GGMtest

Hypothesis testing for conditional independence in gaussian graphical models. This repository is the official implementation of Uniform inference in high-dimensional Gaussian graphical models.

Description

Testing conditional independence in gaussian graphical models becomes a high-dimensional inference problem due to the large number of possible edges. To increase power in contrast to multiple testing adjustments, we employ the double machine learning framework.

Installing GGMtest

To install this package in R, run the following commands:

install.packages("devtools")
library(devtools)
install_github("SvenKlaassen/GGMtest")

Examples

As a small example simulate an example with n = 100 observations of p = 10 dimensional multivariate normal distribution.

rm(list=ls())

library("huge")
library("igraph")
library("GGMtest")

set.seed(42)

# generate data (different graph structures: "random", "hub", "cluster", "band" and "scale-free") 
L <- huge.generator(n = 100, d = 10, graph = "cluster", g = 4) 
# true Graph
true_graph <- graph_from_adjacency_matrix(as.matrix(L$theta), mode='undirected', diag=F)

plot(true_graph, usearrows = FALSE, label=1:10, displaylabels=T, main = "True Graph",
     layout= layout.fruchterman.reingold, edge.width = 2, edge.color = "black")

# index pairs for inference
edgeset_1 <- matrix(c(1,2,2,3,2,4,1,4,4,5), byrow = T, ncol = 2)

# perform test:
ggm_model <- GGMtest(data = L$data,
                     edges = edgeset_1,
                     alpha = 0.05,
                     nuisance_estimaton = "lasso")

# p-value:                     
ggm_model$pvalue_max

# plot the confidence intervals (on a subset of edges)
plot_GGMtest(ggm_model,edges = edgeset_1)

Next lets take a look at a more complex example by increasing n = 2500 and p = 25. This time we will add all egdes (300 since the graph is undirected) for inference and plot the network.

L <- huge.generator(n = 2500, d = 25, graph = "random")

# all indices for inference
edgeset_2 <- t(combn(25,2))
ggm_model_2 <- GGMtest(L$data,edgeset_2, alpha = 0.05,nuisance_estimaton ="lasso")

# true Graph
par(mfrow=c(1,3))
graph_layout <- layout.circle 
#graph_layout <- layout.fruchterman.reingold
true_adj_matrix <- as.matrix(L$theta)
true_graph <- graph_from_adjacency_matrix(true_adj_matrix , mode='undirected', diag=F )
plot(true_graph, usearrows = FALSE, vertex.label=1:25, displaylabels=T, main = "True Graph",
     layout= graph_layout,edge.color = "black",edge.width = 2)

# estimated graph
est_adj_matrix <- adj_GGMtest(ggm_model_2)
est_graph <- igraph::graph_from_adjacency_matrix(est_adj_matrix , mode='undirected', diag=F )
plot(est_graph, usearrows = FALSE, vertex.label=1:25, displaylabels=T, main = "Estimated Graph",
     layout= graph_layout,edge.color = "black",edge.width = 2)

#calculation matrix difference
diff_matrix <- true_adj_matrix - est_adj_matrix 
diff_graph <- igraph::graph_from_adjacency_matrix(diff_matrix  , mode='undirected', diag=F, weighted = T)
E(diff_graph)$color[E(diff_graph)$weight == 1] <- "blue"
E(diff_graph)$color[E(diff_graph)$weight == -1] <- "red"
plot(diff_graph, usearrows = FALSE, vertex.label=1:25, displaylabels=T, main = "Graph Differences",
     layout= graph_layout,edge.width = 2)
legend(x=-1.5, y=-1.1, c("False Positive","False Negative"), pch=21,
       col="#777777", pt.bg=c("red","blue"), pt.cex=2, cex=.8, bty="n", ncol=1)

par(mfrow=c(1,1))

Application

A small application from psychology. The bfi dataset contains n = 2800 oberservations of 25 personality self report intems. For a documentation on the covariables see bfi in the psych package.

library("psych")
library("dplyr")

df <- bfi %>%
  select(1:25) %>%
  na.omit() %>%
  apply(2,function(x) x-mean(x))

# all indices for inference
edgeset_3 <- t(combn(25,2))

ggm_model_3 <- GGMtest(df,edgeset_3,alpha = 0.05,nuisance_estimaton ="lasso")

# estimated graph
est_adj_matrix <- adj_GGMtest(ggm_model_3)
est_graph <- igraph::graph_from_adjacency_matrix(est_adj_matrix , mode='undirected', diag=F )
plot(est_graph, usearrows = FALSE, vertex.label=colnames(df),displaylabels=T,main = "Estimated Graph",
     layout= layout.fruchterman.reingold,edge.color = "black",edge.width = 2)

References

Klaassen, S., Kück, J., Spindler, M., & Chernozhukov, V. (2018). Uniform inference in high-dimensional Gaussian graphical models. arXiv preprint arXiv:1808.10532.
Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/debiased machine learning for treatment and structural parameters.
Belloni, Alexandre, et al. "Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework." Annals of statistics 46.6B (2018): 3643.