# The principles and practice of statistics in biological research

Tags: Continuous Distributions, Interval Estimation, Hypothesis Testing, Normal Distribution, Nonparametric Methods, Skewness and Kurtosis, Graphic Methods, Normally Distributed Data, Analysis of Variance, Confidence Limits, Chi-Square Distribution, Effect Sizes, Effect Size, Frequency Distributions, Poisson Distribution, F. James Rohlf, Descriptive Statistics, Practice of Statistics, Binomial Distribution, Robert R. Sakal, Sample Size, Multiway Analysis of Variance, Factorial Design, Biological Research, Stony Brook University
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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 Some Definitions
1.2 The Development of Biometry
3
1.3 The Statistical Frame of Mind
5
2 Data in Biology
9
2.1 Samples and Populations
9
2.2 Variables in Biology
11
2.3 Accuracy and Precision of Data
13
2.4 Derived Variables
16
2.5 frequency distributions
19
3 Computers and DATA ANALYSIS
33
3.1 Computers
33
3.2 Software
35
3.3 Efficiency and Economy in data processing
37
4 descriptive statistics
39
4.1 The Arithmetic Mean
40
4.2 Other Means
44
4.3 The Median
45
4.4 The Mode
47
4.5 Sample Statistics and Parameters
49
4.6 The Range
49
4.7 The standard deviation
51
4.8 Coding Data Before Computation
54
4.9 The Coefficient of Variation
55
viii Contents
5 Introduction to probability distributions:
Binomial and Poisson
59
5.1 Probability, Random Sampling, and Hypothesis Testing
60
5.2 The Binomial Distribution
68
5.3 The Poisson Distribution
78
5.4 Other Discrete Probability Distributions
87
6 The Normal Probability Distribution
93
6.1 Frequency Distributions of Continuous Variables
93
6.2 Properties of the Normal Distribution
95
6.3 A Model for the Normal Distribution
100
6.4 Applications of the Normal Distribution
102
6.5 Fitting a Normal Distribution to Observed Data
104
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6.6 Skewness and Kurtosis 6.7 Graphic Methods
106 108
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6.8 Other Continuous Distributions
117
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7 Hypothesis Testing and Interval Estimation
119
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7.1 Introduction to Hypothesis Testing: Randomization Approaches 120
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7.2 Distribution and Variance of Means
131
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7.3 Distribution and Variance of Other Statistics
137
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7.4 The t-Distribution
140
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7.5 More on Hypothesis Testing: Normally Distributed Data
142
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7.6 Power of a Test
146
7.7 Tests of Simple Hypotheses Using the Normal and t-Distributions 148
7.8 The Chi-Square Distribution
154
u u6 7.9 Testing the Hypothesis H0: 2 =
156
7.10 Introduction to Interval Estimation (Confidence Limits)
157
7.11 Confidence Limits Using Sample Standard Deviations
162
7.12 Confidence Limits for Variances
167
7.13 The Jackknife and the Bootstrap
168
8 Introduction to Analysis of Variance
177
8.1 Variances of Samples and Their Means
178
8.2 The F-Distribution
182
8.3 The Hypothesis H0 : u~ = u~
187
8.4 Heterogeneity Among Sample Means
190
8.5 Partitioning the Total Sum of Squares
Contents ix
8.6 Modell Anova
200
8.7 Model II Anova
203
9 Single-Classification Analysis of Varia-nc-e - - - - 9.1 Computational Formulas 9.2 General Case: Unequal and Equal n 9.3 Special Case: Two Groups 9.4 Comparisons Among Means in a Modell Anova : Essential Background 9.5 Comparisons Among Means: Special Methods
207 208 208 220 228 246
10 Nested Analysis of-Va-ri-an-c-e ------------ 277
10.1 Nested Anova: Design
277
10.2 Nested Anova: Computation
280
10.3 Nested Anovas with Unequal Sample Sizes
301
11 Two-Way and Multiway Analysis of Variance 11.1 Two-Way Anova: Design 11.2 Two-Way Anova with Equal Replication : Computation 11.3 Two-Way Anova: Hypothesis Testing 11.4 Two-Way Anova Without Replication 11.5 Paired Comparisons 11.6 The factorial design 11.7 A Three-Way Factorial Design 11.8 Higher-Order Factorial Anovas 11.9 Other Designs 11.10 Anova by Computer
319 319 321 331 340 349 354 355 365 370 372
12 Statistical Power and Sample Size in the - -A-nalysis of Variance 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
379 379 382 390 391 395 396
X Contents
13 ··-
-A-ssum--p-tio-n-s·o-f-Analysis
of
Variance
409
13.1 A Fundamental Assumption
410
13.2 Independence
410
13.3 Homogeneity of Variances
413
13.4 Normality
422
13.5 Transformations
426
13.6 The Logarithmic Transformation
427
13.7 The Square Root Transformation
433
13.8 The Box-Cox Transformation
435
13.9 The Arcsine Transformation
438
13.10 Nonparametric Methods in Lieu of Single-Classification
A nova
440
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13.11 Nonparametric Methods in Lieu of Two-Way Anova 14 Linear regression
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14.1 Introduction to Regression
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14.2 Models in Regression
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14.3 The Linear Regression Equation
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14.4 Hypothesis Testing in Regression
E- 14.5 More Than One Value of Y for Each Value of X
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14.6 The Uses of Regression
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14.7 Estimating X From Y
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14.8 Comparing Two Regression Lines
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14.9 Linear Comparisons in Anovas
460 471 472 475 477 485 495 506 511 513 515
14.10 Examining Residuals and Transformations in Regression
524
14.11 Nonparametric Tests for Regression
532
14.12 Model II Regression
535
14.13 Effect Size, Power, and Sample Size in Regression
544
15 Correlation 15.1 Correlation Versus Regression 15.2 The Product-Moment Correlation Coefficient 15.3 Computing the Product-Moment Correlation Coefficient 15.4 The Variance of Sums and Differences 15.5 Hypothesis Tests for Correlations 15.6 Applications of Correlation 15.7 Nonparametric Tests for Association 15.8 Major Axes and Confidence Regions
551 551 554 562 565 567 577 580 t:;RA
Contents xi
16 Multiple and Curvilinear Regression 16.1 Multiple Regression: Computation 16.2 Multiple Regression : Hypothesis Tests 16.3 Path Analysis and Structural Equation Modeling 16.4 Partial and Multiple Correlation 16.5 Selection of independent variables 16.6 Computation of Multiple Regression by Matrix Methods 16.7 Solving Anovas as Regression Problems: General Linear Models 16.8 analysis of covariance (Ancova) 16.9 Curvilinear Regression 16.10 Effect Size, Power, and Sample Size in Multiple Regression 16.11 Advanced Topics in Regression and Correlation
603 604 614 625 644 649 656 659 665 671 685 694
17 Analysis of Frequencies 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
703 704 714 730 739 758 773 793 801
18 Meta~Analysis and Miscellaneous Methods
817
18.1 Synthesis of Prior Research results: Meta-Analysis
817
18.2 Tests for Randomness of Nominal Data: Runs Tests
841
18.3 Isotonic Regression
847
18.4 Application of Randomization Tests to Unconventional Statistics 850
18.5 The Mantel Test of Association Between Two Distance Matrices 852
18.6 The Future of Biometry: Data Analysis
859
Appendices
A. Mathematical Proofs
869
B. Introduction to Matrices
885
Bibliography
891
Author Index
909
Subject Index
915

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