# CHAPTER 5

# AN ADAPTIVE TEST FOR A SUBSET OF COEFFICIENTS IN A LINEAR MODEL

## 5.1 THE GENERAL ADAPTIVE TESTING METHOD

In Chapter 3 we demonstrated that the two-sample test was more powerful than the traditional test when the errors were not normally distributed. In this chapter we consider a test for a subset of coefficients in a regression model. Will the performance that we saw with two-sample tests carry over to the tests that are used with more complex models?

An adaptive test for a subset of coefficients in a linear regression model will be used for all tests in this chapter and the next three chapters. The adaptive test, which is a generalization of the two-sample test that was described in Chapter 3, requires two steps:

- Compute the weights that will normalize the residuals.
- Compute the
*p*-value by using a permutation method.

In this chapter we will carefully estimate the level of significance and the relative power of this adaptive test for testing a slope in a simple linear regression and for testing coefficients in a multiple linear regression. We will also show how these can be performed in R and in SAS. We begin by describing the weighting procedure in detail.

### 5.1.1 Weighting Step

In this section we will use matrix notation to describe the linear models. Let **Y** be the n × 1 vector for the dependent variable, let **X** be the *n* × *q* matrix of independent variables, and let *β* be the *q* × 1 vector of regression coefficients. With this notation the. linear model can he written as ...