Stats 5 95% C'I'

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• Lecture 11
• The paired t-test

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Effects of a diet
10 volunteers are weighed, then they all go on
the same diet for 2 weeks and are weighed
again. Q Does the diet cause a change in weight?
Person Before After Change no.
(Kg) (Kg)
(Kg) 1 70 65.5 -4.5 2 77 71.5 -5.5 3 74.5
73 -1.5 4 95.5 90 -5.5 5
114.5 110.5 -4.0 6 69.5 70.5
1.0 7 92 85 -7.0 8 95.5 94.5
-1.0 9 87 83 -4.0 10 107
103.5 -3.5 Mean 88.2515.50
84.6515.06 -3.602.31
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Difference between means

Low t
t-test

Sample size
Large Variability of data
Initial and final weights are very variable
(15Kg). Tends to reduce t and lead to a
non-significant result.
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Variability
Those subjects that are heaviest at the beginning
are also the heaviest at the end and the lightest
stay the lightest. So, while the starting and
finishing weights are very variable, the changes
in individual weight are much less
variable. Consider the last 2 subjects ...
Person Before After Change no.
(Kg) (Kg) (Kg) 9
87 83 -4.0 10 107
103.5 -3.5
Large differences
Small difference
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Sources of variability

• Can identify Sources of variance
• In this case there are two
• Variability among subjects starting weights -
• LARGE
• Variability in the changes in weight following
• the diet - MUCH SMALLER

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Paired t-test
Ordinary two sample t-test likely to fail because
it is weakened by all the variability among the
starting and finishing weights. Can reduce the
variability that we have to cope with, if we
focus only on the changes in weight. This what
the paired t-test does
Calculate changes in individual weights (i.e.
the effects of the diet)
Non- significant
Calculate 95 C.I. for mean of changes in
individual weights
Yes
Does the C.I. include zero?
No
Significant
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Null hypothesis
In a large population of individuals who used
this diet, the average weight change would be
zero. (Some individuals might loose weight, but
they would be balanced by others who gained
weight.)
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Using Minitab to perform the paired t-test
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Enter the data in 2 columns, as for the 2 sample
t-test
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Follow the menu structure to the Paired t-test
Stat / Basic Statistics / Paired t ...
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Carry out the paired t-test
After First sample and Before Second
sample, so that a weight loss registers as a
negative value.
Optionally, can press Graphs and then select
Dotplot of data. Not essential.
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Minitab output
Paired T-Test and Confidence Interval Paired T
for After - Before N Mean
StDev SE Mean After 10 84.65
15.06 4.76 Before 10 88.25
15.50 4.90 Difference 10 -3.600
2.307 0.730 95 CI for mean difference
(-5.250, -1.950) T-Test of mean difference 0
(vs not 0) T-Value -4.93 P 0.001
If a large number of people went on to use the
diet, we could be 95 confident that their mean
weight loss would be between 1.95 and 5.25 Kg.
This excludes a zero effect. The result is
therefore significant (plt0.05). Exact p value is
0.001
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Dotplot shows the 95 C.I. for the effect of the
diet.
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If we were really stupid ... and did an ordinary
two sample t-test
Two Sample T-Test and Confidence Interval Two
sample T for After vs Before N
Mean StDev SE Mean After 10 84.6
15.1 4.8 Before 10 88.3 15.5
4.9 95 CI for mu After - mu Before ( -18.0,
10.8) T-Test mu After mu Before (vs not ) T
-0.53 P 0.61
The result is nowhere near significant. All the
irrelevant variability among the starting and
final weights has fogged the issue.
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Greater power of the paired t-test
Paired t-test clearly detected the effect of the
diet, whereas ordinary two sample t-test fell far
short. An example of the greater power of the
paired t-test.
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When can a paired t-test be used?
When the data forms natural pairs.
Natural pair - same person
Person Before After Change no.
(Kg) (Kg)
(Kg) 1 70 65.5 -4.5 2 77 71.5 -5.5 3 74.5
73.0 -1.5
Natural pair - same person
etc
This is naturally paired data. The paired t-test
is appropriate.
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Aldostero data is NOT paired
These 2 values have nothing special special in
common
Controls Aldostero
146 149 140 153 152 149
etc
etc
Two separate sets of 15 babies have been studied.
Data is unpaired. It would make no sense to
calculate the difference between each of these
pairs of numbers. Paired t-test would be
inappropriate.
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Using the paired t-test
Whenever the data is naturally paired, the paired
t-test should be used. The increase in test
power only arises if the data is genuinely
paired. If the data is not paired, no advantage
will be gained by using a paired t-test.
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Another example of the use of the paired t-test
Different type of pairing
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Comparison of Leylandii trees from 2 sources
Ten gardens are selected. Two Leylandii trees
planted in each garden (One from source A, other
from B). Their heights are measured after
planting and again a year later. Record the
growth of each plant. Q Is there any
difference in growth rate according to the source
of the plants?
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Growth of Leylandii (Meters) from two sources (A
B)
Garden No. A B Diff (B - A) 1 1.16 1.26 0
.10 2 2.13 2.18 0.05 3 1.90 1.75 -0.15 4 1
.52 1.47 -0.05 5 1.45 1.30 -0.15 6 1.28 1.1
3 -0.15 7 1.97 1.71 -0.26 8 1.54 1.43 -0.11
9 1.40 1.31 -0.09 10 1.63 1.59 -0.04 11 1.
39 1.42 0.03 12 1.56 1.42 -0.14 13 1.96 1.
98 0.02 14 1.23 1.41 0.18 15 1.41 1.51 0
.10 16 1.57 1.50 -0.07 17 1.87 1.85 -0.02 18
2.11 2.08 -0.03 19 1.23 1.45 0.22 20 1.58
1.68 0.10 Mean 1.590.3
1.570.28 -0.0230.12
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Leylandii data is paired
Garden A B Diff (B - A) no.
1 1.16 1.26 0.10 2 2.13 2.18 0.05
3 1.90 1.75 -0.15
Same garden
etc
etc
In this case, the pairing is not based on the
same individual being studied twice.
None-the-less the data is naturally paired.
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Paired t-test of Leylandii data
Paired T-Test and Confidence Interval Paired T
for B - A N Mean
StDev SE Mean B 20 1.5715
0.2806 0.0628 A 20 1.5945
0.2988 0.0668 Difference 20 -0.0230
0.1234 0.0276 95 CI for mean difference
(-0.0807, 0.0347) T-Test of mean difference 0
(vs not 0) T-Value -0.83 P 0.415
There is no significant evidence of any
difference between the two sources of plants.
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Graphical output from analysis of Leylandii data
Difference between the two sources could be zero.
i.e. confirms non-significant result
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Terms with which you should be familiar

• Paired data
• Paired t-test

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What you should be able to do

• Distinguish between paired and unpaired data.
• Select a paired t-test whenever paired data are
  to be analysed
• Describe the greater power of the paired t-test
  when used appropriately
• Use Minitab (or other suitable package) to
  conduct a paired t-test and generate a 95 C.I.
  For the treatment effect.