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The procedures for testing hypotheses discussed in Chapters 3, 4, and 5 apply to experiments in which the control and treatment groups contain different subjects (individuals). It is often possible to design experiments in which each experimental subject can be observed before and after one or more treatments. Such experiments are generally more sensitive because they make it possible to measure how the treatment affects each individual. When the control and treatment groups consist of different individuals, the changes due to the treatment may be masked by variability between experimental subjects. This chapter shows how to analyze experiments in which each subject is repeatedly observed under different experimental conditions.


We will begin with the paired t test for experiments in which the subjects are observed before and after receiving a single treatment. Then, we will generalize this test to obtain repeated measures analysis of variance, which permits testing hypotheses about any number of treatments whose effects are measured repeatedly in the same subjects. We will explicitly separate the total variability in the observations into three components: variability between the experimental subjects, variability in each individual subject's response, and variability due to the treatments. Like all analyses of variance (including t tests), these procedures require that the observations come from normally distributed populations. (Chapter 10 presents methods based on ranks that do not require this assumption.) Finally, we will develop McNemar's test to analyze data measured on a nominal scale and presented in contingency tables.


In experiments in which it is possible to observe each experimental subject before and after administering a single treatment, we will test a hypothesis about the average change the treatment produces instead of the difference in average responses with and without the treatment. This approach reduces the variability in the observations due to differences between individuals and yields a more sensitive test.


Figure 9-1 illustrates this point. Figure 9-1A shows daily urine production in two samples of 10 different people each; one sample group took a placebo and the other took a drug. Since there is little difference in the mean response relative to the standard deviations, it would be hard to assert that the treatment produced an effect on the basis of these observations. In fact, t computed using the methods of Chapter 4 is only 1.33, which comes nowhere near t.05 = 2.101, the critical value for ν = npla + ndrug −2 = 10 + 10−2 = 18 degrees of freedom.

Figure 9-1.
Graphic Jump Location

(A) Daily urine production in two groups of 10 different people. One group of 10 people received the placebo and the other group of 10 people received the drug. The diuretic does not appear to be effective. (B) Daily urine production in a single group of 10 people before and after taking a drug. The ...

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