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QQ_ S\.:L-\ -) t> ,.., . ( -, ~_l f 1.:.' 'l · F ) \ Biometry The Principles and Practice of Statistics . in Biological Research FOURTH EDITION Robert R. Sakal and F. James Rohlf Stony Brook University ·= W.H. Freeman and Company New York ' \._ p Contents Preface xiii Notes on the Fourth Edition xvii 1 Introduction 1.1 1.2 1.3 Some Definitions The Development of Biometry The Statistical Frame of Mind 2 Data in Biology 1 3 5 9 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 9 11 13 16 19 3 Computers and Data Analysis 33 3.1 3.2 3.3 Computers Software Efficiency and Economy in Data Processing 4 Descriptive Statistics 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 The Arithmetic Mean Other Means The Median The Mode Sample Statistics and Parameters The Range The Standard Deviation Coding Data Before Computation The Coefficient of Variation 33 35 37 39 40 44 45 47 49 49 51 54 55 viii Contents 5 Introduction to Probability Distributions: 59 Binomial and Poisson 5.1 5.2 5.3 5.4 Probability, Random Sampling, and Hypothesis Testing The Binomial Distribution The Poisson Distribution Other Discrete Probability Distributions 6 The Normal Probability Distribution [/ c: c ::r E- ~,_ ~ c.,_ E...CJ E< E- CJ cE- > IJ ~ 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 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 7 Hypothesis Testing and Interval Estimation 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 Introduction to Hypothesis Testing: Randomization Approaches Distribution and Variance of Means Distribution and Variance of Other Statistics The t-Distribution More on Hypothesis Testing: Normally Distributed Data Power of a Test Tests of Simple Hypotheses Using the Normal and t-Distributions The Chi-Square Distribution Testing the Hypothesis H0 : 2 = Introduction to Interval Estimation (Confidence Limits) Confidence Limits Using Sample Standard Deviations Confidence Limits for Variances The Jackknife and the Bootstrap u u6 8 Introduction to Analysis of Variance 8.1 8.2 8.3 8.4 8.5 Variances of Samples and Their Means The F-Distribution The Hypothesis H0 : u~ = u~ Heterogeneity Among Sample Means Partitioning the Total Sum of Squares ~nrl nofT~OCH· "f. c .... ..,,-..rl,............., 60 68 78 87 93 93 95 100 102 104 106 108 117 119 120 131 137 140 142 146 148 154 156 157 162 167 168 177 178 182 187 190 Contents 8.6 8.7 200 203 Modell Anova Model II Anova 9 Single-Classification Analysis of Variance ------ 9.1 9.2 9.3 9.4 9.5 Computational Formulas General Case: Unequal and Equal n Special Case: Two Groups Comparisons Among Means in a Modell Anova : Essential Background Comparisons Among Means: Special Methods 10 Nested Analysis of Variance ----------------Nested Anova: Design 10.1 10.2 Nested Anova: Computation 10.3 Nested Anovas with Unequal Sample Sizes 11 Two-Way and Multiway Analysis of Variance 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 Two-Way Anova: Design Two-Way Anova with Equal Replication : Computation Two-Way Anova: Hypothesis Testing Two-Way Anova Without Replication Paired Comparisons The Factorial Design A Three-Way Factorial Design Higher-Order Factorial Anovas Other Designs Anova by Computer 12 Statistical Power and Sample Size in the Analysis of Variance --- ix 12.1 Effect Size 12.2 Noncentral t- and F-Distributions and Confidence Limits for Effect Sizes 12.3 Power in an Anova 12.4 Sample Size in an Anova 12.5 Minimum Detectable Difference 12.6 Post Hoc Power Analysis 207 208 208 220 228 246 277 277 280 301 319 319 321 331 340 349 354 355 365 370 372 379 379 382 390 391 395 396 X Contents 13 Assumptions of Analysis of Variance ··- - - 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 cr. c c ::r E- ~ ..... <:( 1.. E= C/ ..... E<I Er:/ cE> lJ, ~ -- ---·-A Fundamental Assumption Independence Homogeneity of Variances Normality Transformations The Logarithmic Transformation The Square Root Transformation The Box-Cox Transformation The Arcsine Transformation Nonparametric Methods in Lieu of Single-Classification A nova 13.11 Nonparametric Methods in Lieu of Two-Way Anova 14 Linear Regression 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 410 410 413 422 426 427 433 435 438 440 460 471 Introduction to Regression Models in Regression The Linear Regression Equation Hypothesis Testing in Regression More Than One Value of Y for Each Value of X 472 475 477 485 495 The Uses of Regression Estimating X From Y Comparing Two Regression Lines Linear Comparisons in Anovas 506 511 513 515 524 532 535 544 Examining Residuals and Transformations in Regression Nonparametric Tests for Regression Model II Regression Effect Size, Power, and Sample Size in Regression 15 Correlation 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 409 Correlation Versus Regression The Product-Moment Correlation Coefficient Computing the Product-Moment Correlation Coefficient The Variance of Sums and Differences Hypothesis Tests for Correlations Applications of Correlation Nonparametric Tests for Association Major Axes and Confidence Regions 551 551 554 562 565 567 577 580 t:;RA Contents 16 Multiple and Curvilinear Regression 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 Multiple Regression: Computation Multiple Regression : Hypothesis Tests Path Analysis and Structural Equation Modeling Partial and Multiple Correlation Selection of Independent Variables Computation of Multiple Regression by Matrix Methods Solving Anovas as Regression Problems: General Linear Models Analysis of Covariance (Ancova) Curvilinear Regression Effect Size, Power, and Sample Size in Multiple Regression Advanced Topics in Regression and Correlation 17 Analysis of Frequencies 18 xi 603 604 614 625 644 649 656 659 665 671 685 694 703 17.1 Introduction to Tests for Goodness of Fit 17.2 Single-Classification Tests for Goodness of Fit 17.3 Replicated Tests of Goodness of Fit 17.4 Tests of Independence: Two-Way Tables 17.5 Analysis of Three-Way Tables 17.6 Analysis of Proportions 17.7 Randomized Blocks for Frequency Data 17.8 Effect Sizes, Power, and Sample Sizes 704 714 730 739 758 773 793 801 Meta~Analysis and Miscellaneous Methods 817 18.1 18.2 18.3 18.4 18.5 18.6 841 847 850 852 859 Synthesis of Prior Research Results: Meta-Analysis Tests for Randomness of Nominal Data: Runs Tests Isotonic Regression Application of Randomization Tests to Unconventional Statistics The Mantel Test of Association Between Two Distance Matrices The Future of Biometry: Data Analysis Appendices A. Mathematical Proofs B. Introduction to Matrices 817 869 885 Bibliography 891 Author Index 909 Subject Index 915