All models with common latent variables (factors) can be transformed into equivalent network models without common latent variables. This was proven in Molenaar (Reference Molenaar2003); Molenaar et al. (Reference Molenaar, van, Hamaker, Boker and Wenger2007) apply this transformation to the 1-factor model. The networks that are equivalent to latent variable models are directed graphs connecting observed variables and residuals. In contrast, the networks in the target article are undirected graphs connecting observed variables (symptoms). Notwithstanding these differences in detail, the fact that any latent variable model can be transformed into an equivalent network model (where equivalent means the same number of free parameters and goodness of fit to the data) implies that latent variable models can be conceived of as constituting a subset of the set of network models. This, of course, qualifies any presumed contrast between interpretations in terms of networks and latent variables.
The transformation of models with common latent variables into equivalent network models without common latent variables is akin to the transformation of linear state space models into equivalent transfer function models, which is standard practice in computational engineering and signal analysis. A typical state space model is composed of two sub-models: a measurement model linking the observed process to the latent state process, and a dynamic model describing the evolution of the latent state process. In contrast, a transfer function model consists of time-lagged relationships involving only the observed process and residuals. Again, it holds that state space models constitute a subset of the set of transfer function models.
State space models are very popular in applied dynamic systems analysis. Dynamic systems theory is discussed in the concluding section of the target article. Linear state space models are formally equivalent to longitudinal factor models, the latter being typical instances of psychometric latent variable models. Hence, in so far as the authors see an important role for dynamic systems theory within a network perspective, there also should be a role for state space models and longitudinal factor models.
The model equivalences sketched here are suggestive of a rather gradual relationship between network models and latent variable models. The networks presented in the target article are undirected graphs in which the edges reflect associations between observed variables. The analogue for multivariate Gaussian variables would be a covariance matrix depicted as a graph. Such graphs, often augmented by colorings conveying additional information, certainly constitute important tools to display observed relationships among a large number of observed variables. But displays of relationships among observed variables, whether in graphical network form or as arrays of covariances, constitute a low level of modeling. An intermediate level of modeling would include path analysis or transfer function modeling; that is, models involving networks of directed connections among observed variables and residuals. At this level the effects of measurement errors can be taken into account, which appear to be quite important in social-scientific measurements. An example is the generalization of the simplex model to the quasi-simplex model. The highest level includes state space models and psychometric latent variable models. These models explain observed relationships in terms of common causes and measurement error and therefore are ideal for scientific theory formation. They are not as restricted as mentioned in the target article. For instance, measurement errors in state space models can be sequentially dependent and/or can depend upon the latent state process. But obviously they put the strongest demands on the data.
Cramer et al. are correct in stating that not every set of observed variables can be explained in terms of a limited number of common causes, despite the heuristic and scientific value of such an explanation. For instance, with respect to the networks presented in the target article this would explain what the symptoms are symptoms of. Therefore, one would like to have available a set of adequate inductive tools to determine whether or not each symptom (node in a network) has its own etiology. Or, stated more generally, one would like to have adequate inductive tools to progress from the lowest level of modeling (displaying networks of relationships among observed variables) to higher levels.
You have
Access
All models with common latent variables (factors) can be transformed into equivalent network models without common latent variables. This was proven in Molenaar (Reference Molenaar2003); Molenaar et al. (Reference Molenaar, van, Hamaker, Boker and Wenger2007) apply this transformation to the 1-factor model. The networks that are equivalent to latent variable models are directed graphs connecting observed variables and residuals. In contrast, the networks in the target article are undirected graphs connecting observed variables (symptoms). Notwithstanding these differences in detail, the fact that any latent variable model can be transformed into an equivalent network model (where equivalent means the same number of free parameters and goodness of fit to the data) implies that latent variable models can be conceived of as constituting a subset of the set of network models. This, of course, qualifies any presumed contrast between interpretations in terms of networks and latent variables.
The transformation of models with common latent variables into equivalent network models without common latent variables is akin to the transformation of linear state space models into equivalent transfer function models, which is standard practice in computational engineering and signal analysis. A typical state space model is composed of two sub-models: a measurement model linking the observed process to the latent state process, and a dynamic model describing the evolution of the latent state process. In contrast, a transfer function model consists of time-lagged relationships involving only the observed process and residuals. Again, it holds that state space models constitute a subset of the set of transfer function models.
State space models are very popular in applied dynamic systems analysis. Dynamic systems theory is discussed in the concluding section of the target article. Linear state space models are formally equivalent to longitudinal factor models, the latter being typical instances of psychometric latent variable models. Hence, in so far as the authors see an important role for dynamic systems theory within a network perspective, there also should be a role for state space models and longitudinal factor models.
The model equivalences sketched here are suggestive of a rather gradual relationship between network models and latent variable models. The networks presented in the target article are undirected graphs in which the edges reflect associations between observed variables. The analogue for multivariate Gaussian variables would be a covariance matrix depicted as a graph. Such graphs, often augmented by colorings conveying additional information, certainly constitute important tools to display observed relationships among a large number of observed variables. But displays of relationships among observed variables, whether in graphical network form or as arrays of covariances, constitute a low level of modeling. An intermediate level of modeling would include path analysis or transfer function modeling; that is, models involving networks of directed connections among observed variables and residuals. At this level the effects of measurement errors can be taken into account, which appear to be quite important in social-scientific measurements. An example is the generalization of the simplex model to the quasi-simplex model. The highest level includes state space models and psychometric latent variable models. These models explain observed relationships in terms of common causes and measurement error and therefore are ideal for scientific theory formation. They are not as restricted as mentioned in the target article. For instance, measurement errors in state space models can be sequentially dependent and/or can depend upon the latent state process. But obviously they put the strongest demands on the data.
Cramer et al. are correct in stating that not every set of observed variables can be explained in terms of a limited number of common causes, despite the heuristic and scientific value of such an explanation. For instance, with respect to the networks presented in the target article this would explain what the symptoms are symptoms of. Therefore, one would like to have available a set of adequate inductive tools to determine whether or not each symptom (node in a network) has its own etiology. Or, stated more generally, one would like to have adequate inductive tools to progress from the lowest level of modeling (displaying networks of relationships among observed variables) to higher levels.