BiG METRY THE PRINCIPLES AND PRACTICE OF STATISTICS IN BIOLOGICAL RESEARCH THIRD EDITION Robert R. SOKAL and F. James ROHLF State University of New York at Stony Brook as a W. H. FREEMAN AND COMPANY New YorkLibrary of Congress Cataloging-in-Publication Data Sokal, Robert R. Biometry the principles and practice of statistics in biological research / Robert R. Sokal and F. James Rohlf.—3d ed. p.cm. Includes bibliographical references (p. 850) and index. ISBN-13: 978-0-7167-2411-7 ISBN-10: 0-7167-2411-1 1. Biometry. I. Rohlf, F. James, 1936— . II. Title. ZH323.5.863 1995 574'.01'S195—de20 94-11120 cIP © 1995, 1981, 1969 by W. H. Freeman and Company. All rights reserved. No part of this book may be reproduced by an mechanical, photographic, or electronic process, or in the form of a phonographic recording, nor may it be stored in a retrieval system, transmitted, or otherwise copied for public or private use, without written permission from the publisher. Printed in the United States of America Eleventh printing W. 1H. Freeman and Company 41 Madison Avenue New York, NY 10010 Houndmills, Basingstoke, RG21 6XS, England www. whfreeman.comTo our parents of blessed memory Klara and Siegfried Sokal Harriet and Gilbert RohlfCONTENTS PREFACE NOTES ON THE THIRD EDITION 1 INTRODUCTION 1.1 Some Definitions 1.2 The Development of Biometry 1.3 The Statistical Frame of Mind 2 DATA IN BIOLOGY 2.1 Samples and Populations 2.2. Variables in Biology 2.3 Accuracy and Precision of Data 2.4 Derived Variables 2.5 Frequency Distributions 3 THE HANDLING OF DATA 3.1 Computers 3.2 Software 3.3 Efficiency and Economy in Data Processing 4 DESCRIPTIVE STATISTICS 4.1 The Arithmetic Mean 4.2 Other Means 4.3. The Median 4.4 The Mode 4.5 The Range 4.6 The Standard Deviation xiii xviivill “ 5.1 5.2 53 5.4 CONTENTS Sample Statistics and Parameters 52 Coding Data Before Computation 53 Computing Means and Standard Deviations 54 The Coefficient of Variation 57 INTRODUCTION TO PROBABILITY DISTRIBUTIONS: BINOMIAL AND POISSON Probability, Random Sampling, and Hypothesis Testing The Binomial Distribution The Poisson Distribution Other Discrete Probability Distributions 6 THE NORMAL PROBABILITY DISTRIBUTION _ 7.10 Frequency Distributions of Continuous Variables Properties of the Normal Distribution A Model for the Normal Distribution Applications of the Normal Distribution Fitting a Normal Distribution to Observed Data Skewness and Kurtosis Graphic Methods Other Continuous Distributions ESTIMATION AND HYPOTHESIS TESTING Distribution and Variance of Means Distribution and Variance of Other Statistics Introduction to Confidence Limits The t-Distribution Confidence Limits Based on Sample Statistics The Chi-Square Distribution Confidence Limits for Variances Introduction to Hypothesis Testing Tests of Simple Hypotheses Using the Normal and 1-Distributions Testing the Hypothesis Hp: 0? = 03 8 INTRODUCTION TO THE ANALYSIS OF VARIANCE 8) 8.2 83 Variances of Samples and Their Means The F-Distribution The Hypothesis Hy: of= 03 98 98 101 106 ut MW 116 123 127 128 136 139 143 152 154 157 169 175 179 180 184 189CONTENTS 10 R 84 8.5 8.6 8.7 Heterogeneity Among Sample Means Partitioning the Total Sum of Squares and Degrees of Freedom Model I Anova Model I Anova SINGLE-CLASSIFICATION ANALYSIS OF VARIANCE 9.1 9.2 9.3 9.4 95, 9.6 97 98 Computational Formulas General Case: Unequal n Special Case: Equal n Special Case: Two Groups Special Case: A Single Specimen Compared With a Sample Comparisons Among Means: Planned Comparisons Comparisons Among Means: Unplanned Comparisons Finding the Sample Size Required for a Test NESTED ANALYSIS OF VARIANCE Nested Anova: Design Nested Anova: Computation Nested Anovas With Unequal Sample Sizes The Optimal Allocation of Resources TWO-WAY ANALYSIS OF VARIANCE Ml 11.2 11.3 114 115 116 117 Two-Way Anova: Design Two-Way Anova With Equal Replication: Computation Two-Way Anova: Significance Testing Two-Way Anova Without Replication Paired Comparisons Unequal Subclass Sizes Missing Values in a Randomized-Blocks Design MULTIWAY ANALYSIS OF VARIANCE 12.1 12.2 12.3 12.4 12.5 The Factorial Design A Three-Way Factorial Anova Higher-Order Factorial Anovas Other Designs ‘Anovas by Computer 197 201 203 207 208 208 217 219 227 229 260 370 381 385CONTENTS 13 ASSUMPTIONS OF ANALYSIS OF VARIANCE 13.10 13.11 13.12 A Fundamental Assumption Independence Homogeneity of Variances Normality Additivity Transformations The Logarithmic Transformation The Square-Root Transformation The Box-Cox Transformation The Arcsine Transformation Nonparametric Methods in Lieu of Single- Classification Anovas Nonparametric Methods in Lieu of Two-Way Anova Ih LINEAR REGRESSION Is 14.1 Introduction to Regression 14.2. Models in Regression 14.3 The Linear Regression Equation 14.4 Tests of Significance in Regression 14.5 More Than One Value of Y for Each Value of X 14.6 The Uses of Regression 14.7 Estimating X from ¥ 148 Comparing Regression Lines 14.9 Analysis of Covariance 14.10 Linear Comparisons in Anovas 14.11 Examining Residuals and Transformations in Regression 14.12 Nonparametric Tests for Regression 14.13 Model I Regression CORRELATION 15.1 Correlation and Regression 15.2. The Product-Moment Correlation Coefficient 15.3. The Variance of Sums and Differences 15.4 Computing the Product- Moment Correlation Coefficient 15.5. Significance Tests in Correlation 15.6 Applictions of Correlation 15.7 Principal Axes and Confidence Regions 13.8 Nonparametric Tests for Association 392 393 393 396 406 407 409 413 415 417 419 423 440 451 452 455 457 466 476 486 491 493 499 521 531 539 541 555 556 559 567 569 574 583 586 593CONTENTS. 16 MULTIPLE AND CURVILINEAR REGRESSION 16.1 16.2 16.3 16.4 16.5 16.6 16.7 Multiple Regression: Computation Multiple Regression: Significance Tests Path Analysis Partial and Multiple Correlation Choosing Predictor Variables Curvilinear Regression Advanced Topics in Regression and Correlation IT ANALYSIS OF FREQUENCIES WA 17.2 aoe 174 a 17.6 17.7 Introduction to Tests for Goodness of Fit Single-Classification Tests for Goodness of Fit Replicated Tests of Goodness of Fit Tests of Independence: Two-Way Tables Analysis of Three-Way and Multiway Tables Analysis of Proportions Randomized Blocks for Frequency Data 18 MISCELLANEOUS METHODS 18.1 18.2 18.3 18.4 18.5 Combining Probabilities From Tests of Significance Tests for Randomness of Nominal Data: Runs Tests Randomization Tests The Jackknife and the Bootstrap The Future of Biometry: Data Analysis APPENDIX: MATHEMATICAL PROOFS BIBLIOGRAPHY AUTHOR INDEX SUBJECT INDEX 678 685 686 697 at 724 743 718 794 794 197 803 820 825 833 850 871- PREFACE The success of the first two editions of Biometry among teachers, stu- dents, and others who use biological statistics has encouraged us to prepare this extensively revised Third Edition. We wrote and have revised this book because we feel that there is a need for an up-to-date text aimed primarily at the academic biologist—a text that develops the subject from an elementary introduction up to the advanced methods necessary nowadays for biological research and for an understanding of the published litera- ture. Many available texts represent the outlook and interests of agricultural experi- ment stations. This is quite proper in view of the great application of statistics in this field, in fact, modem statistics originated at such institutions. However. personal inclination and the nature of the institution at which we teach cause us to address ourselyes to general biologists, ecologists, geneticists, physiologists, and other biol- ogists working largely on nonapplied subjects in universities, research institutes, and museums. Considerable overlap exists between the needs of these two somewhat artificially contrasted groups and, of necessity. some of our presentation will deal with agricultural experimentation. More broadly, the statistical methods treated here are useful in many applied fields, including medicine and allied health scien- ces. Since it is a well-known pedagogical dictum that people learn best by familiar examples, we have endeavored to make our examples as pertinent as possible for our readers. Much agricultural and biological work is by its very nature experimental. This book, while furnishing ample directions for the analysis of experimental work, also stresses descriptive and analytical statistical study of biological phenomena. These powerful methods are often overlooked and sometimes, by implication, the validity of nonexperimental biological work is put in doubt. We think that descriptive, ana- lytical, and experimental approaches are all of value, and we have tried to strike a balance among them. This approach will be appreciated by workers in applied fields such as the health sciences where ethical, financial, or other considerations may prevent free use of direct experiment. The readers we hope to interest are graduate students in biological departments who require a knowledge of biometry as part of their profesional training and pro- xu