Browse Theses

The purpose of this study was to prepare computational tools, particularly for a microcomputer, to permit application of the unbiased testing technique in standard testing situations. Each situation has been illustrated by comparative studies of point testing and interval testing. A package of programs has been established to facilitate applications.

In the testing of hypotheses, a logical null-region should be an interval instead of a point. Measurements on a continuous random variable cannot be recorded with arbitrary precision. There is always some error (rounding, uncertainty). We express our interval hypotheses in standardized form and restrict the discussion to the normal case. In many standard testing situations such as those for means, linear model parameters, correlations, etc., unbiased tests do exist for both point and interval testing. The formulation of point-hypothesis and interval-hypothesis regions (e.g., in one-way analysis of variance) is as follows: H:$\mu\sb1$ = $\mu\sb2$ = dots = $\mu\sb{\rm k}$ versus K:not H (traditional), and H:$\rm\sum\sbsp{i=1}{k} p\sb{i}(\mu\sb{i}$ $-$ $\mu)\sp2/\sigma\sp2$ $\le$ $\delta\sp2$ versus K:$\rm\sum\sbsp{i=1}{k} p\sb{i}(\mu\sb{i}$ $-$ $\mu)\sp2/\sigma\sp2$ $>$ $\delta\sp2,$ where $\rm p\sb{i}$ is $\rm n\sb{i}$/N (sample size of group i over total sample size), $\mu\sb{\rm i}$'s are treatment means, and $\mu$ = $\rm\sum n\sb{i}\mu\sb{i}$/N. Under the normal assumption, the test statistics are subject to noncentral distributions (noncentral chi-square, t, F, beta, etc.). Although tables or charts for these distributions are available in the literature, their use is limited. Thus, computer programs (programs to find the cumulative distribution functions and inversion programs to find the percentage points and the noncentrality parameters) are desirable to facilitate the evaluation of the noncentral distributions and are developed in this dissertation.

In one-way analysis of variance with equal sample sizes, one can also express an interval hypothesis as H:${\rm Max\atop i,j}\vert(\alpha\sb{\rm i}$ $-$ $\alpha\sb{\rm j})/\sigma\vert\le\delta$ versus K:${\rm Max\atop i,j}\vert(\alpha\sb{\rm i}$ $-$ $\alpha\sb{\rm j})/\sigma\vert$ $>$ $\delta$, (where $\alpha\sb{\rm i}$ = $\mu\sb{\rm i}$ $-$ $\mu$; thus $\sum\alpha\sb{\rm i}$ = 0), and perform range tests when $\sigma\sp2$is known and studentized range tests when $\sigma\sp2$ is unknown. The distributions of the test statistics are noncentral range and noncentral studentized-range distributions, respectively. They are not functions of ${\rm n}\sum\alpha\sbsp{\rm i}{2}/\sigma\sp2$ alone, but functions of any k $-$ 1 of the $\alpha$'s. Furthermore, the tests are not unbiased. However, comparative studies have been included and computer programs have been provided.