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Discriminant Analysis Based on Binary and Continuous Variables Ching-Tsao Tu Iowa State University

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Discriminant Analysis Based on Binary and Continuous Variables Ching-Tsao Tu Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1978 Discriminant analysis based on binary and continuous variables Ching-tsao Tu Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Tu, Ching-tsao, "Discriminant analysis based on binary and continuous variables" (1978). Retrospective Theses and Dissertations. 6431. https://lib.dr.iastate.edu/rtd/6431 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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Unless we meant to delete copyrighted materials that should not have been filmed, you will fînd a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photo­ graphed the photographer has followed a definite method in "sectioning" the material. It is customary to begin Aiming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again-beginning below the first row and continuing on until complete. 4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases we have filmed the best available copy. Universi^ Micrdnlms International 300 N /EEB ROAD, ANN ARBOR. Ml 48106 18 BEDFORD ROW, LONDON WCIR 4EJ, ENGLAND 7907291 TU, CdING-TSA] DISCRIMINAIT ANALYSIS BASED ON BINARY AID CDNTIHUaUS VARIABLES. IOWA STATE UNIVERSITY, PH.D., 1978 Univesiw MicfOTlms International 300 N ZEEB ROAD. ANN ARBOR. Ml 48106 Discriminant analysis based on binary and continuous variables by Ching-tsao Tu A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major; Statistics Approved; Signature was redacted for privacy. in cnârge •or inâjôr wûEK Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. Iowa State University Ames, Iowa 1978 ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 1.1. The Problem of Discriminant Analysis 1 1.2. Discriminant Analysis Based on Binary and Continuous Variables 5 1.3. Objectives 13 1.4. Order of Magnitude 14 2. A SAMPLING SCHEME 18 2.1. The Problem 18 2.2. Double Inverse Sampling From One Population 20 2.3. The Distribution of N^ 25 2.4. The Moments of N^ 29 2.5. Maximum Likelihood Estimator of 6 31 2.6. Extension to Two Populations 40 3. DISCRIMINANT FUNCTIONS AND PROCEDURES 58 3.1. Introduction 58 3.2. The Model 60 3.3. The Discriminant Function When All Parameters are Known 62 3.4. The Sample Discriminant Functions 66 4. AN ASYMPTOTIC EXPANSION FOR THE DISTRIBUTION OF STANDARDIZED WHEN 6^ AND 8^ ARE KNOWN 7 5 4.1. Introduction 75 4.2. Lemmas 76 4.3. The Derivation of an Asymptotic Expansion for the Distribution of the standardized 99 iii Page 5. AN ASYMPTOTIC EXPANSION FOR THE DISTRIBUTION OF STUDENTIZED T^ WHEN ALL PARAMETERS ARE UNKNOWN 122 5.1. Introduction 122 5.2. Lemmas 125 5.3. The Derivation of an Asymptotic Expansion for the Distribution of the Studentized T^ 145 5.4. Probabilities of Misclassification and a Choice of Cut-off Point 157 6. COMPARISONS OF THE THREE CLASSIFICATION PROCEDURES WHEN X AND Y ARE INDEPENDENT 165 6.1. Introduction 16 5 6.2. The Case when all Parameters are Known 167 6.3. The Case when 0, and 8g are Known but y-., and E are Unknown 175 6.4. Discussion and Conclusion 181 7. REFERENCES - 183 8. ACKNOWLEDGMENTS 187 9. APPENDIX A 188 9.1. Partial Derivatives of 188 10. APPENDIX B 196 10.1. Partial Derivatives of Standard Normal Density Function 196 11. APPENDIX C 198 11.1. Partial Derivatives of Cumulative Standard Normal Function 19fi 12. APPENDIX D 200 13. APPENDIX E 203 iv Page 14. APPENDIX F 212 15. APPENDIX G 216 1 1. INTRODUCTION 1.1. The Problem of Discriminant Analysis The problem of classifying an individual into one of two concerned groups (called populations), arises in many areas, typically in anthropology, education, psychology, medical diagnosis, biology, engineering, etc. An anthropometrician may wish to identify ancient human remains in two different racial groups or in two different time periods by measuring certain skull characters (Barnard, 1935). A plant breeder discriminates a desired from an undesirable species by observing some heritable characters (Smith, 1936). A company hires or rejects an applicant frequently based on a certain measurement. Similarly a college accepts or denies a prospective student usually based on his entrance examination scores. In a hospital, a patient may be diagnosed and classi­ fied into a certain potential disease group by a battery of tests. Usually it is assumed that there are two populations, say and iTg, the individual to be classified comes from either or TTg; furthermore, it is assumed that from previous experiments or records we have in our possession the charac­ teristic measurements of n^ individuals who were known to belong to and of ng individuals who were known to belong to ttj. Based on the available data obtained from previous n^+ng individuals and the corresponding character­ 2 istic measurements of a new individual, we would like to classify the new individual into either or by using certain criterion. The case of more than two populations will not be considered in this thesis. As a quick remark of the use of terminology, we shall not in this research distinguish discrimination, classification and identification as some authors do (e.g. Rao, 1965), and shall interchange the use of them freely. The beginning of discriminant analysis is rather diffi­ cult to determine exactly. The coefficient of racial like­ ness proposed by Pearson (1926) is probably one of the earliest statistical applications to discriminatory problems. It was primarily devised for anthropometrics. Fisher (1936, 1938, 1940) in 1936 introduced the linear discriminant func­ tion which marked the advent of modern discriminant analysis. To elaborate a little bit more, let i = 1,2; j = l,...,p and k = l,...,n., be the kth measurement of the jth ^ P _ _ character in population i, and let D = Z A.(Y,. -Y_. ), where nj^ j-2 ] •'•J • 6]' Y,. = Z Y.../n., be a linear function of differences in mean between two samples. Then, Fisher's rationale was to find a set of Xj, j = l,...,p such that E(D)//V(D) is maximized. From analysis of variance point of view, this is equivalent to maximizing the ratio of variation between groups to varia­ tion within groups. 3 The formal mathematical solution to Fisher's reasoning was supplied by Welch (1939). To see his approach, let Y = (Y^,...,Yp)' be the p characters measured on an individual coming from either or Tg, and let f^(Y) and f2(Y) be the probability densities for Y in the two populations. If it is possible to know the prior probabilities for and TTg to be and respectively. Then, by Bayes' Theorem the posterior probabilities for Y to belong to u, and ir, are respectively qifl(Y) qnfgtY) qp^TjfT+q^fjrSr ^en^ the^former is greater than the latter, equivalently then Y should be classified as coming from otherwise, classified as coming from iTg. If it is not possible to know the prior probabilities q^^ and qg, appealing to Neyman-Pearson lemma or applying the likelihood ratio principle, we find that the fn (Y) optimal solution is to assign Y to tt^ if £ > C, and to ng otherwise, where C is a constant chosen to satisfy some given probability level, when and are multivariate normal with means pi and respectively and with the same covariance matrix Z, either Bayes' approach or likelihood ratio principle will give the same linear discriminant function log = Y'Z (u^-yj) - (I'l) the first term of (1.1), namely Y'E "(yj_-y2)f is called Fisher's linear discriminant function. Welch thus mathematically justified Fisher's use of linear discriminant 4 function. The linear discriminant function given in (1.1) was based on the assumption that all parameters are known. In practice and Z are rarely known. Wald (1944) considered linear discriminant function both for large sample and small sample problems. He also introduced the concept of losses or costs of misclassification into the discrimination procedure.
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