variable modeling approach the complexity of a model is usually expressed by the number of freely estimated parameters of a model. The more parameters are allowed to be freely estimated, the more adaptable the model is to the data. The effective degrees of freedom of a regularized partial correlation network model can be estimated as the number of zero-valued edges (Zou, Hastie, Tibshirani, 2007). This number is typically much smaller than the degrees of freedom of a latent variable model on the same set of variables, suggesting that network models are vastly more complex. However, the number of freely estimated parameters does not always capture the complexity of a model. For example, models that have equivalent numbers of freely estimated parameters can differ in their flexibility to fit random data (Preacher, 2006). When comparing a network model and a latent variable model in how they fit to empirical data it is important to account for the model’s flexibility to fit arbitrary data. We will consider several approaches to model complexity from philosophy of science. Our goal is not to provide a technical solution to the question of how to assess model complexity of network models and latent variable models, but to evaluate which possible approaches to model complexity are appropriate when dealing with network models and latent variable models that stem from such different modeling frameworks.

References

Preacher, K. J. (2006). Quantifying parsimony in structural equation modeling.

*Multivariate Behavioral Research, 41*, 227-259.

Zou, H., Hastie, T., & Tibshirani, R. (2007). On the “degrees of freedom” of the

lasso. *The Annals of Statistics, 35*, 2173-2192.

### Abe Hoffman — A comparison of latent variable vs network models using longitudinal data

In psychology the correlational structure between items or subtests is often analyzed with latent variable models and more recently using different network modeling approaches. For example, in the study of intelligence a positive (cross-sectional) correlational structure – the positive manifold – is a well established empirical phenomenon which is often explained by introducing a general intelligence factor. Recent papers have shown that both latent variable and network models can result in the same observed correlational structures (and are in some cases even mathematically equivalent). However, these models differ greatly in their substantive explanations and imply that different mechanisms generated these correlations.

This talk presents a comparison of a network model and a latent variable model using longitudinal data. We use a longitudinal structural equation modeling framework and different specifications of latent change score models that can capture the implied dynamics of different developmental theories. Using data from a large online learning platform for mathematics (Math Garden), we show that the development of learning to do mathematics is best described by a network approach that allows direct links between the development of different domains (e.g. counting and addition).