This tutorial demonstrates how to use the `SEAGLE`

package when the user inputs a matrix ${\bf G}$. We'll begin by loading the `SEAGLE`

package.

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" )

```
library(SEAGLE)
```

As an example, we'll generate some synthetic data for usage in this tutorial. Let's consider a dataset with $n=5000$ individuals and $L=100$ loci, where the first $40$ are causal.

The `makeSimData`

function generates a covariate matrix $\widetilde{\bf X} \in \mathbb{R}^{n \times 3}$, where the first column is the all ones vector for the intercept and the second and third columns are ${\bf X} \sim \text{N}(0,1)$ and ${\bf E} \sim \text{N}(0,1)$, respectively. The last two columns are scaled to have $0$ mean and unit variance.

The `makeSimData`

function additionally generates the genetic marker matrix ${\bf G}$ with synthetic haplotype data from the COSI software. Detailed procedures for generating ${\bf G}$ can be found in the accompanying journal manuscript. Finally, the `makeSimData`

function also generates a continuous phenotype ${\bf y}$ according to the following fixed effects model
$$
{\bf y} = \tilde{\bf X} \boldsymbol{\gamma}*{\widetilde{\bf X}} + {\bf G}\boldsymbol{\gamma}*{G} + \text{diag}(E){\bf G}\boldsymbol{\gamma}*{GE} + {\bf e}.
$$
Here, $\boldsymbol{\gamma}*{\tilde{\bf X}}$ is the all ones vector of length $P=3$, $\boldsymbol{\gamma}*{G} \in \mathbb{R}^{L}$, $\boldsymbol{\gamma}*{GE}\in \mathbb{R}^{L}$, and ${\bf e} \sim \text{N}({\bf 0}, \sigma\, {\bf I}*{n})$. The entries of $\boldsymbol{\gamma}*{G}$ and $\boldsymbol{\gamma}*{GE}$ pertaining to causal loci are set to be $\gamma*{G}$ = `gammaG`

and $\gamma_{GE}$ = `gammaGE`

, respectively. The remaining entries of $\boldsymbol{\gamma}*{G}$ and $\boldsymbol{\gamma}*{GE}$ pertaining to non-causal loci are set to $0$.

dat <- makeSimData(H=cosihap, n=5000, L=100, gammaG=1, gammaGE=0, causal=40, seed=1)

Now that we have our data, we can prepare it for use in the SEAGLE algorithm. We will input our ${\bf y}$, ${\bf X}$, ${\bf E}$, and ${\bf G}$ into the `prep.SEAGLE`

function. The `intercept = 1`

parameter indicates that the first column of ${\bf X}$ is the all ones vector for the intercept.

This preparation procedure formats the input data for the `SEAGLE`

function by checking the dimensions of the input data. It also pre-computes a QR decomposition for $\widetilde{\bf X} = \begin{pmatrix} {\bf 1}*{n} & {\bf X} & {\bf E} \end{pmatrix}$, where ${\bf 1}*{n}$ denotes the all ones vector of length $n$.

objSEAGLE <- prep.SEAGLE(y=dat$y, X=dat$X, intercept=1, E=dat$E, G=dat$G)

Finally, we'll input the prepared data into the `SEAGLE`

function to compute the score-like test statistic $T$ and its corresponding p-value. The `init.tau`

and `init.sigma`

parameters are the initial values for $\tau$ and $\sigma$ employed in the REML EM algorithm.

res <- SEAGLE(objSEAGLE, init.tau=0.5, init.sigma=0.5) res$T res$pv

The score-like test statistic $T$ for the G$\times$E effect and its corresponding p-value can be found in `res$T`

and `res$pv`

, respectively.

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