Relative-importance networks

library(psychnets)

A relative-importance network (method = "relimp") is directed. For each node taken as an outcome, it regresses it on all the others and decomposes that regression’s R-squared into each predictor’s share using the LMG / Shapley method (averaging the predictor’s contribution over every ordering). The edge A -> B is the share of B’s variance explained by A. Unlike a GGM, the weights sum meaningfully: a node’s incoming edges add up to its full-model R-squared.

The Shapley decomposition enumerates predictor subsets, so it is meant for a modest number of nodes. We use the five construct scores in SRL_GPT.

ri <- psychnet(SRL_GPT, method = "relimp")
ri
#> <psychnet> relimp network
#>   nodes: 5   edges: 20   (directed)
#>   optimality (KKT residual): 1.11e-16

The incoming shares per node sum to that node’s R-squared; the certificate confirms this efficiency identity holds to numerical precision:

certificate(ri)
#>   method  certificate       kind certified
#> 1 relimp 1.110223e-16 structural      TRUE

The directed edges – each predictor’s share of an outcome’s variance:

as.data.frame(ri)
#>    from  to      weight
#> 1    IV CSU 0.288505266
#> 2    SE CSU 0.263574096
#> 3    SR CSU 0.274082666
#> 4    TA CSU 0.007326148
#> 5   CSU  IV 0.299473895
#> 6    SE  IV 0.216286692
#> 7    SR  IV 0.258608009
#> 8    TA  IV 0.007032583
#> 9   CSU  SE 0.280588934
#> 10   IV  SE 0.219009605
#> 11   SR  SE 0.220573037
#> 12   TA  SE 0.006045547
#> 13  CSU  SR 0.277393999
#> 14   IV  SR 0.253052645
#> 15   SE  SR 0.212824511
#> 16   TA  SR 0.052422918
#> 17  CSU  TA 0.014154964
#> 18   IV  TA 0.015416846
#> 19   SE  TA 0.012055817
#> 20   SR  TA 0.127772080

Per-node explained variance that the incoming edges reconstruct:

net_predict(ri)
#>   node     type metric predictability accuracy
#> 1  CSU gaussian     R2      0.8334882       NA
#> 2   IV gaussian     R2      0.7814012       NA
#> 3   SE gaussian     R2      0.7262171       NA
#> 4   SR gaussian     R2      0.7956941       NA
#> 5   TA gaussian     R2      0.1693997       NA

Plotting

cograph::splot() draws the directed network with arrows. Because each pair has two distinct edges (A -> B and B -> A), pass directed = TRUE so they are drawn separately, with psych_styling = TRUE (green = positive, red = negative) and the predictability ring via predictability = TRUE.

cograph::splot(ri, directed = TRUE, psych_styling = TRUE, predictability = TRUE)

- Plotting