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
```