Page 76 - Applied Probability

4
Hypothesis Testing and
Categorical Data

4.1 Introduction

Most statistical geneticists are frequentists, and fairly traditional ones at
that. In testing statistical hypotheses, they prefer pure significance tests
or likelihood ratio tests based on large sample theory. Although one could
easily dismiss this conservatism as undue reverence for Karl Pearson and
R. A. Fisher, it is grounded in the humble reality of geneticists’ inability
to describe precise alternative hypotheses and to impose convincing priors.
In the first part of this chapter, we will review by way of example the large
sample methods summarized so admirably by Cavalli-Sforza and Bodmer
[6], Elandt-Johnson [11], and Weir [44]. Then we will move on to modern
elaborations of frequentist tests for contingency tables. Part of the nov-
elty here is in designing tests sensitive to certain types of departures from
randomness. Permutation procedures permit approximation of the exact
p-values for these tests and consequently relieve our anxieties about large
sample approximations [28].

4.2 Hypotheses About Genotype Frequencies

An obvious question of interest to a geneticist is whether a trait satisfies
Hardy-Weinberg equilibrium in a particular population. If the trait is not in
Hardy-Weinberg equilibrium, then two explanations are possible. First, the
genetic model for the trait may be incorrect. For instance, a one-locus model
is inappropriate for a two-locus trait. If the model is basically correct, then
the further population assumptions necessary for Hardy-Weinberg equilib-
rium may not be met. Thus, forces such as selection and migration may be
distorting the Hardy-Weinberg proportions.
Our aim in this section is to discuss simple likelihood methods for testing
Hardy-Weinberg proportions. We emphasize likelihood ratio tests rather
than the usual chi-square tests. The two types of tests are similar, but
likelihood ratio tests extend more naturally to other statistical settings. Our
exposition assumes familiarity with basic notions of large sample theory.
Many books cover the essentials. At an elementary level we recommend
[31] and at an advanced level [14, 26, 36].