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```Comparing Means
from Paired Samples
(Session 13)
Learning Objectives
By the end of this session, you will be able to
• recognise whether two samples are paired
or independent
• explain why there is a gain in precision with
paired samples
• carry out a paired t-test and interpret
results from such a test
• present and report on conclusions from
paired t-tests
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Paired samples - aims
In comparing two samples we would aim to
improve the precision of the comparison
wherever possible, i.e. reduce the standard
error used in the test statistic.
e.g. If the aim is to compare the average
weight of males with that of females amongst
malnourished children, it would be better to
assess pairs of children of the same age,
having one male and one female in each pair.
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Benefits of pairing
The paired approach means that gender
difference would be clearer within a pair of
the same age, thus removing age to age
variability from the comparison.
The situation could be further improved by
matching the children, not only by age, but
other characteristics too, e.g. their location
(rural or urban), levels of poverty, etc.
Matching leads to a study of the differences
between each pair, say di for pair i.
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An example
Suppose we have
data for the mean
annual income (in
1000’s), of doctors
and dentists in the
UK, from 10
different regions.
The data (fictitious)
are given in the
table.
Doctors
Dentists
1
36.1
2
3
37.6
39.2
36.8
37.0
38.6
4
38.1
36.0
5
37.3
34.7
6
37.9
35.5
7
35.4
34.0
8
9
10
36.2
36.8
37.9
35.3
34.1
38.3
Region
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5
Null and alternative hypotheses
The hypotheses to be tested are:
H0: 1 - 2 = 0 versus H1: 1 - 2  0
This is equivalent to
H0: d = 0 versus H1: d  0
where d refers the mean of the population of
differences between the two groups.
Thus, the two-sample case reduces to a single
sample situation. Hence ideas covered in
Session 09 applies to the single sample formed
from differences between each set of pairs.
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Test procedure
The ten differences are found to be:
-0.7 0.6 0.6 2.1 2.6 2.4 1.4 0.9 2.7 -0.4
Further,
d = 1.22, while
2
sd
= 1.497
Hence the t-statistic for testing H0 is:
t=
d
 
s .e . d
=
d
2
= 3.15
s d 10
Comparing with t-tables with 9 d.f. shows this
result is significant at the 2% significance level.
The exact p-value = 0.012
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7
Conclusions and other issues
Conclusion: There is some evidence of a
difference between the mean salaries at a 2%
significance level.
Note: If we had ignored the pairing by region
and conducted an independent samples (or
two-sample t-test), the test statistic is
t= 1.95 on 18 d.f. This is clearly nonsignificant, thus leading to a different
conclusion from above. So take care to
recognise pairing where it occurs.
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8
References
Armitage, P., Matthews J.N.S. and Berry G.
(2002). Statistical Methods in Medical
Research. 4th edn. Blackwell.
Clarke, G.M. and Cooke, D. (2004). A Basic
Course in Statistics. 5th edn. Edward Arnold.
Johnson, R.A. and Bhattacharyya, G.K.
(2001). Statistics Principles and Methods.
4th edn. Wiley.
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9
Some practical work follows…
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10
```
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