Example: Inferred binary causal graph from simulation

title: “BiCausality: Binary Causality Inference Framework” author: “ C. Amornbunchornvej” date: “2023-11-28” output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{BiCausality_demo} %\VignetteEngine{knitr::knitr} \usepackage[utf8]{inputenc}

Example: Inferred binary causal graph from simulation

In the first step, we generate a simulation dataset as an input.

seedN<-2022

n<-200 # 200 individuals
d<-10 # 10 variables
mat<-matrix(nrow=n,ncol=d) # the input of framework

#Simulate binary data from binomial distribution where the probability of value being 1 is 0.5.
for(i in seq(n))
{
  set.seed(seedN+i)
  mat[i,] <- rbinom(n=d, size=1, prob=0.5)
}

mat[,1]<-mat[,2] | mat[,3]  # 1 causes by 2 and 3
mat[,4] <-mat[,2] | mat[,5] # 4 causses by 2 and 5
mat[,6] <- mat[,1] | mat[,4] # 6 causes by 1 and 4

We use the following function to infer whether X causes Y.

# Run the function
library(BiCausality)
resC<-BiCausality::CausalGraphInferMainFunc(mat = mat,CausalThs=0.1, nboot =50, IndpThs=0.05)

## Inferring dependent graph

## Removing confounder(s)

## Inferring causal graph

The result of the adjacency matrix of the directed causal graph is below:

resC$CausalGRes$Ehat

##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
##  [1,]    0    0    0    0    0    1    0    0    0     0
##  [2,]    1    0    0    1    0    0    0    0    0     0
##  [3,]    1    0    0    0    0    0    0    0    0     0
##  [4,]    0    0    0    0    0    1    0    0    0     0
##  [5,]    0    0    0    1    0    0    0    0    0     0
##  [6,]    0    0    0    0    0    0    0    0    0     0
##  [7,]    0    0    0    0    0    0    0    0    0     0
##  [8,]    0    0    0    0    0    0    0    0    0     0
##  [9,]    0    0    0    0    0    0    0    0    0     0
## [10,]    0    0    0    0    0    0    0    0    0     0

The value in the element EValHat[i,j] represents that i causes j if the value is not zero. For example, EValHat[2,1] = 1 implies node 2 causes node 1, which is correct since node 1 have nodes 2 and 3 as causal nodes.

The directed causal graph also can be plot using the code below.

library(igraph)

## 
## Attaching package: 'igraph'

## The following objects are masked from 'package:stats':
## 
##     decompose, spectrum

## The following object is masked from 'package:base':
## 
##     union

net <- graph_from_adjacency_matrix(resC$CausalGRes$Ehat ,weighted = NULL)
plot(net, edge.arrow.size = 0.3, vertex.size =20 , vertex.color = '#D4C8E9',layout=layout_with_kk)

For the causal relation of variables 2 and 1, we can use the command below to see further information.

**Note that the odd difference between X and Y denoted oddDiff(X,Y) is define as |P (X = 1, Y = 1) P (X = 0, Y = 0) −P (X = 0, Y = 1) P (X = 1, Y = 0)|. If X is directly proportional to Y, then oddDiff(X,Y) is close to 1. If X is inverse of Y, then oddDiff(X,Y) is close to -1. If X and Y have no association, then oddDiff(X,Y) is close to zero.

resC$CausalGRes$causalInfo[['2,1']]

## $CDirConfValInv
##  2.5% 97.5% 
##     1     1 
## 
## $CDirConfInv
##      2.5%     97.5% 
## 0.3152526 0.4386415 
## 
## $CDirmean
## [1] 0.371347
## 
## $testRes2
## 
## 	Wilcoxon signed rank test with continuity correction
## 
## data:  abs(bCausalDirDist)
## V = 1275, p-value = 3.893e-10
## alternative hypothesis: true location is greater than 0.1
## 
## 
## $testRes1
## 
## 	Wilcoxon signed rank test with continuity correction
## 
## data:  abs(bSignDist)
## V = 1275, p-value = 3.889e-10
## alternative hypothesis: true location is greater than 0.05
## 
## 
## $sign
## [1] 1
## 
## $SignConfInv
##      2.5%     97.5% 
## 0.0865425 0.1282719 
## 
## $Signmean
## [1] 0.1090915

Below are the details of result explanation.

#This value represents the 95th percentile confidence interval of P(Y=1|X=1). 
$CDirConfValInv
 2.5% 97.5% 
    1     1 
#This value represents the 95th percentile confidence interval of |P(Y=1|X=1) - P(X=1|Y=1)|.
$CDirConfInv
     2.5%     97.5% 
0.3217322 0.4534494 

#This value represents the mean of |P(Y=1|X=1) - P(X=1|Y=1)|.
$CDirmean
[1] 0.3787904

#The test that has the null hypothesis that |P(Y=1|X=1) - P(X=1|Y=1)| below
#or equal the argument of parameter "CausalThs" and the alternative hypothesis
#is that |P(Y=1|X=1) - P(X=1|Y=1)| is greater than "CausalThs".
$testRes2

	Wilcoxon signed rank test with continuity correction

data:  abs(bCausalDirDist)
V = 1275, p-value = 3.893e-10
alternative hypothesis: true location is greater than 0.1


#The test that has the null hypothesis that |oddDiff(X,Y)| below 
#or equal the argument of parameter "IndpThs" and the alternative hypothesis is
#that |oddDiff(X,Y)| is greater than "IndpThs". 
$testRes1

	Wilcoxon signed rank test with continuity correction

data:  abs(bSignDist)
V = 1275, p-value = 3.894e-10
alternative hypothesis: true location is greater than 0.05

#If the test above rejects the null hypothesis with the significance threshold
#alpha (default alpha=0.05), then the value "sign=1", otherwise, it is zero.
$sign
[1] 1

#This value represents the 95th percentile confidence interval of oddDiff(X,Y)
$SignConfInv
      2.5%      97.5% 
0.08670325 0.13693900 

#This value represents the mean of oddDiff(X,Y)
$Signmean
[1] 0.1082242