How to use BGGM to Estimate Bayesian Gaussian Graphical Models

Bayesian version of GGM allows multiple Bayesian techniques to be used in psychological network.

R
Bayesian
Tutorial
Network Analysis
Author

Jihong Zhang

Published

March 5, 2024

This is a quick note illustrate some importance functions of BGGM and illustrate them using one or more example(s).

Overview

  1. The Github page showcases some publications regarding algorithms of GGGM
  2. Donny Williams’s website has more examples

Illustrative Example 1

Click to see the code
pcor_to_df <- function(x, names = paste0("B", 1:5)) {
  colnames(x) = rownames(x) = names
  x_upper <- x
  x_upper[lower.tri(x_upper)] <- 0
  as.data.frame(x_upper) |> 
    rownames_to_column('Start') |> 
    pivot_longer(-Start, names_to = 'End', values_to = 'Mean') |> 
    filter(Mean != 0) |> 
    mutate(Label = paste0(Start,'-', End))  
}

A dataset containing items that measure Post-traumatic stress disorder symptoms (Armour et al. 2017). There are 20 variables (p) and 221 observations (n).

Estimation of Partial Correlation Matrix

The GGM can be estimated using estimate function with some important arguments:

  • type: could be continuous, binary, ordinal, or mixed

  • iter: Number of iterations (posterior samples; defaults to 5000).

  • prior_sd: Scale of the prior distribution, approximately the standard deviation of a beta distribution (defaults to 0.50).

Click to see the code
library(BGGM)
library(tidyverse) # for ggplot2 and dplyr
library(psychonetrics) # the frequentist way for GGM

# data
Y <- ptsd[, 1:5] + 1 # add + 1 to make sure the ordinal variable start from '1'

# ordinal model
fit <- estimate(Y, type = 'ordinal', analytic = FALSE)
fit_cont <- estimate(Y, type = 'continuous', analytic = FALSE)

#summary(fit)
pcor_mat(fit)
      B1     B2    B3     B4    B5
B1 0.000  0.240 0.054  0.338 0.127
B2 0.240  0.000 0.502 -0.061 0.128
B3 0.054  0.502 0.000  0.245 0.190
B4 0.338 -0.061 0.245  0.000 0.353
B5 0.127  0.128 0.190  0.353 0.000

Let’s compare the partial correlation estimates to psychonetrics

Click to see the code
mod_saturated_ord <- ggm(Y, ordered = TRUE)
mod_saturated_cont <- ggm(Y)
getmatrix(mod_saturated_ord, 'omega')
           [,1]       [,2]       [,3]       [,4]      [,5]
[1,] 0.00000000  0.2655136 0.01768286  0.3593433 0.1242903
[2,] 0.26551363  0.0000000 0.55895807 -0.1174681 0.1195668
[3,] 0.01768286  0.5589581 0.00000000  0.2819463 0.1845554
[4,] 0.35934330 -0.1174681 0.28194629  0.0000000 0.3738825
[5,] 0.12429026  0.1195668 0.18455542  0.3738825 0.0000000
Click to see the code
freqEst <- pcor_to_df(getmatrix(mod_saturated_ord, 'omega')) 
freqEst_cont <- pcor_to_df(getmatrix(mod_saturated_cont, 'omega')) 

# transform partial corr. mat. of BGGM into df.
BayesEst <- pcor_to_df(pcor_mat(fit))
BayesEst_cont <- pcor_to_df(pcor_mat(fit_cont)) 

combEst <- rbind(
  freqEst |> mutate(Type = 'psychonetrics_ord'),
  BayesEst |> mutate(Type = 'BGGM_ord'),
  freqEst_cont |> mutate(Type = 'psychonetrics_cont'),
  BayesEst_cont |> mutate(Type = 'BGGM_cont')
) |> 
  mutate(Type = factor(Type, levels = c("psychonetrics_ord", "BGGM_ord", "psychonetrics_cont","BGGM_cont")))
ggplot(combEst) +
  geom_col(aes(x = fct_rev(Label), y = Mean, fill = Type, col = Type), position = position_dodge()) +
  geom_text(aes(x = fct_rev(Label), y = ifelse(Mean>0, Mean+.03, Mean-.03), label = round(Mean, 3), group = Type), position = position_dodge(width = .9)) +
  labs(x = '') +
  coord_flip() +
  theme_classic()

We can find that:

  1. Partial correlations with ordinal responses estimated by psychonetric do not align well with partial correlations with ordinal responses estimated by BGGM
  2. Partial correlations with continuous responses estimated by psychonetric align well with partial correlations with continuous responses estimated by BGGM
  3. Violating the continuous assumption of items responses usually underestimate the partial correlations (see BGGM_cont vs. BGGM_ord)

Regularization

It is easy to conduct regularization using Graph Selection using hypothesis testing

Click to see the code
fit_reg <- BGGM::select(fit, alternative = "greater")
plot(fit_reg)
$plt

Hypothesis testing

Using BGGM, we can perform hypothesis testing like whether edge A is more strong than edge B, whether edge A is “significant” higher or smaller than 0, whether group A and group B have differences in partial correlations.

Testing Edge A vs. Edge B

Click to see the code
# example hypotheses
hyp1 <- c("B2--B3 > B1--B4 > 0") # hypothesis 1
hyp2 <- c("B1--B5 > B2--B5 > 0") # hypothesis 2
hyp3 <- c("B2--B4 < 0") # hypothesis 3
hyp4 <- c("B1--B3 < 0") # hypothesis 4


## Posterior hypothesis probabilities.
extract_php <- function(Y = Y, hypothesis = hyp, type = 'ordinal') {
  x <- confirm(Y = Y, hypothesis = hypothesis, type = type)
  x_info <- x$info
  PHP_confirmatory <- as.matrix(x_info$PHP_confirmatory)
  rownames(PHP_confirmatory) <- c(x$hypothesis, "complement")
  PHP_confirmatory
}

extract_php(Y = Y, hypothesis = hyp1, type = 'ordinal')
                     [,1]
B2--B3 > B1--B4 > 0 0.989
complement          0.011
Click to see the code
extract_php(Y = Y, hypothesis = hyp2, type = 'ordinal')
                     [,1]
B1--B5 > B2--B5 > 0 0.866
complement          0.134
Click to see the code
extract_php(Y = Y, hypothesis = hyp3, type = 'ordinal')
            [,1]
B2--B4 < 0 0.708
complement 0.292
Click to see the code
extract_php(Y = Y, hypothesis = hyp4, type = 'ordinal')
            [,1]
B1--B3 < 0 0.253
complement 0.747

There are four hypothesis tested here:

  1. The first expresses that the relation B2--B3 is larger than B1--B4 . In other words, that the partial correlation is larger for B2--B3. The p(H1|data) = 0.991 means the posterior hypothesis probability (PHP) is 0.991 which provides evidence for the hypothesis: the partial correlation for B2-B3 is larger;
  2. The second expresses that the relation B1--B5 is larger than B2--B5 . The results of PHP suggests the hypothesis can be accepted;
  3. The third expresses that the relation B2--B4 is larger than 0 . The results of PHP suggests the hypothesis can be accepted;
  4. The fourth expresses that the relation B1--B3 is lower than 0 . The results of PHP suggests the hypothesis should be rejected given p(H_1) < p(complement);

Group Comparison

Click to see the code
# data
Y <- bfi

# males
Ymales <- subset(Y, gender == 1, 
                 select = -c(gender, education))

# females
Yfemales <- subset(Y, gender == 2, 
                 select = -c(gender, education))

# model fit
fit <- ggm_compare_estimate(Ymales, Yfemales)

# plot summary
summary(fit)
BGGM: Bayesian Gaussian Graphical Models 
--- 
Type: continuous 
Analytic: FALSE 
Formula:  
Posterior Samples: 5000 
Observations (n):
  Group 1: 805 
  Group 2: 1631 
Nodes (p): 25 
Relations: 300 
--- 
Call: 
ggm_compare_estimate(Ymales, Yfemales)
--- 
Estimates:

 
 Relation Post.mean Post.sd Cred.lb Cred.ub
 A1--A2    0.048    0.039   -0.029   0.125 
 A1--A3   -0.022    0.040   -0.101   0.054 
 A2--A3   -0.033    0.038   -0.111   0.044 
 A1--A4   -0.006    0.041   -0.088   0.075 
 A2--A4   -0.049    0.040   -0.128   0.028 
 A3--A4    0.038    0.039   -0.039   0.114 
 A1--A5    0.000    0.041   -0.082   0.081 
 A2--A5    0.086    0.041    0.004   0.165 
 A3--A5    0.063    0.038   -0.011   0.138 
 A4--A5   -0.024    0.040   -0.102   0.057 
 A1--C1   -0.020    0.041   -0.101   0.058 
 A2--C1   -0.027    0.041   -0.110   0.052 
 A3--C1    0.025    0.041   -0.054   0.106 
 A4--C1    0.010    0.041   -0.070   0.091 
 A5--C1    0.051    0.041   -0.029   0.133 
 A1--C2   -0.025    0.041   -0.105   0.054 
 A2--C2    0.013    0.041   -0.068   0.094 
 A3--C2    0.003    0.041   -0.074   0.082 
 A4--C2    0.046    0.040   -0.036   0.125 
 A5--C2    0.014    0.041   -0.063   0.094 
 C1--C2    0.024    0.038   -0.049   0.100 
 A1--C3    0.012    0.040   -0.066   0.091 
 A2--C3    0.022    0.040   -0.057   0.100 
 A3--C3   -0.032    0.042   -0.113   0.051 
 A4--C3    0.034    0.041   -0.046   0.113 
 A5--C3   -0.019    0.041   -0.101   0.062 
 C1--C3    0.041    0.040   -0.036   0.119 
 C2--C3    0.023    0.040   -0.055   0.101 
 A1--C4    0.015    0.040   -0.061   0.095 
 A2--C4    0.021    0.041   -0.059   0.101 
 A3--C4    0.103    0.040    0.024   0.184 
 A4--C4   -0.038    0.041   -0.119   0.043 
 A5--C4   -0.047    0.040   -0.125   0.031 
 C1--C4    0.062    0.041   -0.017   0.142 
 C2--C4    0.029    0.038   -0.045   0.102 
 C3--C4    0.003    0.040   -0.075   0.083 
 A1--C5   -0.039    0.041   -0.118   0.042 
 A2--C5   -0.005    0.041   -0.085   0.075 
 A3--C5    0.034    0.041   -0.047   0.114 
 A4--C5    0.048    0.040   -0.032   0.125 
 A5--C5    0.048    0.041   -0.032   0.128 
 C1--C5   -0.084    0.041   -0.164  -0.005 
 C2--C5    0.074    0.041   -0.008   0.155 
 C3--C5    0.098    0.040    0.018   0.174 
 C4--C5    0.050    0.038   -0.024   0.123 
 A1--E1    0.051    0.041   -0.030   0.129 
 A2--E1    0.017    0.041   -0.064   0.097 
 A3--E1    0.024    0.041   -0.057   0.102 
 A4--E1    0.001    0.041   -0.078   0.082 
 A5--E1    0.006    0.041   -0.073   0.086 
 C1--E1    0.062    0.041   -0.019   0.143 
 C2--E1    0.005    0.040   -0.075   0.084 
 C3--E1    0.068    0.040   -0.012   0.147 
 C4--E1   -0.007    0.041   -0.087   0.074 
 C5--E1    0.010    0.040   -0.071   0.090 
 A1--E2    0.009    0.041   -0.069   0.089 
 A2--E2    0.041    0.041   -0.040   0.121 
 A3--E2   -0.011    0.041   -0.091   0.066 
 A4--E2   -0.051    0.040   -0.130   0.027 
 A5--E2   -0.027    0.041   -0.106   0.053 
 C1--E2   -0.002    0.041   -0.083   0.079 
 C2--E2    0.039    0.041   -0.042   0.119 
 C3--E2   -0.159    0.040   -0.238  -0.078 
 C4--E2   -0.018    0.041   -0.101   0.063 
 C5--E2    0.081    0.040    0.003   0.159 
 E1--E2    0.012    0.039   -0.064   0.088 
 A1--E3   -0.033    0.041   -0.115   0.045 
 A2--E3   -0.067    0.041   -0.148   0.015 
 A3--E3   -0.037    0.040   -0.115   0.043 
 A4--E3   -0.005    0.040   -0.082   0.074 
 A5--E3   -0.034    0.041   -0.114   0.046 
 C1--E3    0.069    0.041   -0.011   0.149 
 C2--E3   -0.001    0.040   -0.079   0.078 
 C3--E3   -0.034    0.041   -0.114   0.045 
 C4--E3   -0.007    0.041   -0.088   0.071 
 C5--E3    0.000    0.040   -0.080   0.078 
 E1--E3   -0.032    0.041   -0.109   0.049 
 E2--E3   -0.063    0.041   -0.147   0.014 
 A1--E4    0.033    0.041   -0.046   0.115 
 A2--E4    0.031    0.042   -0.051   0.112 
 A3--E4    0.026    0.040   -0.054   0.105 
 A4--E4   -0.063    0.040   -0.143   0.017 
 A5--E4   -0.025    0.039   -0.103   0.052 
 C1--E4   -0.004    0.041   -0.084   0.077 
 C2--E4   -0.077    0.041   -0.159   0.002 
 C3--E4    0.023    0.041   -0.057   0.104 
 C4--E4   -0.026    0.041   -0.106   0.052 
 C5--E4   -0.022    0.041   -0.101   0.057 
 E1--E4   -0.036    0.039   -0.114   0.040 
 E2--E4    0.031    0.039   -0.047   0.107 
 E3--E4    0.029    0.041   -0.051   0.110 
 A1--E5   -0.013    0.041   -0.093   0.065 
 A2--E5    0.022    0.040   -0.057   0.101 
 A3--E5   -0.023    0.040   -0.098   0.056 
 A4--E5    0.121    0.041    0.040   0.200 
 A5--E5   -0.023    0.041   -0.104   0.059 
 C1--E5   -0.018    0.041   -0.099   0.063 
 C2--E5    0.005    0.041   -0.073   0.084 
 C3--E5   -0.028    0.041   -0.107   0.051 
 C4--E5    0.007    0.041   -0.074   0.089 
 C5--E5   -0.024    0.041   -0.102   0.058 
 E1--E5    0.004    0.041   -0.077   0.084 
 E2--E5    0.000    0.041   -0.080   0.080 
 E3--E5    0.063    0.040   -0.015   0.139 
 E4--E5    0.090    0.040    0.011   0.172 
 A1--N1    0.059    0.040   -0.019   0.138 
 A2--N1    0.060    0.041   -0.021   0.141 
 A3--N1   -0.048    0.041   -0.130   0.032 
 A4--N1    0.025    0.041   -0.057   0.105 
 A5--N1   -0.004    0.041   -0.084   0.078 
 C1--N1   -0.039    0.041   -0.119   0.043 
 C2--N1   -0.001    0.041   -0.081   0.076 
 C3--N1    0.069    0.041   -0.013   0.150 
 C4--N1   -0.037    0.041   -0.118   0.041 
 C5--N1   -0.021    0.041   -0.105   0.059 
 E1--N1    0.068    0.041   -0.015   0.148 
 E2--N1   -0.022    0.042   -0.102   0.061 
 E3--N1   -0.010    0.042   -0.091   0.070 
 E4--N1   -0.008    0.041   -0.087   0.074 
 E5--N1   -0.018    0.041   -0.099   0.062 
 A1--N2   -0.041    0.040   -0.119   0.037 
 A2--N2   -0.121    0.041   -0.198  -0.041 
 A3--N2    0.074    0.041   -0.007   0.154 
 A4--N2   -0.049    0.040   -0.127   0.032 
 A5--N2    0.061    0.042   -0.021   0.143 
 C1--N2    0.042    0.040   -0.038   0.121 
 C2--N2   -0.051    0.041   -0.129   0.029 
 C3--N2   -0.030    0.041   -0.110   0.050 
 C4--N2   -0.028    0.041   -0.108   0.053 
 C5--N2    0.037    0.041   -0.042   0.119 
 E1--N2    0.006    0.041   -0.074   0.087 
 E2--N2   -0.034    0.041   -0.114   0.046 
 E3--N2   -0.048    0.041   -0.131   0.031 
 E4--N2    0.043    0.042   -0.039   0.124 
 E5--N2    0.044    0.041   -0.035   0.126 
 N1--N2    0.049    0.028   -0.008   0.103 
 A1--N3   -0.072    0.040   -0.152   0.005 
 A2--N3    0.060    0.042   -0.023   0.141 
 A3--N3   -0.031    0.041   -0.113   0.048 
 A4--N3    0.024    0.041   -0.056   0.103 
 A5--N3    0.051    0.041   -0.031   0.130 
 C1--N3    0.036    0.041   -0.046   0.116 
 C2--N3   -0.007    0.040   -0.084   0.073 
 C3--N3    0.043    0.041   -0.038   0.123 
 C4--N3    0.074    0.041   -0.008   0.153 
 C5--N3    0.042    0.041   -0.037   0.124 
 E1--N3   -0.082    0.041   -0.162  -0.001 
 E2--N3   -0.032    0.040   -0.110   0.047 
 E3--N3   -0.081    0.041   -0.161   0.001 
 E4--N3   -0.117    0.041   -0.197  -0.034 
 E5--N3    0.045    0.041   -0.034   0.126 
 N1--N3   -0.005    0.039   -0.082   0.068 
 N2--N3   -0.100    0.040   -0.177  -0.021 
 A1--N4    0.054    0.041   -0.027   0.135 
 A2--N4    0.018    0.041   -0.062   0.098 
 A3--N4    0.039    0.041   -0.041   0.122 
 A4--N4    0.025    0.042   -0.059   0.106 
 A5--N4   -0.096    0.041   -0.174  -0.013 
 C1--N4    0.054    0.041   -0.025   0.131 
 C2--N4   -0.033    0.041   -0.115   0.048 
 C3--N4   -0.086    0.041   -0.167  -0.006 
 C4--N4   -0.102    0.041   -0.181  -0.023 
 C5--N4    0.034    0.040   -0.046   0.112 
 E1--N4    0.007    0.040   -0.073   0.085 
 E2--N4   -0.016    0.041   -0.098   0.062 
 E3--N4    0.156    0.041    0.075   0.236 
 E4--N4    0.091    0.041    0.012   0.170 
 E5--N4   -0.090    0.041   -0.169  -0.010 
 N1--N4   -0.067    0.041   -0.149   0.012 
 N2--N4    0.066    0.041   -0.014   0.146 
 N3--N4    0.086    0.037    0.014   0.157 
 A1--N5   -0.065    0.040   -0.143   0.012 
 A2--N5   -0.034    0.042   -0.115   0.049 
 A3--N5   -0.010    0.041   -0.092   0.070 
 A4--N5   -0.041    0.041   -0.120   0.039 
 A5--N5   -0.031    0.041   -0.112   0.049 
 C1--N5   -0.016    0.041   -0.097   0.063 
 C2--N5   -0.024    0.041   -0.103   0.054 
 C3--N5    0.045    0.041   -0.035   0.125 
 C4--N5    0.070    0.040   -0.008   0.150 
 C5--N5   -0.106    0.041   -0.185  -0.025 
 E1--N5   -0.016    0.041   -0.097   0.066 
 E2--N5    0.032    0.040   -0.045   0.109 
 E3--N5    0.002    0.041   -0.081   0.081 
 E4--N5   -0.017    0.040   -0.095   0.063 
 E5--N5    0.032    0.040   -0.045   0.110 
 N1--N5   -0.054    0.041   -0.136   0.027 
 N2--N5    0.013    0.041   -0.067   0.094 
 N3--N5   -0.013    0.040   -0.092   0.066 
 N4--N5    0.028    0.040   -0.048   0.108 
 A1--O1   -0.022    0.040   -0.102   0.058 
 A2--O1   -0.008    0.041   -0.088   0.072 
 A3--O1    0.043    0.040   -0.036   0.123 
 A4--O1    0.019    0.041   -0.061   0.099 
 A5--O1   -0.086    0.040   -0.165  -0.009 
 C1--O1   -0.008    0.041   -0.089   0.073 
 C2--O1    0.045    0.040   -0.033   0.123 
 C3--O1   -0.096    0.041   -0.176  -0.016 
 C4--O1    0.001    0.042   -0.081   0.084 
 C5--O1   -0.040    0.041   -0.121   0.041 
 E1--O1   -0.053    0.040   -0.130   0.025 
 E2--O1    0.054    0.041   -0.028   0.135 
 E3--O1   -0.063    0.040   -0.143   0.016 
 E4--O1    0.004    0.040   -0.075   0.083 
 E5--O1   -0.041    0.040   -0.120   0.039 
 N1--O1   -0.034    0.041   -0.114   0.049 
 N2--O1   -0.007    0.041   -0.087   0.074 
 N3--O1    0.065    0.041   -0.014   0.143 
 N4--O1   -0.002    0.041   -0.085   0.078 
 N5--O1   -0.021    0.041   -0.104   0.059 
 A1--O2    0.002    0.040   -0.076   0.080 
 A2--O2    0.036    0.041   -0.042   0.116 
 A3--O2    0.048    0.041   -0.032   0.126 
 A4--O2    0.021    0.040   -0.058   0.100 
 A5--O2   -0.039    0.041   -0.119   0.040 
 C1--O2   -0.014    0.040   -0.096   0.064 
 C2--O2   -0.064    0.041   -0.144   0.015 
 C3--O2   -0.015    0.042   -0.098   0.063 
 C4--O2   -0.008    0.040   -0.087   0.073 
 C5--O2   -0.072    0.041   -0.153   0.007 
 E1--O2    0.072    0.041   -0.010   0.153 
 E2--O2    0.093    0.041    0.014   0.173 
 E3--O2    0.024    0.041   -0.057   0.103 
 E4--O2    0.021    0.040   -0.058   0.099 
 E5--O2    0.038    0.041   -0.041   0.119 
 N1--O2   -0.013    0.041   -0.095   0.065 
 N2--O2   -0.005    0.041   -0.086   0.075 
 N3--O2    0.034    0.040   -0.044   0.112 
 N4--O2   -0.030    0.041   -0.109   0.050 
 N5--O2    0.011    0.041   -0.070   0.092 
 O1--O2    0.049    0.040   -0.031   0.127 
 A1--O3    0.042    0.041   -0.041   0.120 
 A2--O3    0.044    0.041   -0.038   0.126 
 A3--O3    0.032    0.041   -0.048   0.111 
 A4--O3    0.003    0.041   -0.077   0.084 
 A5--O3    0.005    0.041   -0.076   0.084 
 C1--O3   -0.035    0.041   -0.116   0.046 
 C2--O3    0.036    0.040   -0.043   0.114 
 C3--O3    0.012    0.041   -0.068   0.091 
 C4--O3   -0.056    0.041   -0.137   0.023 
 C5--O3    0.034    0.041   -0.046   0.114 
 E1--O3    0.004    0.041   -0.076   0.084 
 E2--O3    0.081    0.041   -0.003   0.161 
 E3--O3    0.020    0.040   -0.059   0.097 
 E4--O3   -0.022    0.041   -0.101   0.056 
 E5--O3   -0.100    0.041   -0.180  -0.021 
 N1--O3    0.051    0.041   -0.028   0.131 
 N2--O3   -0.027    0.042   -0.108   0.054 
 N3--O3   -0.018    0.040   -0.096   0.062 
 N4--O3   -0.040    0.041   -0.121   0.041 
 N5--O3    0.038    0.041   -0.045   0.118 
 O1--O3    0.066    0.039   -0.012   0.144 
 O2--O3    0.015    0.040   -0.063   0.095 
 A1--O4    0.121    0.041    0.040   0.199 
 A2--O4    0.010    0.041   -0.070   0.092 
 A3--O4    0.041    0.040   -0.038   0.120 
 A4--O4    0.049    0.041   -0.033   0.128 
 A5--O4   -0.024    0.041   -0.104   0.056 
 C1--O4   -0.021    0.041   -0.101   0.060 
 C2--O4   -0.032    0.041   -0.111   0.049 
 C3--O4   -0.031    0.042   -0.115   0.049 
 C4--O4    0.012    0.040   -0.066   0.089 
 C5--O4    0.028    0.040   -0.050   0.107 
 E1--O4    0.064    0.041   -0.015   0.142 
 E2--O4    0.024    0.041   -0.056   0.105 
 E3--O4    0.055    0.041   -0.028   0.133 
 E4--O4    0.002    0.040   -0.075   0.081 
 E5--O4    0.011    0.040   -0.070   0.090 
 N1--O4   -0.018    0.041   -0.097   0.063 
 N2--O4    0.018    0.041   -0.062   0.098 
 N3--O4   -0.062    0.041   -0.140   0.016 
 N4--O4   -0.006    0.040   -0.086   0.072 
 N5--O4   -0.016    0.040   -0.097   0.063 
 O1--O4    0.071    0.040   -0.008   0.151 
 O2--O4   -0.043    0.041   -0.123   0.037 
 O3--O4   -0.027    0.040   -0.105   0.052 
 A1--O5    0.059    0.041   -0.024   0.136 
 A2--O5   -0.015    0.041   -0.095   0.064 
 A3--O5   -0.003    0.041   -0.085   0.076 
 A4--O5    0.040    0.040   -0.038   0.118 
 A5--O5    0.069    0.041   -0.009   0.151 
 C1--O5   -0.061    0.041   -0.141   0.019 
 C2--O5    0.041    0.041   -0.038   0.124 
 C3--O5   -0.016    0.041   -0.094   0.065 
 C4--O5    0.006    0.040   -0.073   0.084 
 C5--O5    0.093    0.040    0.012   0.172 
 E1--O5    0.018    0.040   -0.061   0.097 
 E2--O5    0.013    0.041   -0.067   0.093 
 E3--O5    0.020    0.041   -0.060   0.100 
 E4--O5    0.063    0.040   -0.016   0.141 
 E5--O5   -0.063    0.040   -0.141   0.015 
 N1--O5    0.023    0.040   -0.057   0.100 
 N2--O5   -0.039    0.041   -0.119   0.040 
 N3--O5    0.083    0.041    0.005   0.165 
 N4--O5   -0.025    0.041   -0.106   0.054 
 N5--O5    0.009    0.041   -0.071   0.089 
 O1--O5   -0.024    0.041   -0.104   0.059 
 O2--O5   -0.026    0.039   -0.105   0.049 
 O3--O5   -0.008    0.039   -0.084   0.070 
 O4--O5    0.043    0.040   -0.035   0.123 
---

The regularized group difference network plot can be visualized as:

Click to see the code
plot(BGGM::select(fit))
[[1]]

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