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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
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W.H. Freeman and Company
New York
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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
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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
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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
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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
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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
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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