7. Hypothesis Testing

In the preceding chapter, the theoretical basis of estimation theory was presented. Now we turn our interest towards testing issues: we want to test the hypothesis $H_0$ that the unknown parameter $\theta$ belongs to some subspace of $\mathbb{R}^q$. This subspace is called the null set and will be denoted by $\Omega_0 \subset \mathbb{R}^q$.

In many cases, this null set corresponds to restrictions which are imposed on the parameter space: $H_0$ corresponds to a ``reduced model''. As we have already seen in Chapter 3, the solution to a testing problem is in terms of a rejection region $R$ which is a set of values in the sample space which leads to the decision of rejecting the null hypothesis $H_0$ in favor of an alternative $H_1$, which is called the ``full model''.

In general, we want to construct a rejection region $R$ which controls the size of the type I error, i.e. the probability of rejecting the null hypothesis when it is true. More formally, a solution to a testing problem is of predetermined size $\alpha$ if:

$$P(\hbox{Rejecting } H_0 \;\vert\; H_0\hbox{ is true})= \alpha.$$

In fact, since $H_0$ is often a composite hypothesis, it is achieved by finding $R$ such that

$$\sup_{\theta \in \Omega_0} P({\cal{X}}\in R \;\vert\; \theta)= \alpha.$$

In this chapter we will introduce a tool which allows us to build a rejection region in general situations: it is based on the likelihood ratio principle. This is a very useful technique because it allows us to derive a rejection region with an asymptotically appropriate size $\alpha$. The technique will be illustrated through various testing problems and examples. We concentrate on multinormal populations and linear models where the size of the test will often be exact even for finite sample sizes $n$.

Section 7.1 gives the basic ideas and Section 7.2 presents the general problem of testing linear restrictions. This allows us to propose solutions to frequent types of analyses (including comparisons of several means, repeated measurements and profile analysis). Each case can be viewed as a simple specific case of testing linear restrictions. Special attention is devoted to confidence intervals and confidence regions for means and for linear restrictions on means in a multinormal setup.