# CHAPTER 9

# ADAPTIVE MULTIVARIATE TESTS

## 9.1 THE TRADITIONAL LIKELIHOOD RATIO TEST

In the first eight chapters we have restricted our attention to adaptive tests for a subset of coefficients in a linear model having one dependent variable. A test for a subset of coefficients in a linear model that has only one dependent variable will be called a univariate test. In this chapter we extend our methods to multivariate linear models that can include several dependent variables, and we develop an adaptive test for a subset of coefficients in these multivariate linear models. In this section we introduce the notation for multivariate linear models and describe one of the test statistics for traditional (non-adaptive) multivariate tests.

Consider the multivariate linear model in matrix form as **Y** = **X***β* + *ε*, where **Y** is an *n* × *p* matrix of dependent variables, **X** is an *n* × *q* matrix of constants, *β* is a *q* × *p* matrix of regression coefficients, and *ε* is an *n* × *p* error matrix. The error vectors are assumed to be independent and identically distributed. However, for an individual observation the error for one variable may be correlated with the error for another variable. To test a subset of regression coefficients, we will use a reduced model that includes *r* independent variables with 1 ≤ *r* < *q*. Let **X**_{R} be the *n* × *r* matrix consisting of the first *r* columns of **X** and let *β*_{R} be the *r* × *p* matrix consisting of the first *r* rows of *β*. Let **X**_{A} be the *n* × (*q* − *r*) matrix of the last *q* − *r* columns of **X**, and let ...