Methods in Biostatistics 2 (140.652)

Hypothesis testing (cont)

Example 1

# Example data
x <- c(102.5, 106.6,  99.8, 106.5, 103.7, 105.5, 98.2, 104.1,  85.6, 105.5, 114.0, 112.2)
y <- c( 93.7,  90.9, 100.4,  92.0, 100.2, 104.6, 95.4,  96.6,  99.2)

# Two-sided t-test allowing un-equal population SDs
t.test(x,y)

## 
##  Welch Two Sample t-test
## 
## data:  x and y
## t = 2.6041, df = 18.475, p-value = 0.01769
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##   1.30124 12.06543
## sample estimates:
## mean of x mean of y 
##  103.6833   97.0000

# Plot the data with confidence intervals 
source("http://www.biostat.jhsph.edu/~iruczins/teaching/140.615/func/dotplot.r")

# This function plots the data for two samples. 
dotplot(x,y)

Example 2

# One-tailed test example
x <- c(59.4, 52.3, 42.6, 45.1, 65.9, 40.8)
y <- c(82.7, 56.7, 46.9, 67.8, 74.8, 85.7)

# One-tailed t-test
t.test(x,y,alt="less")

## 
##  Welch Two Sample t-test
## 
## data:  x and y
## t = -2.4421, df = 8.6937, p-value = 0.01907
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf -4.454703
## sample estimates:
## mean of x mean of y 
##  51.01667  69.10000

# The dotplot
dotplot(x,y)

Example 3

# another one-tailed test example
x <- c(63.3, 58.6, 59.0, 60.5, 56.3, 57.4)
y <- c(75.6, 65.9, 72.3, 58.0, 64.4, 66.2)
t.test(x,y,alt="less")

## 
##  Welch Two Sample t-test
## 
## data:  x and y
## t = -2.8968, df = 6.5546, p-value = 0.01242
## alternative hypothesis: true difference in means is less than 0
## 95 percent confidence interval:
##       -Inf -2.674212
## sample estimates:
## mean of x mean of y 
##  59.18333  67.06667

dotplot(x,y)

Good plot / bad plot

x <- c(15.1, 13.1, 21.5)
y <- c(35.1, 39.5, 58.8)

par(mar=c(4,4,2,1),mfrow=c(1,2),las=1)

barplot(c(mean(x),mean(y)),width=1,space=c(0.5,0.5),
        col=c("white","gray40"),xlim=c(0,3),names=c("A","B"),
        ylim=c(0,76))
segments(1,mean(x),1,mean(x)+sd(x),lwd=2)
segments(0.8,mean(x)+sd(x),1.2,mean(x)+sd(x),lwd=2)
segments(2.5,mean(y),2.5,mean(y)+sd(y),lwd=2)
segments(2.3,mean(y)+sd(y),2.7,mean(y)+sd(y),lwd=2)
mtext("Bad plot",cex=1.5,line=0.5)

plot(rep(0:1,c(3,3)),c(x,y),xaxt="n",ylim=c(0,76),xlim=c(-0.5,1.5),ylab="",xlab="")
abline(v=0:1,col="gray40",lty=2)
points(rep(0:1,c(3,3)),c(x,y),lwd=2)
mtext("Good plot",cex=1.5,line=0.5)
xci <- t.test(x)$conf.int
yci <- t.test(y)$conf.int
segments(0.25,xci[1],0.25,xci[2],lwd=2,col="blue")
segments(c(0.23,0.23,0.2),c(xci,mean(x)),c(0.27,0.27,0.3),c(xci,mean(x)),lwd=2,col="blue")
segments(1-0.25,yci[1],1-0.25,yci[2],lwd=2,col="red")
segments(1-c(0.23,0.23,0.2),c(yci,mean(y)),1-c(0.27,0.27,0.3),c(yci,mean(y)),lwd=2,col="red")
u <- par("usr")
segments(0:1,u[3],0:1,u[3]-diff(u[3:4])*0.03,xpd=TRUE)
text(0:1,u[3]-diff(u[3:4])*0.08,c("A","B"),xpd=TRUE)

Pre/post example

x <- c(18.6, 14.3, 21.4, 19.3, 24.0)
y <- c(17.8, 24.1, 31.9, 28.6, 40.0)
t.test(y-x)

## 
##  One Sample t-test
## 
## data:  y - x
## t = 3.2936, df = 4, p-value = 0.03011
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##   1.406947 16.513053
## sample estimates:
## mean of x 
##      8.96

End of code