Paired-Sample Hypothesis Testing
In chapter 8, we looked at the types of analysis that are appropriate when there is no relation between the observations in the first sample and the observations in the second sample. In this case each observation in the first sample, is completely random with respect to any observation in the second sample. However, in some cases an observation in the first sample may be related to or correlated with an observation in the second sample. In this case we have what is often called a paired or repeated measures design.
Paired-sample t-test (chapter 9.1)
For example, we might be interested in determining if there is a difference in size between the right hand and the left hand of right-handed people. We could simply measure the width of right and left handed people and do a two-sample t-test on the two sets of observations. However, the size of hands will vary due to factors such as sex, height, weight, or occupation of the person. Thus the variance among hands is greatly inflated by factors that we are not interested in, and this increased variance may make it difficult to reject the null hypothesis, if in fact the null hypothesis is false. Those four factors that I mentioned (sex, height, weight, and occupation) will likely affect both hands of the same person equally. Thus the size of the left hand is related to the size of the right hand, that is a person who has a large right hand will also have a large left hand, and vice versa. Thus if our analysis of these data takes into account the fact that an observation in the left hand set of observations is paired with an observation in the right hand set, then we can remove the effects of this variation in which we are not interested and we will have a more powerful test of our null hypothesis.
The following is a set of data taken of the width in centimeters of human hands across the distal end of the palm at the joint between the metacarpals and the phalanges with the fingers appressed together.
individual right left deviation (right - left)
1 8.7 8.6 0.1
2 7.9 7.6 0.3
3 7.6 7.2 0.4
4 8.5 8.2 0.3
5 9.6 9.7 -0.1
6 8.7 8.4 0.3
7 9.2 9.0 0.2
8 8.4 8.3 0.1
9 8.9 8.7 0.2
10 8.8 8.6 0.2
The null hypothesis is
Ho: mean width of the right hand is equal to the mean width of the left hand.
Another way to consider this is
Ho: mean width of right hand minus the mean width of the left hand equals zero,
which is the same as
Ho: the mean of the deviations is equal to zero.
As the observations from the right hand are paired with observations from the left hand, we can calculate a value called the deviations, which is the right hand observation minus the left hand observation for each person (see table above).
To test the null hypothesis that the mean of the deviations is equal to zero, we calculate a t-value.
t = dbar/sdbar (equation 9.1)
t = sample mean of the deviations divided by the standard error of d
dbar = 0.2, s2 = 0.02, s = 0.14, sdbar = 0.04472
t = 0.2/0.04472 = 4.47
To determine if we reject the null hypothesis, we compare our calculated t to a critical t.
Critical t = talpha(2)=0.05, 9 = 2.262. As our calculated t-value is greater than the critical value, we reject our null hypothesis. Thus we show that the right hand is not the same size as the left hand (t = 4.47, df = 9, 0.002 < P < 0.001).
Use SigmaStat for a paired-sample t-test
We can do one-tailed test, just as we can with other statistical tests (see example 9.2, page 163).
A Paired-sample t-test does not have the assumption that the variances of the two populations are equal nor does it have the assumption that the two population have normal distributions. It does, however, assume that the deviations are distributed normally.
If the data can be paired as above, then a paired-sample t-test will be more powerful than a two-sample t-test. In fact the more variation that can be accounted for by pairing, the more powerful the test because the difference between the two groups becomes more important relative to the unexplained variation in the two populations.
Another common design that employs the paired-sample t-test is the "before and after" type experiment. In this design an organism has two trials each, one an experimental treatment (say some drug) and another trial that serves as the negative control (without the drug). This is an extremely common design in that the before and after observations are paired because they are on the same individual. This is also called a repeated measures design as each individual has repeated measures.
If a "before and after" design is not possible, it may be possible to match individuals in the control and experimental groups. For example, humans may be matched by age, sex, body weight, or any of a number of factors. Thus for each pair of humans, there is one that is in the control group and paired with another individual who is in the treatment group.
Wilcoxon paired-sample test (chapter 9.5)
The Wilcoxon paired-sample test is a nonparametric test than be used if the assumption that the deviations are normally distributed is violated. Again, the paired sample t-test is quite robust so that moderate deviations from normality will have a minor effect on the power of the test. However, if the deviation from normality is large then the Wilcoxon paired-sample test will be a more powerful test of the null hypothesis.
You will not be responsible for the formulas for calculating the Wilcoxon paired-sample test, but you should be very familiar with the output from SigmaStat. See example 9.3 (page 166) for an example of the calculation.
If the observations are paired (as described for the paired-sample t-test) then the Wilcoxon paired-sample test is more powerful than the Mann-Whitney test. The same considerations as to the advantages of pairing (as described above) apply here as well.
Use SigmaStat for a Wilcoxon paired-sample test.
Last updated on 29 September 2000.
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