We began to cover genome wide association analyses (GWAS) in class and this lab will cover the basic concepts of GWAS.

You might wonder why simulating genotypes or phenotypes is necessary since in the real world we only have observational data. The reason why we begin with simulating genotypes and phenotypes is to get an intuitive understanding about the underlying model that is being used in real GWAS.

Letâ€™s begin with simulating genotypes for a single locus

We can use the sample function to simulate genotypes for 2 alleles at a single locus (since we have two possible alleles on each pair of chromosomes)

First we would need to pick 2 nucleotides for a given position like this:

`nucleotides <- sample(c("a","t","g","c"), 2,replace = FALSE)`

- Then we can generate nucleotide pairs (2 alleles) for a given number of samples (in this case N=10).

```
N <- 10
sim.genotypes <- matrix(NA, nrow = N, ncol = 2)
for( i in 1:N){
sim.genotypes[i,] <- sample(nucleotides, 2, replace = TRUE)
}
```

Note that this would create genotypes with a minor allele frequency (MAF) of 0.5 since that is the default. So we would like to change that! The first task for you would be to create genotypes with MAF = 0.3 for 10 samples.

Now from the created sample calculate the MAF and see if it is close to the value that we have set.

The table() function might come in very handy.

Now that we have a simulated set of genotypes, we can code them into the dummy variables Xa and Xd that you have seen in class.

Xa takes values -1,0,1 and Xd takes -1 or 1

- For Xa conversion we could do something like this

`sim.genotypes`

```
[,1] [,2]
[1,] "c" "c"
[2,] "g" "g"
[3,] "g" "g"
[4,] "c" "c"
[5,] "g" "c"
[6,] "c" "c"
[7,] "g" "g"
[8,] "c" "g"
[9,] "c" "g"
[10,] "c" "c"
```

```
minor.allele <- names(which.min(table(sim.genotypes)))
temp <- 1 * (sim.genotypes == minor.allele)
Xa <- temp[,1] + temp[,2] - 1
# Checking if the conversion is correct
cbind(sim.genotypes, Xa)
```

```
Xa
[1,] "c" "c" "-1"
[2,] "g" "g" "1"
[3,] "g" "g" "1"
[4,] "c" "c" "-1"
[5,] "g" "c" "0"
[6,] "c" "c" "-1"
[7,] "g" "g" "1"
[8,] "c" "g" "0"
[9,] "c" "g" "0"
[10,] "c" "c" "-1"
```

- Create an Xd matrix for the simulated genotypes.

Now that we have an Xa and Xd matrix we can simulate phenotypes as well

Matrix calculations will make this process a lot easier.

Lets say for now that we have \(\beta_{\mu} = 0.3\), \(\beta_a = 1.5\), \(\beta_d = 0.05\) and \(\epsilon \sim N(0,1)\). Using these parameters simulate phenotypes following the model: \(Y = \beta_{\mu} + X_a \beta_a + X_d \beta_d + \epsilon\).

Here are some equations that might come in handy for this task.

```
# Note that you have to add a column with 1's to the Xa and Xd matrix for the multiplication of beta_mu
X.mx <- cbind(1, Xa,Xd)
```

- After the simulation the results should look something like this :

```
par(mfrow = c(1,2))
plot(Xa, phenotypes, main = "Xa vs Phenotypes", pch = 19, col = "blue")
plot(Xd, phenotypes, main = "Xd vs Phenotypes", pch = 19, col = "blue")
```

Using the equations given above, calculate the F statistic for the simulated data

As a last step, use the F-statistic and the corresponding degrees of freedom to calculate a p value using the pf() function with option lower.tail = FALSE.

```
# The pseudo code for this task
MLE.beta <- calculate MLE.beta
y.hat <- calculate the estimated values of y given the MLE>beta values
SSM <- calculate SSM
SSE <- calculate SSE
df.M
df.E
MSM <- calculate MSM
MSE <- calculate MSE
Fstatistic <- MSM / MSE
pf(Fstatistic, df.M,df.E,lower.tail =FALSE)
```

Now that we have all the essentials worked out, we can simulate genotypes for multiple samples and generate a manhattan plot from it.

You can review scatter plots from last weeks lab to create manhattan plots.

```
N <- Set number of samples
n.geno <- Set number of genotypes
sim.genotypes <- declare matrix with appropriate dimensions (N x (2*n.geno))
# for the number of genotypes
for( i in 1:n.geno){
nucleotides <- sample two nucleotides for the position
maf <- get a random maf from a uniform distribution (note that 0 < maf =< 0.5)
# for j in 1 to number of samples
for (j in 1:N){
sim.genotypes[j,c(2*i-1,2*i)] <- Generate two genotypes for each sample and use the maf as probability
}
}
Xa <- create a place holding matrix for Xa values
For all genotypes
for ( i in 1: n.geno){
# get minor allele
minor.allele <- names(which.min(table(sim.genotypes[,c(2*i -1, 2*i)])))
temp <- 1 * (sim.genotypes[,c(2*i -1, 2*i)] == minor.allele)
Xa[,i] <- temp[,1] + temp[,2] - 1
}
Xd <- convert Xa into Xd
# Lets make genotype # 20 the significant one
X.mx <- cbind(1, Xa[,20],Xd[,20])
beta.vec <- c(0.3,0.8,0.05)
phenotypes <- X.mx %*% beta.vec + rnorm(10,0,1)
pval <- define vector to save p-values
for (i in 1 : n.geno){
single.Xa <- get single Xa
single.Xd <- get single Xd
X.mx <- define X.mx using single Xa and single Xd
MLE.beta <- calculate MLE.beta
y.hat <- calculate the estimated values of y given the MLE>beta values
SSM <- calculate SSM
SSE <- calculate SSE
df.M
df.E
MSM <- calculate MSM
MSE <- calculate MSE
Fstatistic <- MSM / MSE
pval[i] <- pf(Fstatistic, df.M,df.E,lower.tail =FALSE)
}
```

- Your Manhattan plot should look something like this: