# CHAPTER 6

# MORE APPLICATIONS OF ADAPTIVE TESTS

The general adaptive testing procedure that we described in the last chapter for the multiple regression problem can be used to perform a test of significance for any subset of coefficients in a linear model containing fixed effects. In this chapter we show how an adaptive approach could be used in most of the common testing situations to increase the power of the tests. We will analyze data from a one-way layout, a randomized complete block design, and a two-way layout. We will also show how the residuals can be permuted within blocks in a design that uses blocking. For each of these applications we will give the R and SAS code to perform the test.

## 6.1 THE COMPLETELY RANDOMIZED DESIGN

### 6.1.1 Model Specification

In this section we will describe the adaptive test for group effect in a completely randomized design having one factor with *g* levels. This is the standard one-way layout with *n*_{j} observations at level *j*. The model can be specified as

for *i* = 1,…, *n*, where

Other parameterizations of the model could be used. With this parameterization the mean of the *g*th level is the intercept. To determine if any of the level means are different from any of the other treatment means, we will test *H*_{0} : *β*_{1} = *β*_{2} = … = *β*_{g−1} = 0 against the alternative ...