In psychometrics, the network approach refers to a set of methods employed to model and analyze the relationships among psychological variables. Unlike traditional psychometric approaches, such as the structural equation approach focusing on latent variables, the network approach emphasizes the interconnections among observed variables. Due to the latter emphasis, modeling and analyzing network models to complement structural equation models when developing and evaluating psychometric instruments offers several advantages. Most notably, in undirected weighted network models, a subtype of network models, observed variables are represented by nodes, and associations between observed variables, each assigned a weight that represents the magnitude of associations, are represented by edges. In this perspective, undirected weighted network models provide estimates of the magnitude of associations (i.e., the shared variance) among items of psychometric instruments that structural equation models cannot, providing critical insight into the construct-level content validity of subsets of items of psychometric instruments. To illustrate, if an undirected weighted network model suggests that a subset of items of a psychometric instrument presents a high magnitude of associations with another subset of items, the shared variance of the said subsets of items is therefore high: the content they assess and the information they provide is highly similar. From the standpoint of the latter illustration, undirected weighted network modeling allows for the estimation of whether a subset of items assesses a “narrow” or a “broad” proportion of the construct-level content of a psychometric instrument. Hence, identifying an optimal subset of a desired number of items that assesses the “broadest” proportion of the construct-level content of a psychometric instrument consists of a combinatorial optimization problem.
Consider an undirected weighted network model \(G = (V, E)\), where \(V\) denotes the set of nodes, and \(E\) denotes the set of edges. Each edge \(e_{ij}\) is associated with a positive or negative weight \(w_{ij}\). Let \(S\) be a subset of nodes from \(V\) with a fixed size \(k\), and \(\bar{S}\) denote the complement of \(S\) in \(V\). The objective is to identify the optimal subset \(S\) of size \(k\) such that the sum of the (absolute) values of the edge weights connecting \(S\) with its complement \(\bar{S}\) of size \(n - k\) is maximized. Formally, the combinatorial optimization problem can be expressed as:
\[ \max_{S \subset V, |S| = k} \left( \sum_{i \in S, j \in \bar{S}} |w_{ij}| \right) \]
Solving the combinatorial optimization problem allows identifying what optimal subset of a desired number of items presents the highest magnitude of associations (i.e., the highest shared variance) within the set of items. In this light, combinatorial search algorithms (e.g., brute force search algorithm, simulated annealing search algorithm) allow to identify what optimal subset of a desired number of items should be retained in a short version of a psychometric instrument to assess the “broadest” proportion of the construct-level content of the set of items included in the original version of the said psychometric instrument.
You can install the development version of shortr from GitHub with:
utils::install.packages("remotes")
base::library("remotes")
remotes::install_github("lfourni2/shortr")
base::library("shortr")