Shun-Yi Chen

Single-stage analysis of variance under heteroscedasticity

The procedures of testing the equality of normal means in the conventional analysis of variance (ANOVA) are heavily based on the assumption of the equality of the error variances. Studies have shown that the distribution of the F-test depends heavily on the unknown variances and is not robust under the violation of equal error variances. When the variances are unknown and unequal, Bishop and Dudewicz (1978) developed a design-oriented two-stage procedure for ANOVA, which requires additional samples at the second stage. In this paper we use a single-stage sampling procedure to test the null hypotheses in ANOVA models under heteroscedasticity. The single-stage procedure for ANOVA has an exact distribution and it is a data-analysis-oriented procedure. It does not require additional samples, and can reach a conclusion much earlier, save time and money. Simulation results indicate that the power of the single-stage procedure is better than the two-stage method when the initial sample size is smaller than 6, an...

A range test for the equivalency of means under unequal variances

In this article, we present a range test using a two-stage sampling procedure for testing the hypothesis that the normal means are falling into a practical indifference zone. Both the level and the power associated with the proposed test are controllable and are completely independent of the unknown variances. Tables needed for implementation are given.

One-stage and two-stage statistical inference under heteroscedasticity

This paper presents general one-stage and two-stage sampling procedures for testing the hypotheses of the equality of means when the variances are unknown and unequal. The methods use a new range test and an ANOVA test based on independent t variables for an one-way layout. The one-stage procedure is an exact statistical procedure which provides a feasible solution to the two-stage procedure when the required two-stage sample sizes are not met due to time limitation, budget shortage, or other factors that result in early termination of the experiment. An extension to the two-way layout is also considered and the percentage points of the range test for testing the associated hypotheses are tabulated.

OPTIMAL CONFIDENCE INTERVAL FOR THE LARGEST NORMAL MEAN WITH UNKNOWN VARIANCE

A single-sample procedure for obtaining an optimal confidence interval for the largest or smallest mean of several independent normal populations is proposed. It is assumed that the common variance is unknown. It has been found that the optimal confidence interval is uniformly better than any other existing one-sample confidence interval in the sense of a reduced interval width. This optimal confidence interval is obtained by maximizing the coverage probability with the expected confidence width being fixed at a least favorable configuration of means. Tables of the critical values are given for the optimal confidence interval.

A NEARLY OPTIMAL CONFIDENCE INTERVAL FOR THE LARGEST NORMAL MEAN

A single-stage sampling procedure is proposed for obtaining a nearly optimal confidence interval on the largest (or smallest) mean of k independent normal populations when the common variances are unknown. Numerical methods to obtain the percentage points are thoroughly explained. Tables of these percentage points are given.

An ANOVA test for the equivalency of means under unequal variances

Abstract In this paper, we present a two-stage test procedure for testing the hypothesis that the normal means are falling into a practically indifferent zone. Both the level and the power associated with the proposed test are controllable and are completely independent of the unknown variances. Relation to a single-stage procedure is discussed when the two-stage sampling procedure cannot be completely carried through. An example and tables needed for implementation are given.

A range test for the equality of means when variances are unequal

SYNOPTIC ABSTRACT The level and the power of the usual F-test in analysis of variances are sensitive to unequal variances. In this paper, we present a range test using a two-stage sampling procedure for testing the hypothesis that the normal means are equal against a specific alternative. Both the level and the power associated with the proposed test are controllable at the desired values and are completely independent of the unknown variances. Tables for implementation are given.

Likelihood Ratio Tests for the Non Equivalence of Means

We derive likelihood ratio (LR) tests for the null hypothesis of equivalence that the normal means fall into a practical indifference zone. The LR test can easily be constructed and applied to k ≥ 2 treatments. Simulation results indicate that the LR test might be slightly anticonservative statistically, but when the sample sizes are large, it always produces the nominal level for mean configurations under the null hypothesis. More powerful than the studentized range test, the LR test is a straightforward application that requires only current existing statistical tables, with no complicated computations.

One-sided range test for testing against an ordered alternative under heteroscedasticity

In a one-way fixed effects analysis of variance model, when normal variances are unknown and possibly unequal, a one-sided range test for testing the null hypothesis H 0 : μ 1 = … = μk against an ordered alternative Ha : μ 1 ≤ … ≤ μk by a single-stage and a two-stage procedure, respectively, is proposed. The critical values under H 0 and the power under a specific alternative are calculated. Relation between the one-stage and the two-stage test procedures is discussed. A numerical example to illustrate these procedures is given.

A One-Stage Procedure for Testing Homogeneity of Means Against an Ordered Alternative Under Unequal Variances

Abstract In this paper we use a one-stage procedure described by Chen [Chen, S. Y. (2001). One-stage and two-stage statistical inference under heteroscedasticity. Commun. Statist. Simulation Computat. 30(4):991–1009] for testing the equality of normal means against an ordered alternative in one-way layout when variances are unknown and unequal. Tables of percentage points and the power under a specific alternative needed for implementation are given. The relationship between the one-stage and the two-stage test procedures is discussed. Comparison between the new procedure and the range test [Chen, S. Y. (2000). One-sided range test for testing against an ordered alternative under heteroscedasticity. Commun. Statist. Simulation Computat. 29(4): 1255–1272] by simulation study is also given.