371chapter8f2011a - Chapter 8*Statistical Power(Optional...

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Chapter 8 *Statistical Power (Optional) 8.1 Tests of Hypotheses Revisited As discussed earlier, a standard practice in research consists of: 1. Select null and alternative hypotheses. 2. Specify the significance level of the test, α . (Remember that the significance level is the probability the test rejects a true null hypothesis.) 3. Collect and analyze data; decide whether or not to reject the null. In this chapter we will consider the following two questions. In practice, the first question is hugely more applicable than the second. 1. Just because we fail to reject the null, how good should we feel about continuing to assume it is true? 2. Just because we reject the null, does it really mean the null is no good? Rather than try to explore these questions in a general mathematical way, I will begin with several examples. To fix ideas, consider our Fisher’s test to investigate whether p is constant in a sequence of trials that might be BT. Suppose we obtain the following data. Table 1 Success Failure Total ˆ p First Half 4 1 5 0.80 Second Half 1 4 5 0.20 Total 5 5 10 The P-value for Fisher’s test is 0.2063 and with any popular choice of α the decision would be to fail to reject. But note that the ˆ p ’s are very different! Why did we get such a large P-value when the ˆ p ’s are so different? Well, because we do not have much data. In fact, if the amount of data is small, it can be difficult or impossible to reject the null. For example, the following table gives the largest possible difference between ˆ p ’s, yet the P-value is 0.1000; too large to reject for α < 0 . 10 . 87
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Table 2 Success Failure Total ˆ p First Half 3 0 3 1.00 Second Half 0 3 3 0.00 Total 3 3 6 At the other extreme (and note that these data are a bit silly in practice), consider the following table. Table 3 Success Failure Total ˆ p First Half 50,500 49,500 100,000 0.505 Second Half 50,000 50,000 100,000 0.500 Total 100,500 99,500 200,000 This table gives a P-value of 0.0256, which would lead to rejecting the null for α = 0 . 05 . But I cannot think of any scientific problem for which I would want to conclude that p has changed! The ˆ p ’s are so very close that I believe the assumption of constant p would be scientifically useful.
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