Foundational and Applied Statistics for Biologists Using R

598 Pages·2013·9.06 MB·English

by  Aho

Statistics
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Full of biological applications, exercises, and interactive graphical examples, Foundational and o
S TAT I S T I C S f o r
Applied Statistics for Biologists Using R presents comprehensive coverage of both modern rA

analytical methods and statistical foundations. The author harnesses the inherent properties of B
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the R environment to examine the code of complicated procedures step by step. The graphical I

capabilities of R are used to provide interactive demonstrations of simple to complex statistical Oa B I O L O G I S T S
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The first seven chapters of the book address fundamental topics in statistics, such as the A
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philosophy of science, probability, estimation, hypothesis testing, sampling, and experimental G
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design. The remaining four chapters focus on applications involving correlation, regression, I
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ANOVA, and tabular analyses. S
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Features
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• Covers a wide range of analytical topics, including bootstrapping, Bayesian MCMC E

procedures, regression, model selection, GLMs, GAMs, nonlinear models, ANOVA designs, UD
mixed effects models, and permutation S
• Emphasizes the understanding of statistical foundations
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• Provides R code for all analyses and uses R to generate most of the figures N
• Offers an introduction to R and R code for each chapter on the author’s website
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Unlike classic biometric texts, this book provides you with an understanding of the underlying
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statistics involved in the analysis of biological applications. It shows how a solid statistical
foundation leads to the correct application of procedures, a clear understanding of analyses, and
valid inferences concerning biological phenomena.
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© 2008 Taylor & Francis Group, LLC
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F O U N D AT I O N A L
a n d A P P L I E D
S TAT I S T I C S f o r
B I O L O G I S T S
U S I N G R
© 2008 Taylor & Francis Group, LLC

F O U N D AT I O N A L
a n d A P P L I E D
S TAT I S T I C S f o r
B I O L O G I S T S
U S I N G R
Ken A. Aho
Idaho State University
Pocatello, Idaho, USA
© 2008 Taylor & Francis Group, LLC
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Contents
Preface ...........................................................................................................................................xix
Acknowledgments ......................................................................................................................xxi
Section I Foundations
1. Philosophical and Historical Foundations ........................................................................3
1.1 Introduction ...................................................................................................................3
1.2 Nature of Science ..........................................................................................................3
1.3 Scientific Principles .......................................................................................................4
1.3.1 Objectivity .........................................................................................................4
1.3.2 Realism ..............................................................................................................4
1.3.3 Communalism ..................................................................................................5
1.4 Scientific Method ..........................................................................................................5
1.4.1 A Terse History ................................................................................................6
1.4.1.1 Experimentation ..............................................................................8
1.4.1.2 Induction ...........................................................................................8
1.4.1.3 Probability ........................................................................................8
1.5 Scientific Hypotheses ...................................................................................................9
1.5.1 Falsifiability ......................................................................................................9
1.6 Logic ..............................................................................................................................10
1.6.1 Induction .........................................................................................................10
1.6.2 Deduction ........................................................................................................11
1.6.3 Induction versus Deduction .........................................................................11
1.6.4 Logic and Null Hypothesis Testing ............................................................12
1.6.4.1 Modus Tollens ................................................................................12
1.6.4.2 Reductio Ad Absurdum ...............................................................13
1.7 Variability and Uncertainty in Investigations ........................................................14
1.8 Science and Statistics ..................................................................................................16
1.9 Statistics and Biology ..................................................................................................16
1.10 Summary ......................................................................................................................18
Exercises ..................................................................................................................................18
2. Introduction to Probability .................................................................................................21
2.1 Introduction: Models for Random Variables ..........................................................21
2.1.1 Set Theory Terminology ...............................................................................23
2.1.2 Philosophical Conceptions of Probability ..................................................24
2.2 Classical Probability ...................................................................................................26
2.2.1 Disjoint ............................................................................................................28
2.2.1.1 Boole’s Inequality ..........................................................................30
2.2.2 Independence .................................................................................................31
2.2.2.1 Bonferroni’s Inequality .................................................................31
2.3 Conditional Probability ..............................................................................................32
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viii Contents
2.4 Odds ..............................................................................................................................34
2.4.1 Odds Ratio and Relative Risk ......................................................................34
2.5 Combinatorial Analysis .............................................................................................35
2.5.1 Multiplication Principle ................................................................................35
2.5.2 Permutations ...................................................................................................36
2.5.3 Combinations .................................................................................................37
2.6 Bayes Rule ....................................................................................................................38
2.7 Summary ......................................................................................................................43
Exercises ..................................................................................................................................43
3. Probability Density Functions ...........................................................................................49
3.1 Introduction .................................................................................................................49
3.1.1 How to Read This Chapter ...........................................................................50
3.1.2 So, What Is Density? ......................................................................................50
3.1.3 Cumulative Distribution Function ..............................................................52
3.2 Introductory Examples of pdfs .................................................................................54
3.2.1 Discrete pdfs ...................................................................................................55
3.2.1.1 Bernoulli Distribution ...................................................................55
3.2.1.2 Binomial Distribution ...................................................................55
3.2.2 Continuous pdfs .............................................................................................60
3.2.2.1 Continuous Uniform Distribution ..............................................60
3.2.2.2 Normal Distribution .....................................................................61
3.3 Other Important Distributions..................................................................................65
3.3.1 Other Discrete pdfs .......................................................................................66
3.3.1.1 Poisson Distribution ......................................................................66
3.3.1.2 Hypergeometric Distribution ......................................................69
3.3.1.3 Geometric Distribution .................................................................71
3.3.1.4 Negative Binomial Distribution ..................................................72
3.3.2 Other Continuous pdfs .................................................................................74
3.3.2.1 Chi-Squared Distribution .............................................................74
3.3.2.2 t-Distribution ..................................................................................76
3.3.2.3 F-Distribution .................................................................................77
3.3.2.4 Exponential Distribution ..............................................................79
3.3.2.5 Beta Distribution ...........................................................................81
3.3.2.6 Gamma Distribution .....................................................................82
3.3.2.7 Weibull Distribution .....................................................................83
3.3.2.8 Lognormal Distribution ...............................................................86
3.3.2.9 Logistic Distribution .....................................................................88
3.4 Which pdf to Use? .......................................................................................................90
3.4.1 Empirical cdfs .................................................................................................90
3.5 Reference Tables ..........................................................................................................91
3.6 Summary ......................................................................................................................92
Exercises ..................................................................................................................................95
4. Parameters and Statistics ..................................................................................................101
4.1 Introduction ...............................................................................................................101
4.1.1 How to Read This Chapter .........................................................................102
4.2 Parameters ..................................................................................................................103
4.2.1 Expected Value .............................................................................................103
© 2008 Taylor & Francis Group, LLC
Contents ix
4.2.2 Variance .........................................................................................................105
4.2.3 Chebyshev Inequality .................................................................................106
4.3 Statistics ......................................................................................................................106
4.3.1 Important Considerations ...........................................................................106
4.3.2 Sampling Error .............................................................................................108
4.3.3 Gauging Estimator Effectiveness...............................................................108
4.3.4 Types of Estimators .....................................................................................108
4.3.5 Measures of Location ..................................................................................109
4.3.5.1 Arithmetic Mean .........................................................................109
4.3.5.2 Geometric Mean ..........................................................................110
4.3.5.3 Harmonic Mean ...........................................................................111
4.3.5.4 Mode ..............................................................................................113
4.3.6 Robust Measures of Location .....................................................................113
4.3.6.1 Median ..........................................................................................114
4.3.6.2 Trimmed Mean ............................................................................115
4.3.6.3 Winsorized Mean ........................................................................115
4.3.6.4 M-Estimators ................................................................................116
4.3.6.5 Which Location Estimator to Use? ............................................118
4.3.7 Measures of Scale .........................................................................................119
4.3.7.1 Sample Variance ..........................................................................119
4.3.7.2 Coefficient of Variation ...............................................................121
4.3.8 Robust Estimators of Scale .........................................................................122
4.3.8.1 Interquartile Range .....................................................................123
4.3.8.2 Median Absolute Deviation .......................................................123
4.3.8.3 Which Scale Estimator to Use? ..................................................124
4.3.9 Parameters and Estimators for Distribution Shape ................................124
4.3.9.1 Moment Generating Functions ..................................................124
4.3.9.2 Sample Moments and MOM Estimators ..................................126
4.4 OLS and ML Estimators ...........................................................................................127
4.4.1 Ordinary Least Squares ..............................................................................127
4.4.2 Maximum Likelihood .................................................................................128
4.4.2.1 So, What Is Likelihood? ..............................................................132
4.4.2.2 Likelihood versus Probability ...................................................132
4.4.3 MOM versus OLS versus ML Estimation .................................................133
4.5 Linear Transformations ............................................................................................134
4.5.1 Transformations and Parameters ..............................................................135
4.5.2 Transformations and Statistics ...................................................................136
4.6 Bayesian Applications ..............................................................................................137
4.6.1 Priors ..............................................................................................................138
4.6.1.1 Noninformative Priors ................................................................138
4.6.1.2 Informative Priors .......................................................................138
4.6.2 Conjugacy ......................................................................................................138
4.7 Summary ....................................................................................................................143
Exercises ................................................................................................................................143
5. Interval Estimation: Sampling Distributions, Resampling Distributions,
and Simulation Distributions ..........................................................................................149
5.1 Introduction ...............................................................................................................149
5.1.1 How to Read This Chapter .........................................................................149
© 2008 Taylor & Francis Group, LLC