paper.md

In donaldRwilliams/GGMnonreg: Non-Regularized Gaussian Graphical Models

title: 'GGMnonreg: Non-Regularized Gaussian Graphical Models in R' tags: - Graphical models - partial correlations - Mixed graphical model - Ising model authors: - name: Donald R. Williams affiliation: "1, 2" affiliations: - name: Department of Psychology, University of California, Davis index: 1 - name: NWEA, Portland, USA index: 2 citation_author: Williams date: 08 November 2021 year: 2021 bibliography: inst/REFERENCES.bib

Summary

Studying complex relations in multivariate datasets is a common task across the sciences. Cognitive neuroscientists model brain connectivity with the goal of unearthing functional and structural associations between cortical regions [@ortiz_2015]. In clinical psychology, researchers wish to better understand the intricate web of symptom interrelations that underlie mental health disorders [@mcnally_2016; @borsboom_small_world]. To this end, graphical modeling has emerged as an oft-used tool in the chest of scientific inquiry. The basic idea is to characterize multivariate relations by learning the conditional dependence structure. The cortical regions or symptoms are nodes and the featured connections linking nodes are edges that graphically represent the conditional dependence structure.

Graphical modeling is quite common in fields with wide data, that is, when there are more variables ($p$) than observations ($n$). Accordingly, many regularization-based approaches have been developed for those kinds of data. There are key drawbacks of regularization, including, but not limited to, the fact that obtaining a valid measure of parameter uncertainty is very (very) difficult [@Buhlmann2014] and there can be an inflated false positive rate [see for example, @williams2019nonregularized].

Statement of Need

More recently, graphical modeling has emerged in psychology (Epskamp et al. 2018), where the data is typically long or low-dimensional ($p < n$; @williams2019nonregularized, @williams_rethinking). The primary purpose of GGMnonreg is to provide methods that were specifically designed for low-dimensional data (e.g., those common in the social-behavioral sciences).

Supported Models

• Gaussian graphical model (GGM). The following data types are supported.
• Gaussian
• Ordinal
• Binary
• Ising model [@marsman_2018]
• Mixed graphical model

Additional methods

The following are also included

• Expected network replicability [@williams2020learning]
• Compare Gaussian graphical models
• Measure of parameter uncertainty [@williams_2021_conf]
• Edge inclusion "probabilities" [e.g., Figure 6.4 in @Hastie2015]
• Network visualization
• Constrained precision matrix [the network, given an assumed graph, see p. 631 in @hastie2009elements]
• Predictability [variance explained for each node, @Haslbeck2018]

Gaussian graphical Model

The following estimates a GGM for 5 post-traumatic stress disorder (PTSD) symptoms [@armour2017network]:

fit <- ggm_inference(Y = ptsd[,1:5], 
                     boot = FALSE)

fit
#>           1         2         3         4         5
#> 1 0.0000000 0.2262934 0.0000000 0.3335737 0.1547986
#> 2 0.2262934 0.0000000 0.4993419 0.0000000 0.0000000
#> 3 0.0000000 0.4993419 0.0000000 0.2205442 0.1841798
#> 4 0.3335737 0.0000000 0.2205442 0.0000000 0.3407634
#> 5 0.1547986 0.0000000 0.1841798 0.3407634 0.0000000

Predictability

It is common to then estimate "predictability", which corresponds to $R^2$ for each node in the network. In GGMnonreg, this is implemented with the following code:

predictability(fit)

#>   Estimate Est.Error Ci.lb Ci.ub
#> 1     0.45      0.05  0.35  0.54
#> 2     0.50      0.05  0.41  0.59
#> 3     0.55      0.04  0.47  0.64
#> 4     0.50      0.05  0.41  0.59
#> 5     0.46      0.05  0.37  0.55

Ising Model

An Ising model is for binary data. The PTSD symptoms can be binary, indicating the symptom was either present or absent. This network is estimated with:

# make binary
Y <- ifelse(ptsd[,1:5] == 0, 0, 1)

# fit model
fit <- ising_search(Y, IC = "BIC", 
                    progress = FALSE)

fit
#>          1        2        3        4        5
#> 1 0.000000 1.439583 0.000000 1.273379 0.000000
#> 2 1.439583 0.000000 1.616511 0.000000 1.182281
#> 3 0.000000 1.616511 0.000000 1.716747 1.077322
#> 4 1.273379 0.000000 1.716747 0.000000 1.662550
#> 5 0.000000 1.182281 1.077322 1.662550 0.000000

Network Replicability

Recently, the topic of replicability has captivated the network literature. To this end, I developed an analytic solution to estimate network replicability [@williams2020learning].

The first step is to define a "true" partial correlation network. As an example, I generate a synthetic partial correlation matrix, and then compute expected network replicability.

# edges between 0.05 and 0.25
main <- gen_net(p = 20, 
                lb = 0.05, 
                ub = 0.25)

# enr                
enr(main$pcors, 
    n = 500, 
    replications = 4)

#> Average Replicability: 0.53 
#> Average Number of Edges: 30 (SD = 2.12) 
#> 
#> ----
#> 
#> Cumulative Probability: 
#> 
#>  prop.edges edges Pr(R > prop.edges)
#>         0.0     0               1.00
#>         0.1     6               1.00
#>         0.2    11               1.00
#>         0.3    17               1.00
#>         0.4    23               1.00
#>         0.5    28               0.78
#>         0.6    34               0.02
#>         0.7    40               0.00
#>         0.8    46               0.00
#>         0.9    51               0.00
----
Pr(R > prop.edges):
probability of replicating more than the
correpsonding proportion (and number) of edges

On average, we can expect to replicate roughly half of the edges in four replication attempts, where replication is defined as detecting a given edge in each attempt. Further, the probability of replicating more than 70% of the edges is zero.

Network Visualization

A key aspect of graphical modeling is visualizing the conditional dependence structure. To this end, GGMnonreg makes network plots with ggplot2 [@ggplotpackage].

plot(fit, 
     node_names = colnames(Y), 
     edge_magnify = 2)

Acknowledgements

DRW was supported by a National Science Foundation Graduate Research Fellowship under Grant No. 1650042

References

donaldRwilliams/GGMnonreg documentation built on Nov. 13, 2021, 9:57 a.m.