# The Role of SNP Interactions when Determining Independence of Novel Signals in Genetic Association Studies—An Application to ARG1 and Bronchodilator Response

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}conditional on the SNP X

_{2}. We generated 1000 subjects for 5000 simulations using a significance level of 5*10

^{−8}. SNP X

_{1}is generated from a binomial distribution with a binary genetic coding (i.e., dominant or recessive model) and P(X

_{1}= 1) = 0.5. SNP X

_{2}is generated from a logistic regression based on X

_{1}such that:

_{1}and X

_{2}assume a binary genetic coding for simplicity, the results are generalizable to an additive genetic coding (i.e., X

_{j}= 0, 1, 2 for j = 1, 2). The continuous outcome Y is generated from a normal distribution with a variance of 1 and a mean such that

_{I}varies from 0.3 to 1 by 0.05. We considered additional simulation scenarios for different values for ${\gamma}_{0}$, ${\gamma}_{1}$ in Equation (1) and $\beta $

_{0}, ${\beta}_{1}$, ${\beta}_{2}$ in Equation (2). However, we observed similar results to the presented simulations scenarios; therefore, the results are not shown here.

**Algorithm 0:**Fitting $E\left[Y\right]={\delta}_{0}+{\delta}_{1}{X}_{1}$, we tested ${H}_{0}:{\delta}_{1}=0$ to determine if the SNP ${X}_{1}$ is associated with the trait of interest Y.

**Algorithm 1:**Fitting $E\left[Y\right]={\alpha}_{0}+{\alpha}_{1}{X}_{1}+{\alpha}_{2}{X}_{2},\text{}\mathrm{we}\text{}\mathrm{tested}\text{}{H}_{0}:{\alpha}_{1}=0$ to determine if the SNP ${X}_{1}$ is associated with the trait of interest Y conditional on the SNP ${X}_{2}$.

**Algorithm 2:**Fitting $E\left[Y\right]={\phi}_{0}+{\phi}_{1}{X}_{1}+{\phi}_{2}{X}_{2}+{\phi}_{I}{X}_{1}{X}_{2}$, we tested ${H}_{0}:{\phi}_{I}=0$ to determine if there is an interaction of the 2 SNPs on the trait of interest Y.

**Scenario 1:**Rejected Algorithm 0 ${H}_{0}:{\delta}_{1}=0$, Algorithm 1 ${H}_{0}:{\alpha}_{1}=0$, and Algorithm 2 ${H}_{0}:{\phi}_{I}=0$.

**Scenario 2:**Rejected Algorithm 0 ${H}_{0}:{\delta}_{1}=0$ and Algorithm 1 ${H}_{0}:{\alpha}_{1}=0$, but failed to reject ${H}_{0}$ for Algorithm 2.

**Scenario 3:**Rejected Algorithm 0 ${H}_{0}:{\delta}_{1}=0$ and Algorithm 2 ${H}_{0}:{\phi}_{I}=0$, but failed to reject ${H}_{0}$ for Algorithm 1.

**Scenario 4:**Rejected Algorithm 0 ${H}_{0}:{\delta}_{1}=0$, but failed to reject H

_{0}for Algorithms 1 and 2.

**Scenario 5:**Failed to reject Algorithm 0 ${H}_{0}:{\delta}_{1}=0$

## 3. Results

_{I}closer to 1 in Equation (2)), the majority of simulations concluded scenario 1: rejecting Algorithm 0 ${H}_{0}:{\delta}_{1}=0$, Algorithm 1 ${H}_{0}:{\alpha}_{1}=0$, and Algorithm 2 ${H}_{0}:{\phi}_{I}=0$. These simulations show that there can be a significant association between the SNP ${X}_{1}$ and the trait of interest Y in Algorithm 0, and the SNP ${X}_{1}$ is still significantly associated with the trait of interest Y when conditioning on the SNP ${X}_{2}$ in Algorithm 1. However, there is a significant interaction between the two SNPs in Algorithm 2. This shows that if a researcher were to use Algorithm 1 to conclude that the two SNPs are independent since the SNP ${X}_{1}$ is significantly associated with the trait of interest Y conditional on the SNP ${X}_{2}$, a false conclusion would be reached because there is a significant interaction of the two SNPs on the trait Y in Algorithm 2 and as generated by the data using Equation (2), such that ${\beta}_{I}\ne 0$. These simulations demonstrate that it is not sufficient to consider independence of two genetic signals by considering Algorithm 1: $E\left[Y\right]={\alpha}_{0}+{\alpha}_{1}{X}_{1}+{\alpha}_{2}{X}_{2}\text{}\mathrm{and}\text{}\mathrm{testing}\text{}{H}_{0}:{\alpha}_{1}=0$. One needs to also consider if there is a significant SNP by SNP interaction by fitting Algorithm 2: $E\left[Y\right]={\phi}_{0}+{\phi}_{1}{X}_{1}+{\phi}_{2}{X}_{2}+{\phi}_{I}{X}_{1}{X}_{2}$ and testing ${H}_{0}:{\phi}_{I}=0$.

## 4. Data Analysis

**Algorithm 1:**$E\left[Y\right]={\alpha}_{0}+{\alpha}_{1}{X}_{1}+{\alpha}_{2}{X}_{2}+{\alpha}_{C}{\mathrm{C}}^{T}\text{}\mathrm{and}\text{}{H}_{0}:{\alpha}_{1}=0$

**Algorithm 2:**$E\left[Y\right]={\phi}_{0}+{\phi}_{1}{X}_{1}+{\phi}_{2}{X}_{2}+{\phi}_{I}{X}_{1}{X}_{2}+{\phi}_{C}{C}^{T}$ and ${H}_{0}:{\phi}_{I}=0,$

^{2}= 0.995) and rs2781665 (r

^{2}= 0.891) and rs185631674 is a rare variant in a study with a small sample size (N = 892).

^{2}. This also shows that special consideration needs to be given to rare variants.

## 5. Discussion

^{2}or D’. Also, prior knowledge, for example protein–protein interactions or biological pathways, should be considered when examining SNP by SNP interactions [23].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**${\beta}_{1}=0.3$ and ${\beta}_{2}=0.3$ in Equation (2) for the plot on the left and ${\beta}_{1}=0$ and ${\beta}_{2}=0$ in Equation (2) for the plot on the right, where both y-axes are the proportion of simulations where the null hypothesis was rejected. For the plot on the left, when the interaction was simulated to be weaker (i.e., β

_{I}closer to 0.3 in Equation (2)), the majority of simulations concluded scenario 2: rejecting Algorithm 0 ${H}_{0}:{\delta}_{1}=0$ and Algorithm 1 ${H}_{0}:{\alpha}_{1}=0$, but failing to reject ${H}_{0}$ for Algorithm 2 (i.e., there was not a signification SNP by SNP interaction). For the plot on the right, when the interaction was simulated to be weaker (i.e., β

_{I}closer to 0.3 in Equation (2)), the majority of simulations concluded scenario 5: failing to reject Algorithm 0 ${H}_{0}:{\delta}_{1}=0$ (i.e., the SNP ${X}_{1}$ was not associated with the trait of interest Y). For both plots, when a stronger interaction between the 2 SNPs was simulated (i.e., β

_{I}closer to 1 in Equation (2)), the majority of simulations concluded scenario 1: rejecting Algorithm 0 ${H}_{0}:{\delta}_{1}=0$, Algorithm 1 ${H}_{0}:{\alpha}_{1}=0$, and Algorithm 2 ${H}_{0}:{\phi}_{I}=0$.

**Table 1.**Algorithm 1 considers the association of rs2781659 with bronchodilator response (BDR) conditioning on the SNPs in column 1 and Algorithm 2 considers the interaction of rs2781659 with the SNPs in column 1 on BDR. MAF denotes the minor allele frequency and SD denotes the standard deviation in the table below.

Algorithm 1 | Algorithm 2 | ||||||||
---|---|---|---|---|---|---|---|---|---|

SNP | Position (Hg38) | r^{2} | MAF | Beta | SD | p-Value | Beta | SD | p-Value |

rs2781663 | 131571207 | 0.995 | 0.32 | −0.15 | 0.05 | 0.003 | 0.14 | 0.08 | 0.06 |

rs2781665 | 131572107 | 0.891 | 0.31 | −0.18 | 0.63 | 0.78 | 0.14 | 0.08 | 0.06 |

rs185631674 | 131570984 | 0.002 | 0.001 | −0.15 | 0.05 | 0.004 | −1.75 | 0.83 | 0.03 |

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**MDPI and ACS Style**

Walsh, R.; Voorhies, K.; McDonald, M.-L.; McGeachie, M.; Sordillo, J.E.; Lange, C.; Wu, A.C.; Lutz, S.M.
The Role of SNP Interactions when Determining Independence of Novel Signals in Genetic Association Studies—An Application to *ARG1* and Bronchodilator Response. *J. Pers. Med.* **2021**, *11*, 145.
https://doi.org/10.3390/jpm11020145

**AMA Style**

Walsh R, Voorhies K, McDonald M-L, McGeachie M, Sordillo JE, Lange C, Wu AC, Lutz SM.
The Role of SNP Interactions when Determining Independence of Novel Signals in Genetic Association Studies—An Application to *ARG1* and Bronchodilator Response. *Journal of Personalized Medicine*. 2021; 11(2):145.
https://doi.org/10.3390/jpm11020145

**Chicago/Turabian Style**

Walsh, Ryan, Kirsten Voorhies, Merry-Lynn McDonald, Michael McGeachie, Joanne E. Sordillo, Christoph Lange, Ann Chen Wu, and Sharon M. Lutz.
2021. "The Role of SNP Interactions when Determining Independence of Novel Signals in Genetic Association Studies—An Application to *ARG1* and Bronchodilator Response" *Journal of Personalized Medicine* 11, no. 2: 145.
https://doi.org/10.3390/jpm11020145