Differences

The key to analysing paired data is to recognise that the differences between X and Y hold all the information about whether their means are the same. Writing

D = Y - X

the hypotheses

H0 :   μX = μY
HA :   μXμY

can be expressed as

H0 :   μD = 0
HA :   μD ≠ 0

This reduces the paired data set to a univariate data set of differences. The test also becomes a simpler hypothesis test about the mean of these differences.

Blood pressure and the pill

The increase in blood pressure for each subject is shown in the final column below.

  Blood pressure  
Subject Before pill After pill Difference
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
70
80
72
76
76
76
72
78
82
64
74
92
74
68
84
68
72
62
70
58
66
68
52
64
72
74
60
74
72
74
-2
-8
-10
-6
-18
-10
-4
-26
-18
8
0
-32
0
4
-10

Is the mean of the differences zero?

Twin studies

The final column below shows the difference in IQ for each pair (good minus poor)

  IQ  
Family Poor environment Good environment Difference
1
2
3
4
5
6
7
8
9
10
100
65
60
125
85
145
55
180
60
135
125
95
100
120
120
185
80
210
105
175
25
30
40
-5
35
40
25
30
45
40

Is the mean of the differences zero?

Snail shell temperature

The final column shows how much warmer the brown shell in each pair was.

  Temperature  
Pair   Yellow  
shell
  Brown  
shell
Difference
1
2
3
4
5
6
7
8
9
10
25.6
27.8
26.3
25.9
28.0
25.4
24.6
28.9
27.2
26.0
25.5
27.5
27.3
27.3
29.2
25.3
26.4
28.5
28.1
26.4
-0.1
-0.3
1.0
1.4
1.2
-0.1
1.8
-0.4
0.9
0.4

Is the mean of the differences zero?


Analysis of paired data

By taking differences, much of the variability between the individuals is eliminated. This provides considerably more information to help assess the null and alternative hypotheses.

The benefits of pairing will be explained more fully in a later chapter about experimental design.

Snail shell temperature

The diagram below shows the temperatures of the yellow and brown shells. The two distributions overlap considerably due to variability in the exposure to sunlight, so it initially appears that there will be little evidence against equal means.

Click on individual crosses to show the difference between the temperatures for each pair. Although the yellow shell is sometimes a little warmer, the brown shell is usually considerably warmer.

Click Show Pairing to draw lines between the pairs of crosses and display the differences in a jittered dot plot. The differences give much clearer evidence that the mean temperature is higher for the brown shells — it seems that the mean difference is positive.