We introduce the network model as a formal psychometric model, conceptualizing 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, establishing network modeling as parts of the more general framework of structural equation modeling (SEM). 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. We show in simulation studies that these search algorithms perform adequately in identifying the structure of the relevant residual or latent networks. We further demonstrate the utility of these generalizations in an empirical example on a personality inventory dataset.
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Throughout this paper, vectors will be represented with lowercase boldfaced letters and matrices will be denoted by capital boldfaced letters. Roman letters will be used to denote observed variables and parameters (such as the number of nodes), and Greek letters will be used to denote latent variables and parameters that need to be estimated. The subscript i will be used to denote the realized response vector of subject i, and omission of this subscript will be used to denote the response of a random subject.
We make use here of the convenient all-y notation and do not distinguish between exogenous and endogenous latent variables (Hayduk, 1987).
A saturated GGM is also called a partial correlation network because it contains the sample partial correlation coefficients as edge weights.
To our knowledge, the GGM has not yet been framed in this form. We chose this form because it allows for clear modeling and interpretation of the network parameters.
We use the CFA framework instead of the SEM framework here as the main application of this framework is in exploratively estimating relationships between latent variables.
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Epskamp, S., Rhemtulla, M. & Borsboom, D. Generalized Network Psychometrics: Combining Network and Latent Variable Models. Psychometrika 82, 904–927 (2017). https://doi.org/10.1007/s11336-017-9557-x
- network models
- structural equation modeling
- simulation study