Paired t-Tests

Author

Andrew Dalby

Background

This is a set of data from Statistical Methods in Biology, 2nd Edition N.J.T Bailey, Hodder and Stoughton, London, 1981 on the properties of two analgesics. The data is paired for 8 patients showing the number of hours of relief with each analgesic.

The first step is to create the data.

Patient <- c(rep(1:8,2))
Hours <- c(3.2,1.6,5.7,2.8,5.5,1.2,6.1,2.9,3.8,1.0,8.4,3.6,5.0,3.5,7.3,4.8)
Treatment <- c(rep("A",8),rep("B",8))
analgesic <- data.frame(Patient,Hours,Treatment)

It is always a good idea to summarise your data graphically before you do any analysis.

library(ggpubr)

Loading required package: ggplot2

ggboxplot(analgesic, x="Treatment" ,y= "Hours", 
          color = "Treatment", palette= c("#4B0082","#5c3317"),
          order = c("A","B"),
          ylab = "Hours of relief", xlab = "Treatment")

There is not a strong apparent different between the two groups based on a summary of the data but the paired t-test might find a difference if the effect size is smaller than the population variability. To check that the t-test is valid we can check the normality of the differences with a Shapiro-Wilk test.

d <- with(analgesic, Hours[Treatment == "A"]- Hours[Treatment =="B"])
shapiro.test(d)


    Shapiro-Wilk normality test

data:  d
W = 0.94176, p-value = 0.6284

results <- t.test(Hours ~ Treatment, data=analgesic, paired = TRUE)
results


    Paired t-test

data:  Hours by Treatment
t = -2.4342, df = 7, p-value = 0.04514
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.07000426 -0.02999574
sample estimates:
mean difference 
          -1.05

Carrying out a pairwise data plot would have been more revealing than the boxplots in this case.

A <- subset(analgesic, Treatment == "A", Hours)
B <- subset(analgesic, Treatment == "B", Hours)
library(PairedData)

Loading required package: MASS

Loading required package: gld

Loading required package: mvtnorm

Loading required package: lattice


Attaching package: 'PairedData'

The following object is masked from 'package:base':

    summary

pd <- paired(A,B)
plot(pd, type = "profile") + theme_cleveland()

Most patients see an upward movement between A and B however 2 see slight falls and there are large differences in the slope of the connections.