################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean)/kmin)
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2)
################# Linear model: Least-Squares Fit #################
#g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.breaks <- g.hist$breaks # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.breaks[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean)/kmin)
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean)/kmin)
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
alpha.LM
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean/kmin))
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1, lwd=5)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean/kmin))
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1, lwd=3)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean/kmin))
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4)
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1, lwd=3)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2, lwd=3)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
#plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3, lwd=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean/kmin))
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4, lwd=3)
plot(yeast)
hist(yeast)
hist(g.vec)
g.pois
g.mean
alpha.LM
alpha.ML
degree(g)
sort(degree(g))
sort(degree(g),decreasing=FALSE)
sort(degree(g),decreasing=F)
sort(degree(g),decreasing=false)
sort(degree(g), decreasing = TRUE)
head(sort(degree(g), decreasing = TRUE))
stddev(degree(g))
sd(degree(g))
tail(sort(degree(g), decreasing = TRUE))
plot(log(g.breaks.clean), log(g.probs.clean))
# Homework 4 for the University of Tulsa' s CS-7863 Network Theory Course
# Degree Distribution
# Professor: Dr. McKinney, Spring 2022
# Noah Schrick - 1492657
library(igraph)
library(igraphdata)
data(yeast)
g <- yeast
g.netname <- "Yeast"
################# Set up Work #################
g.vec <- degree(g)
g.hist <- hist(g.vec, freq=FALSE, main=paste("Histogram of the", g.netname,
" Network"))
legend("topright", c("Guess", "Poisson", "Least-Squares Fit",
"Max Log-Likelihood"), lty=c(1,2,3,4), col=c("#40B0A6",
"#006CD1", "#E66100", "#D35FB7"))
g.mean <- mean(g.vec)
g.seq <- 0:max(g.vec) # x-axis
################# Guessing Alpha #################
alpha.guess <- 1.5
lines(g.seq, g.seq^(-alpha.guess), col="#40B0A6", lty=1, lwd=3)
################# Poisson #################
g.pois <- dpois(g.seq, g.mean, log=F)
lines(g.seq, g.pois, col="#006CD1", lty=2, lwd=3)
################# Linear model: Least-Squares Fit #################
g.breaks <- g.hist$breaks[-c(1)] # remove 0
g.probs <- g.hist$density[-1] # make lengths match
# Need to clean up probabilities that are 0
nz.probs.mask <- g.probs!=0
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
plot(log(g.breaks.clean), log(g.probs.clean))
g.fit <- lm(log(g.probs.clean)~log(g.breaks.clean))
summary(g.fit)
alpha.LM <- coef(g.fit)[2]
lines(g.seq, g.seq^(-alpha.LM), col="#E66100", lty=3, lwd=3)
################# Max-Log-Likelihood #################
n <- length(g.breaks.clean)
kmin <- g.breaks.clean[1]
alpha.ML <- 1 + n/sum(log(g.breaks.clean/kmin))
alpha.ML
lines(g.seq, g.seq^(-alpha.ML), col="#D35FB7", lty=4, lwd=3)
plot(log(g.breaks.clean), log(g.probs.clean))
g.breaks.clean <- g.breaks[nz.probs.mask]
g.probs.clean <- g.probs[nz.probs.mask]
plot(log(g.breaks.clean), log(g.probs.clean))
## Turn into function to repeat for all SNPs
LR.fn <- function(i){
lr <- glm(pheno.factor~genotypes.df[[i]],family=binomial)
td.lr <- tidy(lr)
pval_vec <- td.lr$p.value # vector of $p.value from td.lr
coef_vec <- td.lr$estimate # vector of $estimate
cbind(snp.ids[i], coef_vec[1], coef_vec[2], coef_vec[3], pval_vec[1], pval_vec[2], pval_vec[3])
}
# apply Logistic Regression model to all SNPs
LRresults.df <- data.frame(t(sapply(1:ncol(genotypes.df), LR.fn)))
# Lab 6 for the University of Tulsa's CS-6643 Bioinformatics Course
# GWAS
# Professor: Dr. McKinney, Fall 2022
# Noah L. Schrick - 1492657
## Set Working Directory to file directory - RStudio approach
setwd(dirname(rstudioapi::getActiveDocumentContext()$path))
#### Part 0: PLINK
if (!require("BiocManager")) install.packages("BiocManager")
library(BiocManager)
if (!require("snpStats")) BiocManager::install("snpStats")
library(snpStats)
ex.data <- read.pedfile(file="extra.ped", snps="extra.map")
ex.data$fam
phenotype <- ex.data$fam$affected-1  # change pheno from 1/2 to 0/1
genotypes <- ex.data$genotypes       # encoded as AA/AB/BB
snp.ids <- as.character(ex.data$map$snp.names)
genotypes.df <- data.frame(as(genotypes, "character"))
colnames(genotypes.df) <- snp.ids
# observed contingency table for SNP rs630969
table(phenotype,genotypes.df$rs630969,
dnn=c("phenotype","genotype")) # dnn dimension names of table
dim(genotypes.df)
#### Part A: Chi-Square Test
# creates list of observed contingency tables for all SNPs
# sapply acts on each column of genotypes.df
observed.tables.list <- sapply(genotypes.df, function(x)
table(phenotype,x,dnn=c("phenotype","genotype")))
test.table <- observed.tables.list$rs634228
genoMarg.vec  <-  colSums(test.table)           # margin vector
phenoMarg.vec <- rowSums(test.table)            # margin vector
totalSubj <- sum(genoMarg.vec)                  # total subjects
expect.test <- outer(phenoMarg.vec,genoMarg.vec/totalSubj,'*')
## Fisher Test
# Fisher exact test (chi-square test) for all SNPs
fish_fn <- function(i){
cbind(snp.ids[i], fisher.test(observed.tables.list[[i]])$p.value)
}
# apply fisher exact test to all SNPs
fish.df <- data.frame(t(sapply(1:ncol(genotypes.df), fish_fn)))
colnames(fish.df) <- c("rs", "p_value")
# sort SNPs by Fisher exact p-value
if (!require("dplyr")) install.packages("dplyr")
library(dplyr)
fish.results <- fish.df                            %>%
mutate_at("p_value", as.character) %>%
mutate_at("p_value", as.numeric)   %>%
arrange(p_value)
print(fish.results)
#### Part B: Logistic regression with genotypes
if (!require("ggplot2")) BiocManager::install("ggplot2")
library(ggplot2)
i<-8
A1<-ex.data$map$allele.1[i]
A2<-ex.data$map$allele.2[i]
geno.labels <- c(paste(A1,A1,sep=""),paste(A1,A2,sep=""),paste(A2,A2,sep=""))
## Plot with ggplot2
# data from the one SNP
oneSNP.df <- data.frame(cbind(genotypes.df[[i]],as.numeric(phenotype)))
colnames(oneSNP.df) <- c("genotypes","phenotypes")
lr.plot <- ggplot(oneSNP.df, aes(x=genotypes, y=phenotypes)) +
geom_point(position = position_jitter(w = 0.1, h = 0.1)) +
# stat_smooth plots the probability based on the model
stat_smooth(method="glm", method.args = list(family = "binomial")) #+
#xlim(geno.labels) + ggtitle(snp.ids[i])
print(lr.plot)
## Fit a logistic regression model of phenotype with SNP in the 8th column
if (!require("broom")) install.packages("broom")
library(broom) # for tidy function
pheno.factor <- factor(phenotype,labels=c(0,1))
i<-8
lr <- glm(pheno.factor~genotypes.df[[i]],family=binomial)
td.lr <- tidy(lr)
pval_vec <- td.lr$p.value # vector of $p.value from td.lr
coef_vec <- td.lr$estimate # vector of $estimate
cbind(snp.ids[i], coef_vec[1], coef_vec[2], coef_vec[3], pval_vec[1], pval_vec[2], pval_vec[3])
## Turn into function to repeat for all SNPs
LR.fn <- function(i){
lr <- glm(pheno.factor~genotypes.df[[i]],family=binomial)
td.lr <- tidy(lr)
pval_vec <- td.lr$p.value # vector of $p.value from td.lr
coef_vec <- td.lr$estimate # vector of $estimate
cbind(snp.ids[i], coef_vec[1], coef_vec[2], coef_vec[3], pval_vec[1], pval_vec[2], pval_vec[3])
}
# apply Logistic Regression model to all SNPs
LRresults.df <- data.frame(t(sapply(1:ncol(genotypes.df), LR.fn)))
# add column names to results data frame
colnames(LRresults.df) <- c("rs", "AAintercept", "ABcoef", "BBcoef", "AA.pval", "AB.pval", "BB.pval")
# The following sorts LR results by the p-value of the BB
# homozygous coefficient. tidy made $p_value a factor and when you try to
# convert directly to numeric (as.numeric) turns factors into integer and
# this messes up sorting especially with scientific notation
lr.results.sorted <- LRresults.df    %>%
mutate_at("BB.pval", as.character) %>%  # convert to char before numeric
mutate_at("BB.pval", as.numeric)   %>%  # convert to numeric for arrange
arrange(BB.pval)                        # sort
as.matrix(lr.results.sorted %>% pull(rs,BB.pval))
fish.df
fish.results
fish.results$rs
lr.results.sorted
lr.results.sorted$rs
table(fish.results$rs, lr.results.sorted$rs)
expand.grid(fish=fish.results$rs, lr=lr.results.sorted$rs)
CJ(fish=fish.results$rs, lr=lr.results.sorted$rs)
table(fish=fish.results$rs, lr=lr.results.sorted$rs)
as.table(fish=fish.results$rs, lr=lr.results.sorted$rs)
setNames(fish.results$rs, lr.results.sorted$rs)
as.table(setNames(fish.results$rs, lr.results.sorted$rs))
data.frame(fish.results$rs, lr.results.sorted$rs)
tmp <- data.frame(fish.results$rs, lr.results.sorted$rs)
?plot
length(fish.results$rs)
length(lr.results.sorted$rs)
?plot(x=1:length(fish.results$rs))
?plot(y=1:length(fish.results$rs), x=fish.results$rs)
plot(y=1:length(fish.results$rs), x=fish.results$rs)
plot(fish.results)
plot(fish.results$rs)
fish.results$rs
class(fish.results$rs)
as.vector(fish.results$rs)
plot(as.vector(fish.results$rs))
as.data.frame(fish.results$rs)
plot(as.data.frame(fish.results$rs))
ggplot(as.data.frame(fish.results$rs))
ggplot(as.data.frame(fish.results$rs))
plot(as.data.frame(fish.results$rs))
tmpy<-1:length(fish.results$rs)
tmpy
yvals <- 1:length(fish.results$rs)
xfish <- fish.results$rs
xfish
which(fish.results$rs == "rs17786052")
which(fish.results$rs == "rs17786052")[[1]]
snp.ids
snp.ids[i]
match(fish.results$rs, snp.ids)
yvals <- 1:length(fish.results$rs)
xvals <- snp.ids
fishvals <- match(fish.results$rs, snp.ids)
lrvals <- match(lr.results.sorted$rs, snp.ids)
plot(xvals, fishvals)
fishvals
plot(xlimxvals, fishvals, type="l")
plot(xvals, fishvals, type="l")
plot(1:17, yvals, type=l, xaxt="none", xlab="snp")
plot(1:17, yvals, type="l", xaxt="none", xlab="snp")
axis(1, axTicks(1), labels=snp.ids)
snp.ids
axis(labels=snp.ids)
?axis
axis(labels=snp.ids, side=1)
axis(labels=snp.ids, side=1, at=1)
axis(labels=snp.ids, side=1, at=17)
axis(labels=snp.ids, side=1, at=17)
length(snp.ids)
length(snp.ids[1])
xlabels <- c(snp.ids)
xlabels
axis(xlabels=snp.ids)
axis(labels=xlabels)
axis(labels=xlabels, side=1)
axis(side = 1, at = 1:length(fish.results$rs), labels=snp.ids)
plot(1:17, fishvals, type="l", xaxt="none", xlab="snp", ylab="rank")
axis(side = 1, at = 1:length(fish.results$rs), labels=snp.ids)
lines(1:length(lr.results.sorted$rs), lrvals, type="l", col="red", lwd =2, lty=1)
plot(1:length(fish.results$rs), fishvals, type="l", xaxt="none",
xlab="snp", ylab="rank", lty=1, col=1)
axis(side = 1, at = 1:length(fish.results$rs), labels=snp.ids)
lines(1:length(lr.results.sorted$rs), lrvals, type="l", col=2, lty=2)
legend(x="topright", legend=c("Chi-Square Fisher Test", "Logistic Regression"),
lty=c(1,2), col=c(1,2))
plot(1:length(fish.results$rs), fishvals, type="l", xaxt="none",
xlab="snp", ylab="rank", lty=1, col=1)
axis(side = 1, at = 1:length(fish.results$rs), labels=snp.ids)
lines(1:length(lr.results.sorted$rs), lrvals, type="l", col=2, lty=2, lwd=2)
legend(x="topright", legend=c("Chi-Square Fisher Test", "Logistic Regression"),
lty=c(1,2), col=c(1,2))