Session04

stats04

# CHAPTER 6:

# TWO SAMPLES

# The classical tests:
# - Comparing two variances: Fisher's F test
var.test
# - Comparing two sample means with normal errors: Student's t test
t.test
# - Comparing two means with non-normal errors: Wilcoxon's test
wilcox.test
# - Comparing two proportions: binomial test
prop.test
# - Comparing two variables: Pearson's or Spearman's rank correlation
cor.test
# - Testing for independence in contingency tables using chi-squared
chisq.test
# - Testing small samples for correlation with Fisher's exact test
fisher.test

# COMPARING TWO VARIANCES 

# Before carrying out a test to compare two sample means we need to test whether the sample variances are significantly different
# Thus, Fisher's F test
# We divide the larger variance by the smaller variance, the larger variance goes on top

qf(0.975,9,9)
# This calculates the critical value of Fisher's F test

f.test.data<-read.csv("c:\\MSc\\Statistics\\Data\\f.test.data.csv")
attach(f.test.data)
names(f.test.data)

var(gardenB)
var(gardenC)
# this computes the two variances
# gradenC has the larger variance

F.ratio<-var(gardenC)/var(gardenB)
F.ratio
# the test shows that the variance in gardenC is more than 10 times as big as the variance in gardenB
# since the critical value is 4.026 - since the test statistic ios larger than the critical value we reject tje null hypothesis
# the variances are significantly different

2*(1-pf(F.ratio,9,9))
var.test(gardenB,gardenC)
detach(f.test.data)

# COMPARING TWO MEANS

qt(0.975,18)

t.test.data <- read.csv("c:\\MSc\\Statistics\\Data\\t.test.data.csv")
attach(t.test.data)
names(t.test.data)

ozone <- c(gardenA,gardenB)
label <- factor(c(rep("A",10),rep("B",10)))
boxplot(ozone∼label,notch=T,xlab="Garden",
  ylab="Ozone pphm",col="lightblue")

s2A <- var(gardenA)
s2B <- var(gardenB)

s2A/s2B

(mean(gardenA)-mean(gardenB))/sqrt(s2A/10+s2B/10)

2*pt(-3.872983,18)

t.test(gardenA,gardenB)

# WILCOXON RANK-SUM TEST

ozone <- c(gardenA,gardenB)
ozone

label <- c(rep("A",10),rep("B",10))
label

combined.ranks <- rank(ozone)
combined.ranks

tapply(combined.ranks,label,sum)

wilcox.test(gardenA,gardenB)

# TEST ON PAIRED SAMPLES

streams <- read.csv("c:\\MSc\\Statistics\\Data\\streams.csv")
attach(streams)
names(streams)

t.test(down,up)

t.test(down,up,paired=T)

d <- up-down
t.test(d)

# THE BINOMIAL TEST

binom.test(1,9)

# BINOMIAL TESTS TO COMPARE TWO PROPORTIONS

prop.test(c(4,196),c(40,3270))

# CHI-SQUARED CONTINGENCY TABLES

qchisq(0.95,1)

count <- matrix(c(38,14,11,51),nrow=2)
count

chisq.test(count)

chisq.test(count,correct=F)

# FISHER'S EXACT TEST

factorial(8)*factorial(12)*factorial(10)*factorial(10)/
  (factorial(6)*factorial(2)*factorial(4)*factorial(8)*factorial(20))

factorial(8)*factorial(12)*factorial(10)*factorial(10)/
  (factorial(7)*factorial(3)*factorial(1)*factorial(9)*factorial(20))

factorial(8)*factorial(12)*factorial(10)*factorial(10)/
  (factorial(8)*factorial(2)*factorial(0)*factorial(10)*factorial(20))
  
0.07501786 + 0.009526078 + 0.000352279

2*(0.07501786+0.009526078+0.000352279)

x <- as.matrix(c(6,4,2,8))
dim(x) <- c(2,2)
x

fisher.test(x)

table <- read.csv("c:\\MSc\\Statistics\\Data\\fisher.csv")
attach(table)
head(table)

fisher.test(tree,nests)

# CORRELATION AND COVARIANCE

data <- read.csv("c:\\MSc\\Statistics\\Data\\twosample.csv")
attach(data)
plot(x,y,pch=21,col="blue",bg="orange")

var(x)

var(y)

var(x,y)

var(x,y)/sqrt(var(x)*var(y))

cor(x,y)

# CORRELATION AND THE VARIANCE OF DIFFERENCES BETWEEN VARIABLES

paired <- read.csv("c:\\MSc\\Statistics\\Data\\water.table.csv ")
attach(paired)
names(paired)

cor(Summer, Winter)

cor.test(Summer, Winter)

varS <- var(Summer)
varW <- var(Winter)
varD <- var(Summer-Winter)

(varS+varW-varD)/(2*sqrt(varS)*sqrt(varW))

varD

varS + varW

varS + varW - 2 * 0.8820102 * sqrt(varS) * sqrt(varW)

# SCALE-DEPENDENT CORRELATIONS

data <- read.csv("c:\\MSc\\Statistics\\Data\\productivity.csv")
attach(data)
names(data)

plot(productivity,mammals,pch=16,col="blue")

cor.test(productivity,mammals,method="spearman")

plot(productivity,mammals,pch=16,col=as.numeric(region))