1 and 2 focus on the probabilistic relationships and further casual relationships among variables. 3 and 4 focuses on network structure and node importance. 5 focus on the regression coefficients of structural and measurement model.
All 5 analysis methods have a network-shaped diagram. Graphical modeling is a more “general” term that can comprise of the other network models.
3.2 Examples of Varied Networks
Figure 1: Bayesian Network (Briganti et al., 2023)
Figure 2: Facebook friendship network in a single undergraduate dorm (Lewis et al., 2008)
Figure 3: Factor Analysis and Psychological Network (Borsboom et al., 2021)
3.3 Research Aims
Bayesian Network (BN) aims to derive the casual relations between variables
Social Network aims to examine the network structure (community, density or centrality) of individuals
Factor Analysis aims to identify latent variables
Psychological Network aims to examines the associations among observed variables (topological structures) and their positions in the network.
3.4 Network Psychometrics
Network psychometrics is a novel area that allows representing complex phenomena of measured constructs in terms of a set of elements that interact with each other.
It is inspired by the so-called mutualism model and research in ecosystem modeling (Kan, Maas, and Levine 2019).
A mutualism model proposes that basic cognitive abilities directly and positively interact during development.
Psychometric network arises as dynamics or reciprocal causation among variables are getting more attentions.
For example, individual differences in depression could arise from, and could be maintained by, vicious cycles of mutual relationships among symptoms.
A depression symptom such as insomnia can cause another symptom, such as fatigue, which in turn can determine concentration problems and worrying, which can result in more insomnia and so on.
Kan, Kees-Jan, Han L. J. van der Maas, and Stephen Z. Levine. 2019. “Extending Psychometric Network Analysis: Empirical Evidence Against g in Favor of Mutualism?”Intelligence 73 (March): 52–62. https://doi.org/10.1016/j.intell.2018.12.004.
3.5 Comparison to factor analysis
Factor analysis (common factor model) assumes the associations between observed features can be explained by one or more common factors.
For example, higher “depression” level leads to increased frequency of depressive behaviors
Psychometric network, however, assumes the associations between observed features ARE the reason of the development of depression. Or “depression” is the network itself.
unavoidable cycle of “fatigue -> worrying -> insomnia -> fatigue” will leads to higher “depression”
3.6 Utility of Psychometric Network
Explain the pathways of certain psychological phenomenon
Identify the most important problem that needed to be intervene in the treatment procedure
Examine group differences in interactions among observed features
Examine density of network: more dense network indicates more dynamic of certain problems
Examine clusters/communities of observed features: some symptoms are more likely to concur than other symptoms
3.7 Terminology I - Overall procedure
Network structure estimation: the application of statistical models to assess the structure of pairwise (conditional) associations in multivariate data.
Network description: characterization of the global topology and the position of individual nodes in that topology.
Psychometric network analysis: the analysis of multivariate psychometric data using network structure estimation and network description.
3.8 Terminology II - Network description
Node: psychometric variables that are selected in the network
such as responses to questionnaire items, symptom ratings, and cognitive test scores, background variables such as age and gender, experimental interventions.
Edge (conditional association): associations between variables taking into account other variables that may explain the association
Edge weight: parameter estimates that represent the strength of conditional association between nodes
Node centrality: the relative importance of a node in the context of other nodes, that can be calculated using different statistics
3.9 Terminology III - Network structure estimation
Node selection: the choice of which variables will function as nodes in the network model.
Network stability analysis: the assessment of estimation precision and robustness to sampling error of psychometric networks.
Pairwise Markov random field (PMRF): an undirected network that represents variables as nodes and conditional associations as edges, in which unconnected nodes are conditionally independent.
3.10 Exploratory Nature
Psychometric network is exploratory by nature. To obtain a meaningful network structure, psychometric networks need to drop weak edges but keep strong edges.
This procedure is typically called edge selection. One popular edge selection method is regularization.
Original network structure without regularization is called saturated network; vice versa regularized network
Emotion regulation (Awareness), Interpersonal problems and eating disorder
3.11 Workflow of psychometric network
Psychometric network analysis methodology includes steps of network structure estimation (to construct the network), network description (to characterize the network) and network stability analysis (to assess the robustness of results).
3.12 Types of data and network models
Cross-sectional data (N = large, T = 1)
Ising model for categorical variables
Gaussian graphical model (GGM; Foygel & Drton, 2010) for continuous variables
Mixed graphical model (MGM) for mixed types of variables: include both categorical variables and continuous variables
Panel data (N >> T)
Multilevel Graphical vector autoregressive model (GVAR)
i.e., longitudinal data, repeated measures
Time-series data (N \geq 1, T = large)
Graphical vector autoregression
i.e., ecological momentary assessment, conducted via smartphones
4 Network Estimation
4.1 Gaussian graphical model
GGM is one type of Pairwise Markov random field (PMRF) when data are continuous.
In a PMRF, the joint likelihood of multivariate data is modeled through the use of pairwise conditional associations, leading to a network representation that is undirected.
For p-dimensional data following multivariate normal distribution:
Where K is a inverse covariance matrix of \boldsymbol{X} (K = \Sigma^{-1}), also known as precision/concentration matrix.
To obtain sparse network structure, the ith row and jth column element of \boldsymbol{K}, k_{ij}=0 when edge \{j, k\} is not included in the network G,
It means X^{(i)} and X^{(j)} are independent conditional on the other variables .
4.2 Partial correlation networks
GGM can be standardized as the partial correlation network, in which each edge of GGM representing partial correlations between two nodes.
Network stability: The assessment of estimation precision and robustness to sampling error of psychometric networks. Assess whether edge weights and node centrality changes with case dropping subset bootstrap.
Network sensitivity: The sensitivity analysis of the network by adding covariates (age, gender, hukou, education, marital status, and self-rated health) to the model to control their confounding effects. Report whether some central nodes are no longer central after adding some variables.
Node-level metrics:
Total number of node
Node centrality: the position of individual nodes within the network. A generic term that subsumes a family of measures that aim to assess how central a node is in a network topology, such as node strength, betweenness and closeness.
Node bridge strength: the degree to which one node connects two nodes from different facets. The bridge nodes (nodes with high bridge strength) that connect different communities of nodes (e.g., Neurotic factors is bridge nodes between personality traits and depressive symptoms Li and Zhang (2024)).
Node centrality differences between groups: node A is central in group 1’s network but not so central in group 2’s network.
Edge-level metrics:
Total Number of edges: proportions of non-zero edges
Edge weight: edge weights typically are parameter estimates that represent the strength of the conditional association between nodes. Average edge weight can be used to assess the overall strength of network connections.
Li, Jia, and Jihong Zhang. 2024. “Personality Traits and Depressive Symptoms Among Chinese Older People: A Network Approach.”Journal of Affective Disorders 351 (April): 74–81. https://doi.org/10.1016/j.jad.2024.01.215.
70.6% edges are nonzero using EBICglasso, while only 63.6% edges are nonzero using BF method and 62.0% edges are kept using \alpha =.01 significance testing
BGGMfit <- BGGM::explore(bfi[,1:25], type ="continuous", iter =1000, analytic =FALSE) |>select()density_nonzero_edge(BGGMfit$pcor_mat_zero)
[1] 0.62
5.2 Network Structure
EBICglasso
Sig. Test
BF
5.3 Centrality - Strength
Strength centrality measures suggest that C4 and E4 or N1 have highest centrality indicating they play most imporatant roles in the networks.
Method 1: EBICglasso
Method 2: Sig. Test
Method 3: Bayes Factor
5.4 Centrality - Bridge
E4 and E5 has highest bridge strength, indicating they serve as bridges linking communities of personality. They are important elements connecting varied types of personality
Method 1: EBICglasso
Method 2: Sig. Test
Method 3: Bayes Factor
5.5 Wrapping up
Network analysis is an alternative way of modeling dependency among variables.
Gaussian graphical model is used for multivariate continuous data.
The goal is to estimate network structure, node importance, and stability.
---title: "Lecture 13: Confirmatory Factor Analysis and Psychological Network"subtitle: ""author: "Jihong Zhang"institute: | Educational Statistics and Research Methods (ESRM) Program* University of Arkansasdate: "2024-11-22"sidebar: falseexecute: echo: true eval: true warning: false message: falseformat: html: page-layout: full toc: true toc-depth: 2 toc-expand: true lightbox: true code-fold: false uark-revealjs: scrollable: true chalkboard: true embed-resources: false code-fold: false number-sections: false footer: "Lecture 13: Confirmatory Factor Analysis and Network Psychometrics" slide-number: c/t tbl-colwidths: auto output-file: slides-index.htmlbibliography: references.bib---## Today's Objectives1. Understand the relationship between Exploratory and Confirmatory Factor Analysis2. Understand what is network analysis and confirmatory factor analysis3. Understand how network model can be applied into real examples# Confirmatory Factor Analysis## CFA Approach to EFA- EFA and CFA are same things except that one is to explore the factor structure (EFA) and one is to confirm pre-determined factor structure (CFA).- But ... we can also conduct **exploratory analysis** using a CFA model<<<<<<< HEAD======= - Need to set the right number of constraints for identification - We set the value of factor loadings for a few items on a few of the factors - Typically to zero - Sometimes to one (Brown, 2002) - We keep the factor covariance matrix as an identity - Uncorrelated factors (as in EFA) with variances of one- Benefits of using CFA for exploratory analyses: - CFA constraints remove rotational indeterminacy of factor loadings – no rotating is needed (or possible) - Defines factors with potentially less ambiguity - Constraints are easy to see - For some software (SAS and SPSS), we get much more model fit information - For some software (SAS and SPSS), we get much more model fit information ## CFA Example- We can use `lavaan` to do CFA... here is the syntax for the one factor model - The `~=` is the symbol represent factors and their loaded items - We can see that the one-factor CFA has same loading with EFA```{r}library(lavaan)data02a =read.csv(file="gambling_lecture12.csv", header=TRUE)head(data02a, 3)# One-factor CFA ModelCFA_1factor.syntax ="factor1 =~ X1 + X3 + X5 + X9 + X10 + X13 + X14 + X18 + X21 + X23"#for comparison with EFA we are using standardized factors (var = 1; mean = 0)CFA_1factor.model =cfa(model = CFA_1factor.syntax, data = data02a, estimator ="MLR", std.lv =TRUE)summary(CFA_1factor.model, fit.measures =TRUE, standardized =TRUE)```## CFA Example: Two-factor model```{r}#two factor CFA: one item removed from factor 2 and zero covariance between factorsCFA_2factor.syntax ="factor1 =~ X1 + X3 + X5 + X9 + X10 + X13 + X14 + X18 + X21 + X23factor2 =~ X3 + X5 + X9 + X10 + X13 + X14 + X18 + X21 + X23factor1 ~ 0*factor2"#for comparison with EFA we are using standardized factors (var = 1; mean = 0)CFA_2factor.model =cfa(model = CFA_2factor.syntax, data = data02a, estimator ="MLR", std.lv =TRUE)summary(CFA_2factor.model, fit.measures =TRUE, standardized =TRUE)```## Wrapping up1. CFA is an alternative model with EFA. 2. The difference is that we have more control over the model structure using CFA, such as we can specify which item(s) loaded on which factors. 3. EFA is not necessary and CFA can be the first step. You just keep revising the model structure until you get acceptable model fitting.4. Another alternative modeling for dependency among items is network modeling.>>>>>>> a79e7de (local changes) - Need to set the right number of constraints for identification - We set the value of factor loadings for a few items on a few of the factors - Typically to zero - Sometimes to one (Brown, 2002) - We keep the factor covariance matrix as an identity - Uncorrelated factors (as in EFA) with variances of one<<<<<<< HEAD- Benefits of using CFA for exploratory analyses:- CFA constraints remove rotational indeterminacy of factor loadings – no rotating is needed (or possible) - Defines factors with potentially less ambiguity - Constraints are easy to see- For some software (SAS and SPSS), we get much more model fit information## Example of CFASee the [example](https://www.lavaan.ugent.be/tutorial/cfa.html) from lavaan package# Background=======# Network Analysis>>>>>>> a79e7de (local changes)## Network AnalysisNetwork analysis is a broad area. It has many names in varied fields:1. Graphical Models (Computer Science, Machine Learning)2. Bayesian Network (Computer Science, Educational Measurement)3. Social Network (Sociology)4. Psychological/Psychometric Network (Psychopathology, Psychology)5. Structural Equation Model, Path Analysis (Psychology, Education)1 and 2 focus on the probabilistic relationships and further casual relationships among variables. 3 and 4 focuses on network structure and node importance. 5 focus on the regression coefficients of structural and measurement model.All 5 analysis methods have a network-shaped diagram. [**Graphical modeling**]{.underline} is a more "general" term that can comprise of the other network models.## Examples of Varied Networks::: {layout="[[40,40], [100]]" layout-halign="bottom"}{#fig-BN-DAG}{#fig-social-network}{#fig-GGM}:::## Research Aims1. Bayesian Network (BN) aims to derive the casual relations between variables2. Social Network aims to examine the network structure (community, density or centrality) of [individuals]{.underline}3. Factor Analysis aims to identify latent variables4. Psychological Network aims to examines the associations among observed variables (topological structures) and their positions in the network.## Network Psychometrics1. Network psychometrics is a novel area that allows representing complex phenomena of measured constructs in terms of a set of elements that interact with each other.2. It is inspired by the so-called [mutualism model]{.underline} and research in ecosystem modeling [@kan2019]. - A mutualism model proposes that basic cognitive abilities directly and positively interact during development.3. Psychometric network arises as [dynamics or reciprocal causation]{.underline} among variables are getting more attentions.4. For example, individual differences in depression could arise from, and could be maintained by, vicious cycles of mutual relationships among symptoms.5. A depression symptom such as insomnia can cause another symptom, such as fatigue, which in turn can determine concentration problems and worrying, which can result in more insomnia and so on.## Comparison to factor analysis[**Factor analysis**]{.underline} (common factor model) assumes the associations between observed features can be explained by one or more common factors.- For example, higher "depression" level leads to increased frequency of depressive behaviors[**Psychometric network**]{.underline}, however, assumes the associations between observed features ARE the reason of the development of depression. Or "depression" is the network itself.- unavoidable cycle of "fatigue -\> worrying -\> insomnia -\> fatigue" will leads to higher "depression"## Utility of Psychometric Network1. Explain the pathways of certain psychological phenomenon2. Identify the most important problem that needed to be intervene in the treatment procedure3. Examine group differences in interactions among observed features4. Examine density of network: more dense network indicates more dynamic of certain problems5. Examine clusters/communities of observed features: some symptoms are more likely to concur than other symptoms## Terminology I - Overall procedure1. [Network structure estimation]{.underline}: the application of statistical models to assess the structure of pairwise (conditional) associations in multivariate data.2. [Network description]{.underline}: characterization of the global topology and the position of individual nodes in that topology.3. [Psychometric network analysis]{.underline}: the analysis of multivariate psychometric data using network structure estimation and network description.## Terminology II - Network description::: columns::: {.column width="50%"}1. [Node]{.underline}: psychometric variables that are selected in the network - such as responses to questionnaire items, symptom ratings, and cognitive test scores, background variables such as age and gender, experimental interventions.2. [Edge (conditional association)]{.underline}: associations between variables taking into account other variables that may explain the association3. [Edge weight]{.underline}: parameter estimates that represent the strength of conditional association between nodes4. [Node centrality]{.underline}: the relative importance of a node in the context of other nodes, that can be calculated using different statistics:::::: {.column width="50%"}::::::## Terminology III - Network structure estimation1. [Node selection]{.underline}: the choice of which variables will function as nodes in the network model.2. [Network stability analysis]{.underline}: the assessment of estimation precision and robustness to sampling error of psychometric networks.3. [Pairwise Markov random field (PMRF)]{.underline}: an undirected network that represents variables as nodes and conditional associations as edges, in which unconnected nodes are conditionally independent.## Exploratory NaturePsychometric network is exploratory by nature. To obtain a meaningful network structure, psychometric networks need to drop weak edges but keep strong edges.This procedure is typically called edge selection. One popular edge selection method is [regularization]{.underline}.- Original network structure without regularization is called saturated network; vice versa regularized network::: columns::: {.column width="50%"}{fig-align="center"}:::::: {.column width="50%"}::::::## Workflow of psychometric network::: columns::: {.column width="60%"}{fig-align="center"}:::::: {.column width="40%"}Psychometric network analysis methodology includes steps of network structure estimation (to construct the network), network description (to characterize the network) and network stability analysis (to assess the robustness of results).::::::## Types of data and network models1. Cross-sectional data (N = large, T = 1) - **Ising** model for categorical variables - Gaussian graphical model (**GGM**; Foygel & Drton, 2010) for continuous variables - Mixed graphical model (**MGM**) for mixed types of variables: include both categorical variables and continuous variables2. Panel data (N \>\> T) - Multilevel Graphical vector autoregressive model (**GVAR**) - i.e., longitudinal data, repeated measures3. Time-series data ($N \geq 1$, T = large) - Graphical vector autoregression - i.e., ecological momentary assessment, conducted via smartphones# Network Estimation## Gaussian graphical modelGGM is one type of Pairwise Markov random field (PMRF) when data are continuous.In a PMRF, the joint likelihood of multivariate data is modeled through the use of pairwise conditional associations, leading to a network representation that is undirected.For $p$-dimensional data following multivariate normal distribution:$$\boldsymbol{X} \sim \mathcal{N}(\mu, \boldsymbol{K}^{-1})$$Where $K$ is a inverse covariance matrix of $\boldsymbol{X}$ ($K = \Sigma^{-1}$), also known as *precision*/*concentration* matrix.To obtain sparse network structure, the $i$th row and $j$th column element of $\boldsymbol{K}$, $k_{ij}=0$ when edge $\{j, k\}$ is not included in the network $G$,- It means $X^{(i)}$ and $X^{(j)}$ are independent conditional on the other variables .## Partial correlation networksGGM can be standardized as the partial correlation network, in which each edge of GGM representing partial correlations between two nodes.$$\rho_{ij}=\rho(X^{(i)}, X^{(j)}|\boldsymbol{X}^{-(i,j)}) = -\frac{k_{ij}}{\sqrt{k_{ii}}\sqrt{k_{jj}}}$$Assume there are three variables: fatigue, insomnia, concentration::: columns::: {.column width="50%"}```{r}#| code-summary: 'True Covariance Matrix - $\boldsymbol{\Sigma}$'Sigma =matrix(c(1, -.26, .31,-.26, 1, -.08, .31, -.08, 1), ncol =3, byrow = T)Sigma``````{r}#| code-summary: 'True Precision Matrix - K'K =solve(Sigma)round(K, 2)```:::::: {.column width="50%"}```{r}#| code-summary: 'Partial correlation matrix - R'R = Kfor (i in1:nrow(R)) {for (j in1:ncol(R)){if (i != j) { R[i, j] =- K[i, j] / (sqrt(K[i, i])*sqrt(K[j, j])) }else{ R[i, j] =1 } }}round(R, 2)```::::::## Network Interpretation1. Someone who is tired is also more likely to suffer from concentration problems and insomnia.2. Concentration problems and insomnia are conditional independent given the level of fatigue - Or the correlation between insomnia and concentration can be totally explained by the relationships of both variables with fatigue```{r}#| fig-width: 7#| fig-height: 3qgraph::qgraph(R, labels =c("Fatigue", "Concentration", "Insomnia"))```## Estimation of partial correlation model for cross-sectional dataFactor analysis model:$$\boldsymbol{X}\sim\mathcal{N}(\mu, \boldsymbol{\Lambda\Psi\Lambda^\text{T}+\Phi})$$GGM with partial correlation matrix:$$\boldsymbol{X} \sim \mathcal{N}(0, \boldsymbol{\Delta(I-\Omega)^{-1}\Delta})$$Where1. $\boldsymbol{\Delta}$ is a diagonal scaling matrix that controls the variances2. $\boldsymbol{\Omega}$ is a square symmetrical matrix with $0$s on the diagonal and partial correlation coefficients on the off diagonal.## `psychonetrics`: Partial correlation matrix estimation::: columns::: {.column width="50%"}```{r}#| code-summary: 'Population covariance matrix - $\Sigma$'Sigma |>round(3)``````{r}#| code-summary: 'Population partial correlation matrix - R'R |>round(3)```:::::: {.column width="50%"}```{r}#| code-summary: 'Estimated sample partial correlation matrix - $\hat\Omega$'library(psychonetrics)fit =ggm(covs = Sigma, nobs =50) |>runmodel()Omega <- fit |>getmatrix("omega") Omega |>round(3)# estimated Omega``````{r}#| code-summary: 'Estimated sample scaling matrix - $\hat\Delta$'Delta <- fit |>getmatrix(matrix ="delta") Delta |>round(3)``````{r}#| code-summary: 'Estimated sample covaraicne matrix - $\hat S$'S = Delta %*%solve(diag(1, 3) - Omega) %*% DeltaS |>round(3)```::::::## `BGGM`: Bayesian approach```{r}#| code-summary: 'Posterior means of partial correlation matrix'library(BGGM)set.seed(1234)dat <- mvtnorm::rmvnorm(500, mean =rep(0, 3), sigma = Sigma)fit_bggm <- BGGM::estimate(dat, type ="continuous", iter =1000, analytic =FALSE)fit_bggm$pcor_mat |>round(3)``````{r}#| code-summary: 'Posterior distributions of partial correlations'summary(fit_bggm)```## Edge Selection: RegularizationMultiple procedure and software can be used to perform edge selection:1. `prune` function in `psychonetrics` package uses stepdown model search by pruning non-significant parameters - a edge with significance level lower than $\alpha$ will be removed and re-fit the model until no nonsignificant edge existed - $p$ values of edges needed to be adjusted2. `EBICglasso` in `qgraph` and `glasso` package uses Extended Bayesian Information Criterion (EBIC) to select best model $$ \text{EBIC}=-2\text{L}+E(\log(N))+4\gamma E(log(P)) $$3. `select` function in [`BGGM`](https://osf.io/preprints/psyarxiv/ypxd8) package uses Bayesian Hypothesis Testing — Bayes Factor (BF) to select model - $\mathcal{H}_0: \rho_{ij}=0$ - $\mathcal{H}_1: \rho_{ij}\in(-1, 1)$ - $$ BF = \frac{p(\boldsymbol{X}|\mathcal{H}_0)}{p(\boldsymbol{X}|\mathcal{H}_1)} $$ - By default, edges with BF \< 3 will be included## Network Description1. Network-level metrics: - **Network stability**: The assessment of estimation precision and robustness to sampling error of psychometric networks. Assess whether edge weights and node centrality changes with case dropping subset bootstrap. - **Network sensitivity**: The sensitivity analysis of the network by adding covariates (age, gender, hukou, education, marital status, and self-rated health) to the model to control their confounding effects. Report whether some central nodes are no longer central after adding some variables.2. Node-level metrics: - **Total number of node** - **Node centrality**: the position of individual nodes within the network. A generic term that subsumes a family of measures that aim to assess how central a node is in a network topology, such as node strength, betweenness and closeness. - **Node bridge strength**: the degree to which one node connects two nodes from different facets. The bridge nodes (nodes with high bridge strength) that connect different communities of nodes (e.g., Neurotic factors is bridge nodes between personality traits and depressive symptoms @liPersonalityTraitsDepressive2024). - **Node centrality differences between groups**: node A is central in group 1's network but not so central in group 2's network.3. Edge-level metrics: - **Total Number of edges**: proportions of non-zero edges - **Edge weight**: edge weights typically are parameter estimates that represent the strength of the conditional association between nodes. **Average edge weight** can be used to assess the overall strength of network connections.# Example: Big Five Personality Scale## Data Information25 self-reported personality items representing 5 factors.70.6% edges are nonzero using EBICglasso, while only 63.6% edges are nonzero using BF method and 62.0% edges are kept using $\alpha =.01$ significance testing```{r}#| code-summary: 'EBICglasso'library(psych)library(qgraph)data(bfi)big5groups <-list(Agreeableness =1:5,Conscientiousness =6:10,Extraversion =11:15,Neuroticism =16:20,Openness =21:25)CorMat <-cor_auto(bfi[,1:25])EBICgraph <-EBICglasso(CorMat, nrow(bfi), 0.5, threshold =TRUE)## densitydensity_nonzero_edge <-function(pcor_matrix){ N_nonzero_edge = (sum(pcor_matrix ==0) -ncol(pcor_matrix)) /2 N_all_edge =ncol(pcor_matrix)*(ncol(pcor_matrix)-1)/2 N_nonzero_edge/N_all_edge}density_nonzero_edge(EBICgraph)``````{r}#| code-summary: 'Significance testing'PruneFit <-ggm(bfi[,1:25]) |>prune(alpha = .01)density_nonzero_edge(getmatrix(PruneFit, "omega"))``````{r}#| code-summary: 'Bayes Factor'BGGMfit <- BGGM::explore(bfi[,1:25], type ="continuous", iter =1000, analytic =FALSE) |>select()density_nonzero_edge(BGGMfit$pcor_mat_zero)```## Network Structure::: {layout-nrow="2"}::: column```{r}#| echo: false#| fig-cap: "EBICglasso"library(qgraph)qgraph(EBICgraph, groups = big5groups)```:::::: column```{r}#| echo: false#| fig-cap: "Sig. Test" prune_omega <-getmatrix(PruneFit, "omega")colnames(prune_omega) <-colnames(bfi[,1:25])qgraph(prune_omega, labels=colnames(bfi[,1:25]), groups = big5groups)```:::::: {.column width="50%"}```{r}#| echo: false#| fig-cap: "BF"BGGMfit_omega <- BGGMfit$pcor_mat_zerocolnames(BGGMfit_omega) <-colnames(bfi[,1:25])qgraph(BGGMfit_omega, labels=colnames(bfi[,1:25]), groups = big5groups)```::::::## Centrality - StrengthStrength centrality measures suggest that `C4` and `E4` or `N1` have highest centrality indicating they play most imporatant roles in the networks.```{r}#| fig-subcap:#| - "Method 1: EBICglasso"#| - "Method 2: Sig. Test"#| - "Method 3: Bayes Factor"#| layout-ncol: 3#| fig-height: 12#| echo: falselibrary(ggplot2)p1 <-centralityPlot(EBICgraph, print =FALSE) p2 <-centralityPlot(prune_omega, print =FALSE)p3 <-centralityPlot(BGGMfit_omega, print =FALSE)p1$layers[[2]] <-NULLp2$layers[[2]] <-NULLp3$layers[[2]] <-NULLhl1 =!(colnames(bfi[,1:25])%in%c("C4", "E4"))hl2 =!(colnames(bfi[,1:25])%in%c("C4", "E4"))hl3 =!(colnames(bfi[,1:25])%in%c("N1", "C4"))p1 +geom_point(aes(color = hl1), size =10)+theme(text =element_text(size =30), legend.position="none")p2 +geom_point(aes(color = hl2), size =10)+theme(text =element_text(size =30), legend.position="none")p3 +geom_point(aes(color = hl3), size =10)+theme(text =element_text(size =30), legend.position="none")```## Centrality - Bridge`E4` and `E5` has highest bridge strength, indicating they serve as bridges linking communities of personality. They are important elements connecting varied types of personality```{r}#| fig-subcap:#| - "Method 1: EBICglasso"#| - "Method 2: Sig. Test"#| - "Method 3: Bayes Factor"#| layout-ncol: 3#| fig-height: 12#| echo: falselibrary(tidyverse)library(networktools)p1 <-bridge(EBICgraph, communities = big5groups)[[1]]p2 <-bridge(prune_omega, communities = big5groups)[[1]]p3 <-bridge(BGGMfit$pcor_mat_zero, communities = big5groups)[[1]]bridge_plot <-function(p) {data.frame(Node =colnames(bfi[,1:25]),Bridge =as.numeric(p) ) |>ggplot(aes(y = Node, x = Bridge)) +geom_path(group =1) +geom_point(group =1, size =10) +theme_bw()+theme(text =element_text(size =30), legend.position="none")}bridge_plot(p1)bridge_plot(p2)bridge_plot(p3)```## Wrapping up1. Network analysis is an alternative way of modeling dependency among variables.2. Gaussian graphical model is used for multivariate continuous data.3. The goal is to estimate network structure, node importance, and stability.## Other Materials:1. [Jihong's post: how to choose network analysis estimation](https://jihongzhang.org/notes/2024-04-04-Network-Estimation-Methods)## Reference
