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The Combination of Statistical Tests of Significance Wendell Carl Smith Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1974 The combination of statistical tests of significance Wendell Carl Smith Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Smith, Wendell Carl, "The ombc ination of statistical tests of significance " (1974). Retrospective Theses and Dissertations. 5120. https://lib.dr.iastate.edu/rtd/5120 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 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Xerox University Microfilms 300 North Zeeb Road Ann Arbor, Michigan 48106 75-3335 'I SiriH, Wendell Carl, 1944- THE CCMBINATION OF STATISTICAL TESTS OF SIGNIFICANCE. Iowa State University, Ph.D., 1974 Statistics Xerox University Microfilms, Ann Arbor, Michigan 48106 THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED. The ccanbination of statistical tests of significance by Wendell Carl Smith A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of She Requirements for the Degree of DOCTOR OF PHILOSOPHÏ Major: Statistics Approved: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Graduâie College Iowa State University Ames, Iowa 197^ ii TABLE OF CONTENTS Page I. IÏÏTRODUCTIOIT 1 A. Concept of Combining Tests of H^otheses 1 B. Statement of the Problems h II. REVIEW OF COMPUTATIOML PROCEDURES 6 A. Univariate Central Distributions 6 1. Chi square 6 2. F distribution 8 3- Student's t 10 B. IMivariate Non-central Distributions 12 1. Non-central chi square 12 2. Non-central F l4 3- Non-central Student's t l6 C. A Miltivariate F Distribution 17 III. COMBIHATIOK OF INDEPENDENT TESTS OF SIGNIFICANCE 21 A. Introduction 21 B. Review of the Literature 21 C. Description of the Test Procedures 25 1. Fisher's method 27 2. Equiprobable test 28 3- Likelihood ratio test 28 4. Studentized test 31 D. Conç>arison of the Critical Regions for Combining Two Tests 33 1. Critical regions for equal sample sizes 3^ 2. Critical regions for unequal sample sizes 35 E. Computational Procedures for Power 4^4- 1. Fisher's test kh 2. Equiprobable test 51 3. Modified likelihood ratio test 52 4. Studentized test 55 5. A note on the evaluation of the integrals 56 iii Page F. Power Comparison 58 1. Equal sample sizes (ni =•112) 59 2. Unequal sample sizes (ni < ng) 60 3. Monotonicity of power curves 62 G. Conclusions Gk rV. FAMILY ERROR RATE FOR DEPEEDENT TESTS OF SIGNIFICANCE 83 A. Introduction 83 B. Review of the Literature 83 C. A Bivariate Distribution Involving Ratios of Chi Square Variates 87 1. V2 even 87 2. V3 even 99 D. Applications I06 1. Family error rates for two tests I06 2. Testing a contrast after testing for the equality of k means IO8 V. PROBABILITY OF SELECTING THE "CORRECT" MODEL IN MODEL BUILDING 121 A. Introduction 121 B. Assumptions and Notation 123 C. Description of the Test Procedures 125 1. Never pool procedure 125 2. Always pool procedure 125 D. Exact Probability Expressions 126 1. Never pool procedure 127 2. Always pool procedure I3I E. Patnaik's Approximation to the Probabilities of Correct Selection 132^. F. Comparison of the Two Test Procedures I36 1. œ 136 iv Page 137 3* n—CO 138 G. Sample Size Considerations 145 1. "Least favorable" concept l46 2. Results for the general polynomial model ikQ 3' Application to other linear models I59 K. Concluding Remarks I60 VI. BIBLIOGRAPHY I63 VII. ACKN0WIiE2)GMENTS I69 1 I. nraRODucnoN A. Concept of Combining Tests of I^otbeses In the classical theoiy of hypothesis testing, a single null hypoth­ esis is tested against a specified alternative with the objective to obtain a decision (accept-reject) about the hypothesis . If a test rejects when the hypothesis is true, the resulting error is called a type I error. If a test fails to reject "sdien the hypothesis is false, the error is referred to as a type II error. Ideally, the researcher desires then a test procedure for which the probability of both the type I and type II error are zero. Unfortunately, this cannot be achieved since, for a fixed sample size, the modification of a test to decrease the probability of one of the errors vill increase the probability of the other. ïïierefore, the general procedure is to control or bound the prob­ ability of a type I error, and to use a test that has smallest type II error. In practice, a researcher may wish to consider simultaneously more than one null hypothesis. It is convenient then to think of the individual hypotheses as members in a "family" of hypotheses. In addition to the type I and type II errors for the individual hypotheses, there are numerous other errors that can be defined for the problem of hypothesis testing in this family framework. MUer (1966) discusses and gives an interpretation of many of these error concepts. Die error most relevant to the problems we consider is the "family type I error; " A family type I error results whenever a null hypothesis about the family is rejected when all the 2 individual null hypotheses are true. To illustrate the concept of a family type I error and also three different viewpoints that a given researcher can take of the family of hypotheses, assume we have a family consisting of k individual hypotheses (i=l,...,k) The first viewpoint is where the researcher desires a single decision atout the family as an aggregate and is not concerned with a decision about the individual members (hypotheses) of the family. In this context the problem is not greatly different from the classical problem of hypoth­ esis testing. Bie only distinction between the two is that here the hypothesis is viewed as consisting of a family of individual hypotheses. Bae evidence against the individual hypotheses , derived from sample data, is then "combined" to obtain a single decision about the family. In many cases, an "optimal" test exists for the when they are consid­ ered separately. A test of the hypotheses as a family is then derived by combining the individual test statistics or the significance level of the sample data computed from the known distributions of these test statistics. Note that the family type I error in this situation is committed when the family null hypothesis is rejected and the are all true. dhe second viewpoint is where a decision is desired about the indiv­ idual hypotheses, or some grouping of them, but the probability of a family type I error, called the family error rate, is required to be less than seme specified level. A family type I error occurs in this case if any of the individual are rejected when they are all true. In this situation, we are not concerned with combining the individual hypotheses to reach a single decision, as in the earlier case, but the tests are 3 considered simultaneously with the individual levels of significance altered so that the family error rate is controlled. Finally, the researcher may wish to make a decision about the individ­ ual hypotheses simultaneously, but he takes the viewpoint that there is no reason to alter the significance levels of an individual test simply because other tests are also performed. However, despite his reluctance to control the family error rate, he may still be interested in knowing its level when the tests are considered together. The problem of hypothesis testing when considered from the first two viewpoints falls into the category of simultaneous inference.
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