The MR.RGM R package presents a crucial advancement in Mendelian randomization (MR) studies, providing a robust solution to a common challenge. While MR has proven invaluable in establishing causal links between exposures and outcomes, its traditional focus on single exposures and specific outcomes can be limiting. Biological systems often exhibit complexity, with interdependent outcomes influenced by numerous factors. MR.RGM introduces a network-based approach to MR, allowing researchers to explore the broader causal landscape.

With two available functions, RGM and NetworkMotif, the package offers versatility in analyzing causal relationships. RGM primarily focuses on constructing causal networks among response variables and between responses and instrumental variables. On the other hand, NetworkMotif specializes in quantifying uncertainty for given network structures among response variables.

RGM accommodates both individual-level data and two types of summary-level data, making it adaptable to various data availability scenarios. This adaptability enhances the package’s utility across different research contexts. The outputs of RGM include estimates of causal effects, adjacency matrices, and other relevant parameters. Together, these outputs contribute to a deeper understanding of the intricate relationships within complex biological networks, thereby enriching insights derived from MR studies.

You can install MR.RGM R package from CRAN with:

```
install.packages("MR.RGM")
```

Once the MR.RGM package is installed load the library in the R work-space.

```
library("MR.RGM")
```

We offer a concise demonstration of the capabilities of the RGM function within the package, showcasing its effectiveness in computing causal interactions among response variables and between responses and instrumental variables using simulated data sets. Subsequently, we provide an example of how NetworkMotif can be applied, utilizing a specified network structure and Gamma_Pst acquired from executing the RGM function.

```
# Model: Y = AY + BX + E
# Set seed
set.seed(9154)
# Number of data points
n = 10000
# Number of response variables and number of instrument variables
p = 5
k = 6
# Create d vector
d = c(2, 1, 1, 1, 1)
# Initialize causal interaction matrix between response variables
A = matrix(sample(c(-0.1, 0.1), p^2, replace = TRUE), p, p)
# Diagonal entries of A matrix will always be 0
diag(A) = 0
# Make the network sparse
A[sample(which(A!=0), length(which(A!=0))/2)] = 0
# Initialize causal interaction matrix between response and instrument variables
B = matrix(0, p, k)
# Initialize m
m = 1
# Calculate B matrix based on d vector
for (i in 1:p) {
# Update ith row of B
B[i, m:(m + d[i] - 1)] = 1
# Update m
m = m + d[i]
}
# Create variance-covariance matrix
Sigma = 1 * diag(p)
Mult_Mat = solve(diag(p) - A)
Variance = Mult_Mat %*% Sigma %*% t(Mult_Mat)
# Generate instrument data matrix
X = matrix(runif(n * k, 0, 5), nrow = n, ncol = k)
# Initialize response data matrix
Y = matrix(0, nrow = n, ncol = p)
# Generate response data matrix based on instrument data matrix
for (i in 1:n) {
Y[i, ] = MASS::mvrnorm(n = 1, Mult_Mat %*% B %*% X[i, ], Variance)
}
# Print true causal interaction matrices between response variables and between response and instrument variables
A
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.0 -0.1 0.0 0.0 0.1
#> [2,] 0.1 0.0 -0.1 0.1 0.1
#> [3,] 0.0 -0.1 0.0 0.0 0.1
#> [4,] 0.0 -0.1 0.0 0.0 0.0
#> [5,] 0.0 0.1 0.0 0.0 0.0
B
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 0 0 0 0
#> [2,] 0 0 1 0 0 0
#> [3,] 0 0 0 1 0 0
#> [4,] 0 0 0 0 1 0
#> [5,] 0 0 0 0 0 1
```

We will now apply RGM based on individual level data, summary level data and Beta, Sigma_Hat matrices to show its functionality.

```
# Apply RGM on individual level data with Threshold prior
Output1 = RGM(X = X, Y = Y, d = c(2, 1, 1, 1, 1), prior = "Threshold")
# Calculate summary level data
S_YY = t(Y) %*% Y / n
S_YX = t(Y) %*% X / n
S_XX = t(X) %*% X / n
# Apply RGM on summary level data for Spike and Slab Prior
Output2 = RGM(S_YY = S_YY, S_YX = S_YX, S_XX = S_XX,
d = c(2, 1, 1, 1, 1), n = 10000, prior = "Spike and Slab")
# Calculate Beta and Sigma_Hat
# Centralize Data
Y = t(t(Y) - colMeans(Y))
X = t(t(X) - colMeans(X))
# Calculate S_XX
S_XX = t(X) %*% X / n
# Generate Beta matrix and Sigma_Hat
Beta = matrix(0, nrow = p, ncol = k)
Sigma_Hat = matrix(0, nrow = p, ncol = k)
for (i in 1:p) {
for (j in 1:k) {
fit = lm(Y[, i] ~ X[, j])
Beta[i, j] = fit$coefficients[2]
Sigma_Hat[i, j] = sum(fit$residuals^2) / n
}
}
# Apply RGM on S_XX, Beta and Sigma_Hat for Spike and Slab Prior
Output3 = RGM(S_XX = S_XX, Beta = Beta, Sigma_Hat = Sigma_Hat,
d = c(2, 1, 1, 1, 1), n = 10000, prior = "Spike and Slab")
```

We get the estimated causal interaction matrix between response variables in the following way:

```
Output1$A_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.00000000 -0.11140615 0.0000000 0.0000000 0.10761964
#> [2,] 0.09927277 0.00000000 -0.1111652 0.1000334 0.10912908
#> [3,] 0.00000000 -0.09637386 0.0000000 0.0000000 0.09759288
#> [4,] 0.00000000 -0.10203596 0.0000000 0.0000000 0.00000000
#> [5,] 0.00000000 0.09928744 0.0000000 0.0000000 0.00000000
Output2$A_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.000000000 -0.11262237 0.0009547800 0.003000252 0.10640122
#> [2,] 0.100675105 0.00000000 -0.1121416975 0.100621628 0.10774403
#> [3,] -0.002229899 -0.09529822 0.0000000000 0.001269346 0.10016839
#> [4,] -0.003257993 -0.10542234 -0.0008876829 0.000000000 0.01154689
#> [5,] 0.001466929 0.10033516 -0.0072119898 -0.002885797 0.00000000
Output3$A_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0.000000000 -0.08916909 0.040323704 0.0004407186 0.09569033
#> [2,] 0.111429981 0.00000000 -0.113294802 0.0967872414 0.13633727
#> [3,] 0.016102229 -0.09532651 0.000000000 0.0103318146 0.11898981
#> [4,] -0.006334381 -0.11169543 0.002591126 0.0000000000 0.01366949
#> [5,] -0.002296943 0.13176051 0.012056218 -0.0010876040 0.00000000
```

We get the estimated causal network structure between the response variables in the following way:

```
Output1$zA_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 1 0 0 1
#> [2,] 1 0 1 1 1
#> [3,] 0 1 0 0 1
#> [4,] 0 1 0 0 0
#> [5,] 0 1 0 0 0
Output2$zA_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 1 0 0 1
#> [2,] 1 0 1 1 1
#> [3,] 0 1 0 0 1
#> [4,] 0 1 0 0 0
#> [5,] 0 1 0 0 0
Output3$zA_Est
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 0 1 0 0 1
#> [2,] 1 0 1 1 1
#> [3,] 0 1 0 0 1
#> [4,] 0 1 0 0 0
#> [5,] 0 1 0 0 0
```

We observe that the causal network structures inferred in the three outputs mentioned are identical. To gain a clearer understanding of the network, we compare the true network structure with the one estimated by RGM. Since the networks derived from all three outputs are consistent, we plot a single graph representing the estimated causal network.

```
# Define a function to create smaller arrowheads
smaller_arrowheads <- function(graph) {
igraph::E(graph)$arrow.size = 0.60 # Adjust the arrow size value as needed
return(graph)
}
# Create a layout for multiple plots
par(mfrow = c(1, 2))
# Plot the true causal network
plot(smaller_arrowheads(igraph::graph.adjacency((A != 0) * 1,
mode = "directed")), layout = igraph::layout_in_circle,
main = "True Causal Network")
# Plot the estimated causal network
plot(smaller_arrowheads(igraph::graph.adjacency(Output1$zA_Est,
mode = "directed")), layout = igraph::layout_in_circle,
main = "Estimated Causal Network")
```

We get the estimated causal interaction matrix between the response and the instrument variables from the outputs in the following way:

```
Output1$B_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.9935119 1.008009 0.000000 0.0000000 0.0000000 0.0000000
#> [2,] 0.0000000 0.000000 0.997496 0.0000000 0.0000000 0.0000000
#> [3,] 0.0000000 0.000000 0.000000 0.9998662 0.0000000 0.0000000
#> [4,] 0.0000000 0.000000 0.000000 0.0000000 0.9995511 0.0000000
#> [5,] 0.0000000 0.000000 0.000000 0.0000000 0.0000000 0.9982094
Output2$B_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.9937211 1.0075 0.0000000 0.0000000 0.0000000 0.000000
#> [2,] 0.0000000 0.0000 0.9964755 0.0000000 0.0000000 0.000000
#> [3,] 0.0000000 0.0000 0.0000000 0.9990566 0.0000000 0.000000
#> [4,] 0.0000000 0.0000 0.0000000 0.0000000 0.9987975 0.000000
#> [5,] 0.0000000 0.0000 0.0000000 0.0000000 0.0000000 1.002271
Output3$B_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.9902686 1.004035 0.0000000 0.0000000 0.0000000 0.0000000
#> [2,] 0.0000000 0.000000 0.9928588 0.0000000 0.0000000 0.0000000
#> [3,] 0.0000000 0.000000 0.0000000 0.9988565 0.0000000 0.0000000
#> [4,] 0.0000000 0.000000 0.0000000 0.0000000 0.9970532 0.0000000
#> [5,] 0.0000000 0.000000 0.0000000 0.0000000 0.0000000 0.9965687
```

We get the estimated graph structure between the response and the instrument variables from the outputs in the following way:

```
Output1$zB_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 0 0 0 0
#> [2,] 0 0 1 0 0 0
#> [3,] 0 0 0 1 0 0
#> [4,] 0 0 0 0 1 0
#> [5,] 0 0 0 0 0 1
Output2$zB_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 0 0 0 0
#> [2,] 0 0 1 0 0 0
#> [3,] 0 0 0 1 0 0
#> [4,] 0 0 0 0 1 0
#> [5,] 0 0 0 0 0 1
Output3$zB_Est
#> [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 1 1 0 0 0 0
#> [2,] 0 0 1 0 0 0
#> [3,] 0 0 0 1 0 0
#> [4,] 0 0 0 0 1 0
#> [5,] 0 0 0 0 0 1
```

We can plot the log-likelihoods from the outputs in the following way:

```
plot(Output1$LL_Pst, type = 'l', xlab = "Iterations", ylab = "Log-likelihood")
```

```
plot(Output2$LL_Pst, type = 'l', xlab = "Iterations", ylab = "Log-likelihood")
```

```
plot(Output3$LL_Pst, type = 'l', xlab = "Iterations", ylab = "Log-likelihood")
```

Next, we present the implementation of the NetworkMotif function. We begin by defining a random subgraph among the response variables. Subsequently, we collect Gamma_Pst arrays from various outputs and proceed to execute NetworkMotif based on these arrays.

```
# Start with a random subgraph
Gamma = matrix(0, nrow = p, ncol = p)
Gamma[5, 2] = Gamma[3, 5] = Gamma[2, 3] = 1
# Plot the subgraph to get an idea about the causal network
plot(smaller_arrowheads(igraph::graph.adjacency(Gamma,
mode = "directed")), layout = igraph::layout_in_circle,
main = "Subgraph")
```

```
# Store the Gamma_Pst arrays from outputs
Gamma_Pst1 = Output1$Gamma_Pst
Gamma_Pst2 = Output2$Gamma_Pst
Gamma_Pst3 = Output3$Gamma_Pst
# Get the posterior probabilities of Gamma with these Gamma_Pst matrices
NetworkMotif(Gamma = Gamma, Gamma_Pst = Gamma_Pst1)
#> [1] 1
NetworkMotif(Gamma = Gamma, Gamma_Pst = Gamma_Pst2)
#> [1] 0.3795
NetworkMotif(Gamma = Gamma, Gamma_Pst = Gamma_Pst3)
#> [1] 0.485125
```

Yang Ni. Yuan Ji. Peter Müller. “Reciprocal Graphical Models for Integrative Gene Regulatory Network Analysis.” Bayesian Anal. 13 (4) 1095 - 1110, December 2018. https://doi.org/10.1214/17-BA1087

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