Generalized Network Psychometrics: Combining Network and Latent Variable Models

Formalizing network models

• The GGM models inverse covariance structures and can be represented in SEM as either networks among latent variables or networks among residuals, enabling model fitting and comparison.This establishes network modeling as a formal psychometric approach within a framework familiar to psychometricians and methodologists.Source: Introduction · p. 2

Latent Network Modeling

• LNM enables exploratory estimation of conditional-independence relationships among latent variables through model-search algorithms, addressing equivalent-model difficulties in SEM.SEM exploratory estimation is complicated by the presence of equivalent models.Source: Introduction · pp. 2–3

Residual Network Modeling

• RNM models residuals as a constrained inverse-covariance network, allowing correlated residuals while accounting for covariance attributable to latent factors.This relaxes SEM’s local-independence assumption without adopting the GGM assumption that item covariance is entirely unrelated to latent variables.Source: Introduction · p. 3

Exploratory methods and implementation

• The paper presents stepwise edge-search and penalized maximum-likelihood algorithms for exploratory sparse network estimation, evaluates them in four simulation studies, and implements them in the free-to-use lvnet package.The package is illustrated with an empirical example using personality inventory items.Source: Introduction · p. 3

• Centered responses are assumed to follow a multivariate Gaussian distribution with variance–covariance matrix Σ, represented by the model-implied estimate ˆΣ.Source: Modeling Multivariate Gaussian Data · p. 3
• Given N independent observations, the sample covariance matrix S is computed from the N × P data matrix Y and used in maximum likelihood estimation.S = 1/(N − 1)Y⊤Y.Source: Modeling Multivariate Gaussian Data · p. 3
• A saturated model achieves perfect fit when ˆΣ = S, using K = P(P + 1)/2 parameters equal to the number of unique elements in S.Source: Modeling Multivariate Gaussian Data · p. 4
• The modeling goal is instead a properly identified model with positive degrees of freedom, K < P(P + 1)/2, whose ˆΣ closely resembles S.Source: Modeling Multivariate Gaussian Data · p. 4

Structural Equation Modeling

• SEM extends confirmatory factor analysis by modeling latent-variable variances through structural linear relations, while path analysis models direct effects among observed variables by setting Λ = I and Θ = O.The latent-variable structural model uses regression coefficients B and residual covariance Ψ = Var(ζ).Source: Structural Equation Modeling · p. 4
• Local independence requires Θ to be sufficiently restricted, typically diagonal, so indicators are independent after conditioning on the latent variables; otherwise the model can become saturated or unidentified.A zero in Θ indicates local independence between two variables conditional on the latent variables, making the assumption critical to CFA and SEM (Lord et al., 1968; Holland and Rosenbaum, 1986).Source: Structural Equation Modeling · p. 4
• Psychometric indicators may systematically violate local independence through direct causal effects, semantic overlap, reciprocal interactions, or prerequisite relationships, leaving standard CFA and SEM misspecified.Examples include fatigue affecting concentration independently of depression and walking-distance items whose responses are inherently dependent.Source: Structural Equation Modeling · pp. 4–5
• Although residual covariances are often freed to improve fit, one-by-one modification-index decisions are post hoc and uncontrolled, making the path to the final model difficult to determine.Network models can systematize connecting residuals on top of a latent-variable structure.Source: Structural Equation Modeling · p. 5

Network representation

• Pairwise Markov Random Fields represent indicators as nodes whose missing edges encode conditional independence after conditioning on all remaining nodes.Figure 1 illustrates that Y1 and Y3 can be conditionally independent given Y2 while all three variables remain correlated.Source: Network Modeling · p. 6, Figure 1 · p. 6
• Network edges represent conditional interactions, not merely correlations: strongly correlated variables may be unconnected, while uncorrelated variables may be strongly connected.A network with paths connecting every node can still imply a fully populated covariance matrix.Source: Network Modeling · p. 7

Gaussian Graphical Model

• For multivariate Gaussian data, the Gaussian Graphical Model uses partial correlations as edge weights, with zero partial correlations indicating conditional independence and no edge.The network is represented by a symmetric p × p weight matrix Ω, while the partial correlations are obtained by standardizing the inverse covariance, or precision, matrix.Source: Network Modeling · p. 7
• GGM structures can be tested and compared with saturated and independence models using χ2 fit statistics and indices including RMSEA and CFI.This provides confirmatory model testing for network structures on psychometric data.Source: Network Modeling · p. 7

Assumptions and limitations

• The GGM assumes that covariances among observed variables are not caused by latent or unobserved variables; if a latent factor generated the data, the estimated GGM would generally be saturated.Thus, missing edges indicate conditional independence given all indicators, an assumption that may be implausible when unobserved traits contribute to item responses.Source: Network Modeling · pp. 7–8

Latent Network Modeling

• LNM models the latent variables’ variance–covariance matrix as a Gaussian graphical model, representing their conditional independencies through an undirected network rather than directed SEM effects.This framework is designed for exploratively estimating relationships between latent variables.Source: Generalizing Factor Analysis and… · p. 8

Residual Network Modeling

• RNM models the residual variance–covariance matrix as a Gaussian graphical model, treating residual associations as pairwise interactions rather than correlations.It can fit a confirmatory factor structure even when local independence is systematically violated and all residuals are correlated.Source: Generalizing Factor Analysis and… · p. 8

Modeling Frameworks

• The generalized frameworks encompass SEM, LNM, and RNM models, including equivalent representations in which network interactions replace conventional covariance or directed-effect structures.Figure 2 illustrates SEM, network, equivalent LNM, and residual-network configurations; undirected edges denote pairwise interactions rather than covariances.Source: Generalizing Factor Analysis and… · p. 8, Figure 2 · p. 9

Latent Network Modeling

• LNM models the latent variance–covariance matrix of a CFA model as a GGM, allowing latent-variable conditional independencies without assuming directionality or acyclicity.Unlike DAG-based SEM representations, LNM does not require directed edges or an acyclic generating model.Source: Latent Network Modeling · p. 10
• GGMs offer a middle ground between correlations and DAGs, representing some structures DAGs cannot, while failing to reproduce certain collider structures with the same number of parameters.Figure 3 illustrates both equivalences and differences between DAG and GGM representations, including a cycle representable by a GGM but not a DAG.Source: Latent Network Modeling · p. 10, Figure 3 · p. 11
• GGM representations avoid equivalent undirected models, making exploratory estimates of latent-variable relationships unique and unambiguous.Equivalent DAGs can encode the same conditional independencies while implying different causal interpretations, whereas LNM provides one undirected model for the retrieved structure.Source: Latent Network Modeling · pp. 10–11
• LNM enables network construction while accounting for measurement error by modeling small concepts measured with multiple indicators.Earlier network approaches relied on single indicators and did not remediate measurement error.Source: Latent Network Modeling · p. 11

Residual Network Modeling

• RNM models SEM residual structure as a Gaussian graphical model, with residual interactions represented in Ω_Θ after controlling for latent structure.The model is defined by equation (6), which combines latent-variable structure with a residual network.Source: Residual Network Modeling · pp. 11–12
• RNM preserves the specified factor structure while allowing correlated residuals, so local independence can be violated without changing latent-factor interpretation.This provides a compromise between factor modeling and network analysis when local independence is implausible but a purely sparse indicator network is too restrictive.Source: Residual Network Modeling · p. 12
• RNM enables sparse residual-network estimation while accounting for covariation caused by latent variables, avoiding the saturated model that can result when latent structure is ignored.The residual network captures structured error correlations rather than treating all unexplained item covariation as unrestricted.Source: Residual Network Modeling · p. 12
• Unlike ESEM (Marsh et al. 2014) and LASSO-regularized SEM (Jacobucci et al., 2016), RNM improves confirmatory-model fit by adding residual interactions instead of cross-loadings.Because the factor structure remains intact, latent-factor interpretation is unchanged, which is valuable when theory strongly specifies the latent structure.Source: Residual Network Modeling · p. 12

Confirmatory Network Testing

• The lvnet R package supports confirmatory RNM, LNM, and network-model estimation, returning fit indices, parameter estimates, and model-comparison tests.Confirmatory models are specified through fixed and free model-matrix elements and estimated using OpenMx.Source: Exploratory Network Estimation · pp. 12–13

Exploratory Search Algorithms

• Exploratory estimation uses step-wise model search and penalized maximum likelihood estimation for unknown residual- or latent-level network structures.Both algorithms are evaluated for both modeling frameworks in simulation studies.Source: Exploratory Network Estimation · p. 13

Simulation Evaluation

• Sensitivity measures detected true edges, whereas specificity measures correctly identified absent edges and limits false-positive network connections.Sensitivity is the true-positive rate, while specificity is the true-negative rate.Source: Exploratory Network Estimation · p. 13
• Estimation procedures should maintain high specificity to favor sparse, interpretable models, while sensitivity should increase with sample size.High specificity indicates that few truly zero edges are incorrectly estimated as nonzero.Source: Exploratory Network Estimation · p. 13

Simulation construction

• Networks were generated by assigning random signed weights, scaling diagonal entries, row-normalizing, and symmetrizing the inverse-covariance matrix to ensure positive definiteness.Weights were drawn uniformly from 0.5 to 1 and made negative with 50% probability; diagonal elements were set to 1.5 times each row’s absolute-weight sum, or 1 when the sum was zero.Source: Simulating Gaussian Graphical models · p. 13
• In the simulated chain graphs, nonzero partial correlations had a mean of 0.33 and standard deviation of 0.04.Source: Simulating Gaussian Graphical models · p. 13

Stepwise Model Search

• LNM searches start from a fully populated latent network, whereas RNM searches start from an empty residual network because dense residual networks would be over-identified.The lvnetSearch function implements both search algorithms.Source: Stepwise Model Search · pp. 13–14
• Stepwise network estimation uses χ2 difference testing or AIC, BIC, or EBIC optimization; χ2 search adds edges that significantly improve fit or removes edges that do not significantly worsen it.Among eligible changes, it adds the edge improving fit most or removes the edge worsening fit least, using α = 0.05. χ2 search stops when no addition significantly improves fit and every removal significantly worsens it.Source: Stepwise Model Search · p. 14

Simulation Studies

• Both simulations used five sample sizes from 50 to 1 000, four evaluation criteria, 1 000 replications per condition, and 20 000 datasets overall.The LNM simulation used four latent factors with three indicators each in a chain network; the RNM simulation used two latent variables with five indicators each and a chain residual network.Source: Stepwise Model Search · pp. 14, 15, 17, Figure 4 · p. 15

Simulation Study 1: Latent Network Modeling

• In LNM, sensitivity improved with sample size, AIC performed best, EBIC worst, specificity was high, and all criteria showed high sensitivity and specificity above sample size 500.The procedure was conservative: missing edges were adequately detected, while true edges could go unnoticed in smaller samples.Source: Stepwise Model Search · pp. 15, 17, Figure 5 · p. 16

Simulation Study 2: Residual Network Modeling

• In RNM, sensitivity increased with sample size, AIC performed best, EBIC worst, specificity was very high, and true edges were generally recovered above sample size 250.EBIC favored caution, whereas AIC favored discovery; all four criteria performed well overall.Source: Stepwise Model Search · pp. 17, 19, Figure 7 · p. 18

Motivation and method

• LASSO addresses the step-wise algorithms’ slow performance as network size grows, especially for residual network models with more than 10 indicators.It optimizes a penalized likelihood using an absolute-parameter penalty, encouraging sparse structures with many relationships estimated as zero.Source: LASSO Regularization · p. 19

Implementation

• lvnet applies LASSO to selected model matrices and treats parameter estimates below ϵ = 0.0001 as zero when counting parameters and degrees of freedom.The lvnetLasso procedure tests 20 logarithmically spaced tuning parameters from 0.01 to 1 by default, selects the best AIC, BIC, or EBIC model, and refits it without LASSO.Source: LASSO Regularization · pp. 19–20

Simulation Study 3: Latent Network Modeling

• In latent network simulations, AIC showed relatively poor specificity but high sensitivity, whereas EBIC performed well with sample sizes of 500 and higher.The study used 8 latent variables, 24 observed variables, five sample sizes, and 1,000 datasets per sample size.Source: LASSO Regularization · p. 20

Simulation Study 4: Residual Network Modeling

• In residual network simulations, LASSO showed good recovery of the residual structure, with AIC performing worst in specificity and EBIC best.The simulations used four latent variables with five indicators each and residuals linked by a chain graph.Source: LASSO Regularization · p. 20

• The analysis used 2,800 observations on 25 Big Five personality items, applying LASSO across 100 tuning parameters with EBIC to estimate residual and latent networks.The data contained five items per trait, and the analysis proceeded from CFA to RNM and then RNM+LNM estimation.Source: Empirical Example: Personality Inventory · p. 20
• The RNM model fit substantially better than the poorly fitting CFA model, while the RNM+LNM model removed five latent-network edges after accounting for residual interactions.Table 1 compares the CFA, RNM, and RNM+LNM models; the CFA was theory-based, whereas RNM was selected through high-dimensional model search.Source: Empirical Example: Personality Inventory · p. 23, Table 1 · p. 23
• In the final RNM+LNM model, Agreeableness was connected only to Extraversion, which was the sole trait directly linked to all other traits.The residual network contained 30% of all possible edges and yielded 176 degrees of freedom, despite a fully populated residual covariance structure.Source: Empirical Example: Personality Inventory · p. 23, Figure 10 · p. 23
• Because the procedures were highly explorative and RNM was discovered through model search rather than theory alone, the reported structure should be replicated in independent samples.The empirical interpretation of the network in Figure 10 therefore remains provisional.Source: Empirical Example: Personality Inventory · p. 23

Contributions

• The LNM and RNM frameworks extend psychometric network modeling to latent-variable structures and are implemented in the freely available lvnet R package.LNM models networks among latent variables, whereas RNM models networks among indicator residuals, enabling exploratory latent conditional-independence searches and network estimation despite measurement error or latent causes.Source: Conclusion · p. 24

Simulation findings

• Simulation studies found high specificity and increasing sensitivity with sample size, with AIC favoring sensitivity and EBIC favoring specificity.The results were comparable to state-of-the-art network estimation techniques under plausible psychological sample and model sizes, but depend on the model setup and sample-size ranges.Source: Conclusion · p. 24

Extensions and limitations

• The framework can combine latent and residual networks or add directed regression paths, although the paper focuses on multivariate normal data and lvnet lacks more advanced implementations.Future work could extend these combinations to Ising models involving binary latent variables or residual networks with binary indicators.Source: Conclusion · p. 24

Data limitations and perspective

• The current estimation expressions require complete data because full-information maximum likelihood for incomplete data has not yet been implemented in lvnet.The authors characterize the framework as versatile and promising while emphasizing GGMs as a systematic, nondirectional representation of associations when causal direction is uncertain.Source: Conclusion · p. 25

Introduction

• The GGM models the inverse covariance matrix and provides an alternative constrained covariance-structure approach closely related to SEM.Both GGM and SEM models imply constrained covariance structures, and GGM models can be obtained through SEM reparameterization.Source: Introduction · p. 32
• The framework yields two generalizations: Latent Network Modeling connects latent variables, while Residual Network Modeling connects residuals within a fitted SEM.Both forms can be modeled and estimated in SEM, allowing network analysis in a framework familiar to psychometricians and methodologists.Source: Introduction · p. 32
• LNM enables exploratory estimation of conditional-independence relationships among latent variables, addressing SEM’s difficulty with equivalent models (MacCallum et al., 1993).The latent network is estimated through model-search algorithms.Source: Introduction · p. 32
• RNM permits structured residual dependence, avoiding local-independence assumptions while accounting for covariance partly attributable to latent factors.All residuals may be correlated under constraints on the inverse residual covariance matrix.Source: Introduction · pp. 32–33
• The paper introduces stepwise edge-search and penalized maximum-likelihood algorithms, evaluates them in four simulation studies, and implements them in the free-to-use lvnet package.The package is illustrated with personality inventory items.Source: Introduction · p. 33

Modeling Multivariate Gaussian Data

• In maximum likelihood estimation, S is used to compute and minimize the objective, while the model-implied ˆΣ is estimated as closely as possible to S; a saturated model has ˆΣ = S.Source: Modeling Multivariate Gaussian Data · p. 34
• Psychometric items can remain dependent after conditioning on latent traits, yet excessive covariance is often ignored or handled by ad-hoc, post hoc freeing of residual correlations.This uncontrolled model-search process makes it difficult to determine how reported final models were obtained.Source: Modeling Multivariate Gaussian Data · p. 36
• Network models are proposed to optimize and systematize residual connections on top of latent-variable models.Source: Modeling Multivariate Gaussian Data · p. 37
• In network psychometrics, indicators are nodes connected by edges representing pairwise interactions and conditional associations that cannot be explained by other variables.Undirected network edges encode conditional independence structure rather than marginal correlations; absent edges indicate conditional independence after conditioning on the remaining nodes.Source: Modeling Multivariate Gaussian Data · pp. 37–39
• For multivariate Gaussian data, the Gaussian Graphical Model uses partial correlations to represent conditional independence, with zero partial correlations corresponding to absent network edges.Partial correlations can serve directly as edge weights and are obtained from the inverse variance-covariance, or precision, matrix.Source: Modeling Multivariate Gaussian Data · pp. 39–40
• The proposed model enables confirmatory testing of GGM structures and comparison with saturated and independence models using χ2 fit statistics and indices such as RMSEA and CFI.The introduction states that these fit-assessment methods had not previously been used in network psychometrics.Source: Modeling Multivariate Gaussian Data · p. 40
• A GGM would generally be saturated when data are generated by a latent factor model, although assuming no latent causes may be implausible for psychological indicators.Missing GGM edges indicate conditional independence given all other indicators, but indicators may also reflect unobserved traits.Source: Modeling Multivariate Gaussian Data · pp. 40–41

Generalizing Factor Analysis and Network Modeling

• The authors propose two generalizations that incorporate network structures into SEM: Latent Network Modeling (LNM) and Residual Network Modeling (RNM).LNM models latent-variable covariance as an undirected network, whereas RNM models the residual covariance structure as a GGM.Source: Generalizing Factor Analysis and… · p. 41
• RNM models residual associations as pairwise interactions, enabling confirmatory factor structures even when local independence is systematically violated and all residuals are correlated.The network formulation replaces covariance representation with interactions, distinguishing these models from typical SEM representations.Source: Generalizing Factor Analysis and… · p. 41
• LNM represents conditional independence relationships among latent variables without imposing directionality or acyclicity, addressing limitations of exploratory DAG estimation.DAGs can yield multiple equivalent structures with different causal interpretations, whereas each latent conditional-independence structure corresponds to one GGM.Source: Generalizing Factor Analysis and… · pp. 43, 45
• GGMs provide a middle ground between zero-order correlations and DAGs: each set of zero-order correlations implies exactly one GGM, while GGMs can represent some cyclic structures that DAGs cannot.Some DAG structures, including colliders, cannot be represented by GGMs with the same number of parameters, so the frameworks are not universally equivalent.Source: Generalizing Factor Analysis and… · p. 45
• LNM identifies a unique exploratory latent network when equivalent SEM structures encode the same conditional independences, whose number grows without bound as latent variables increase.The LNM representation has no equivalent model and can clearly portray relationships that exploratory SEM cannot uniquely distinguish.Source: Generalizing Factor Analysis and… · p. 46
• LNM constructs networks from latent concepts measured by multiple indicators, thereby accounting for measurement error rather than relying on single-indicator networks.This allows network analysts to model relationships among graspable small concepts while controlling for measurement error.Source: Generalizing Factor Analysis and… · p. 46
• RNM models residual indicator structure as a GGM, allowing sparse residual interactions after latent structure is controlled while retaining the specified factor structure and its interpretation.Residual interactions can represent pairwise effects or partial overlap between indicators that remain after conditioning on latent variables and other indicators.Source: Generalizing Factor Analysis and… · pp. 46–47
• RNM offers a compromise between network and factor modeling by allowing local-independence violations without assuming that latent traits are absent.This is useful when theory strongly specifies latent variables but indicators plausibly retain residual dependence.Source: Generalizing Factor Analysis and… · pp. 46–47

Exploratory Network Estimation

• Both LNM and RNM support confirmatory network testing, implemented in lvnet through OpenMx with model-fit indices, parameter estimates, and model-comparison tests.LNM confirmatory models set Λ = I and Θ = O, whereas RNM models omit latent variables.Source: Exploratory Network Estimation · p. 48
• The package provides step-wise model search and penalized maximum-likelihood estimation for recovering unknown latent or residual network structures, evaluated through sensitivity and specificity.Step-wise searches can use χ2 difference tests or information criteria including AIC, BIC, and extended BIC.Source: Exploratory Network Estimation · pp. 48–49
• Exploratory searches add or remove one edge at a time, selecting changes by χ2 difference testing or AIC, BIC, or EBIC improvement until no further improvement occurs.LNM searches start from a fully populated network, whereas RNM searches start from an empty network; lvnetSearch implements both algorithms.Source: Exploratory Network Estimation · pp. 50–51

Simulation Study 1: Latent Network Modeling

• With sample sizes over 500, all four LNM estimation methods achieved high sensitivity and specificity, although AIC had the highest sensitivity and EBIC the highest specificity.Sensitivity improved with sample size, while the procedure remained conservative and could miss true edges in smaller samples.Source: Figure 5 · p. 53, Simulation Study 1: Latent Network… · p. 54

Simulation Study 2: Residual Network Modeling

• In RNM, all four criteria performed well, with specificity very high across sample sizes, EBIC most cautious, and the method generally discovering true edges above sample size 250.AIC produced the best sensitivity, whereas EBIC produced the best specificity; false positives were therefore very low.Source: Simulation Study 2: Residual Network… · pp. 54, 57, Figure 7 · p. 56
• Stepwise selection can recover the correct network structure but becomes very slow when networks exceed about 10 nodes, especially for RNM with many indicators.This limitation motivates faster high-dimensional approaches such as LASSO-based penalized likelihood estimation.Source: Simulation Study 2: Residual Network… · p. 57
• LASSO produces sparse models with better cross-validation performance, and lvnet selects and refits models using information criteria after thresholding negligible parameters.The exploratory procedure tests tuning parameters, selects the best AIC, BIC, or EBIC model, and refits it without LASSO.Source: Simulation Study 2: Residual Network… · pp. 57–58

Simulation Study 4: Residual Network Modeling

• LASSO performed well in recovering residual networks; AIC had the worst specificity, while EBIC had the best.The residual-network simulation used four latent variables, five indicators per latent variable, and chain-graph residual links across 5,000 datasets.Source: Simulation Study 4: Residual Network… · p. 59, Figure 9 · p. 61

Empirical Example: Personality Inventory

• In the personality inventory example, RNM substantially improved fit over CFA, and RNM+LNM removed five latent-network edges after accounting for residual interactions.The final network showed Extraversion connected directly to all other traits, Agreeableness connected only to Extraversion conditionally, and a residual network containing 30% of possible edges.Source: Empirical Example: Personality Inventory · p. 62

Conclusion

• The conclusion presents LNM as a network among latent variables and RNM as a network among indicator residuals, extending network modeling within latent-variable frameworks.The authors describe both frameworks as offering benefits over conventional latent-variable and network modeling, with LNM supporting exploratory searches for latent-variable relationships.Source: Conclusion · p. 63
• LNM models conditional-independence structures among latent variables while accounting for measurement error, whereas RNM estimates residual networks without assuming local independence, even when nodes have latent causes.Both frameworks combine network structures with latent-variable modeling and are implemented in the freely available lvnet R package.Source: Conclusion · pp. 63–64
• Simulation studies found high specificity, rapidly increasing sensitivity with sample size, and performance comparable to state-of-the-art network estimation techniques in psychologically plausible settings.The algorithms detected more true edges as sample size increased and did not often produce false positives.Source: Conclusion · p. 64
• AIC produced the best sensitivity and EBIC the best specificity across the four simulation studies, but criterion choice depends on whether discovery or caution is prioritized.These results alone do not establish one universally preferable information criterion.Source: Conclusion · p. 64
• The reported simulation results are specific to the studied model setups and sample sizes, while incomplete-data estimation would require FIML, which was not implemented in lvnet.Binary indicators require an Ising model, and future work could extend these ideas to binary latent variables and residual networks.Source: Conclusion · pp. 64–65
• The framework supports additional combinations of latent and residual networks or directed regression paths, although the paper focuses on the distinct benefits of residual and latent networks.More advanced models were possible but not yet implemented in lvnet, which was described here for multivariate normal data.Source: Conclusion · p. 64
• The authors present GGMs as a versatile interface between correlation and causality that can systematically chart variable connections without committing to a particular causal interpretation.They argue that GGMs are especially suitable when temporal information or experimental interventions are absent and associations may arise from diverse mechanisms.Source: Conclusion · p. 65