# 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 *H*_{0} : *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 denote the confidence interval for *β*. If , then the test of *H*_{0} : *β* = *c* versus *H*_{a} : *β* ≠ *c* would produce a two-sided *p*-value that is greater than *α*. If *c* ∉ [*β*_{L}, *β*_{U}], 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 ...