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Adaptive Tests of Significance Using Permutations of Residuals with R and SAS by Thomas W. O'Gorman

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CHAPTER 12

ADAPTIVE CONFIDENCE INTERVALS AND ESTIMATES

12.1 THE RELATIONSHIP BETWEEN TESTS AND CONFIDENCE INTERVALS

Often researchers publish confidence intervals, instead of p-values from tests of significance, because they provide more useful information to the reader. Tests are used to determine if a value c, when used in a test of H0 : β = c, is reasonably consistent with the data, whereas confidence intervals give a range of values of c that are consistent with the data. Although tests and confidence intervals are often used in different situations, there is a close relationship between the two approaches.

One way to construct confidence intervals is to determine the interval that contains values of c that, when used in a test of significance, would lead to acceptance of the null hypothesis. To illustrate this idea, suppose we want to construct a 100(1 − α)% confidence interval for a regression parameter β in a regression model. Let images denote the confidence interval for β. If images, then the test of H0 : β = c versus Ha : βc would produce a two-sided p-value that is greater than α. If c ∉ [βL, βU], then the p-value would be less than α.

In the first ten chapters of this book we have described how to calculate p-values for many kinds of adaptive tests, so we should be able to construct ...

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