Biostatistics for Animal Science

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Preface This book was written to serve students and researchers of the animal sciences, with the primary purpose of helping them to learn about and apply appropriate experimental designs and statistical methods. Statistical methods applied to biological sciences are known as biostatistics or biometrics, and they have their origins in agricultural research. The characteristic that distinguishes biometrics within statistics is the fact that biological measurements are variable, not only because of measurement error, but also from their natural variability from genetic and environmental sources. These sources of variability must be taken into account when making inferences about biological material. Accounting for these sources of variation has led to the development of experimental designs that incorporate blocking, covariates and repeated measures. Appropriate techniques for analysis of data from these designs and others are covered in the book. Early in the book, readers are presented basic principles of statistics so they will be able to follow subsequent applications with familiarity and understanding, and without having to switch to another book of introductory statistics. Later chapters cover statistical methods most frequently used in the animal sciences for analysis of continuous and categorical variables. Each chapter begins by introducing a problem with practical questions, followed with a brief theoretical background and short proofs. The text is augmented with examples, mostly from animal sciences and related fields, with the purpose of making applications of the statistical methods familiar. Some examples are very simple and are presented in order to provide basic understanding and the logic behind calculations. These examples can be solved using a pocket calculator. Some examples are more complex, especially those in the later chapters. Most examples are also solved using SAS statistical software. Both sample SAS programs and SAS listings are given with brief explanations. Further, the solutions are often given with sufficient decimal digits, more than is practically necessary, so that readers can compare results to verify calculation technique. The first five chapters of the book are: 1) Presenting and Summarizing Data; 2) Probability; 3) Random Variables and Their Distributions; 4) Population and Sample; and 5) Estimation of Parameters. These chapters provide a basic introduction to biostatistics including definitions of terms, coverage of descriptive statistics and graphical presentation of data, the basic rules of probability, methods of parameter estimation, and descriptions of distributions including the Bernoulli, binomial, hypergeometric, Poisson, multinomial, uniform, normal, chi-square, t, and F distributions. Chapter 6 describes hypothesis testing and includes explanations of the null and alternate hypotheses, use of probability or density functions, critical values, critical region and P values. Hypothesis tests for many specific cases are shown such as population means and proportions, expected and empirical frequency, and test of variances. Also, the use of confidence intervals in hypothesis testing is shown. The difference between statistical and practical significance, types of errors in making conclusions, power of test, and sample size are discussed. Chapters 7 to 10 present the topics of correlation and regression. The coverage begins with simple linear regression and describes the model, its parameters and assumptions. Least squares and maximum likelihood methods of parameter estimation are shown. The concept of partitioning the total variance to explained and unexplained sources in the analysis of variance table is introduced. In chapter 8 the general meaning and definition of xii
Preface xiii the correlation coefficient, and the estimation of the correlation coefficient from samples and testing of hypothesis are shown. In chapters 9 and 10 multiple and curvilinear regressions are described. Important facts are explained using matrices in the same order of argument as for the simple regression. Model building is introduced including the definitions of partial and sequential sum of squares, test of model adequacy using a likelihood function, and Conceptual Predictive and Akaike criteria. Some common problems of regression analysis like outliers and multicollinearity are described, and their detection and possible remedies are explained. Polynomial, nonlinear and segmented regressions are introduced. Some examples are shown including estimating growth curves and functions with a plateau such as for determining nutrient requirements. One-way analysis of variance is introduced in chapter 11. In this chapter a one-way analysis of variance model is used to define hypotheses, partition sums of squares in order to use an F test, and estimate means and effects. Post-test comparison of means, including least significant difference, Tukey test and contrasts are shown. Fixed and random effects models are compared, and fixed and random effects are also shown using matrices. Chapters 12 to 21 focus on specific experimental designs and their analyses. Specific topics include: general concepts of design, blocking, change-over designs, factorials, nested designs, double blocking, split-plots, analysis of covariance, repeated measures and analysis of numerical treatment levels. Examples with sample SAS programs are provided for each topic. The final chapter covers the special topic of discrete dependent variables. Logit and probit models for binary and binomial dependent variables and loglinear models for count data are explained. A brief theoretical background is given with examples and SAS procedures. We wish to express our gratitude to everyone who helped us produce this book. We extend our special acknowledgement to Matt Lucy, Duane Keisler, Henry Mesa, Kristi Cammack, Marijan Posavi and Vesna Luzar-Stiffler for their reviews, and Cyndi Jennings, Cinda Hudlow and Dragan Tupajic for their assistance with editing. Zagreb, Croatia Miroslav Kaps Columbia, Missouri William R. Lamberson March 2004
Page 1: Biostatistics for Animal Science
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Chapter 3 Random Variables and thei
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The expectation and variance of the
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Chapter 4 Population and Sample A p
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Chapter 4 Population and Sample 55
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5.2 Maximum Likelihood Estimation C
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Chapter 5 Estimation of Parameters
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By setting both terms to zero the e
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Chapter 5 Estimation of Parameters
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Chapter 6 Hypothesis Testing The fo
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µ 0 y − z ≈ s n Chapter 6 Hypo
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Known values are: y = 4000 kg σ =
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α -zα Figure 6.7 The critical val
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s( y 1− y2 ) = 2 s1 s + n n 1 2 2
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The estimated pooled variance is: s
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Chapter 6 Hypothesis Testing 77 tha
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( y − y2 ) − 0 ( 11. 857 − 21
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Normal Approximation Z 2.0626 One-S
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pˆ − pˆ 1 The estimator has var
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2 χ [ yi E( yi ) ] E( y ) ∑ −
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∑ ∑ y i i p0 = i = 1,..., k n i
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Chapter 6 Hypothesis Testing 89 Exp
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Chapter 6 Hypothesis Testing 91 An
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Decision of a statistical test H 0
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Chapter 6 Hypothesis Testing 95 nor
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Chapter 6 Hypothesis Testing 97 One
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0.45 0.40 0.35 0.30 0.25 0.20 0.15
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SAS output: alpha n mi0 mi1 stdev d
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6.12 Sample Size Chapter 6 Hypothes
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SAS program: DATA a; DO n = 2 TO 10
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Chapter 6 Hypothesis Testing 107 PR
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Chapter 7 Simple Linear Regression
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The measurements of the dependent v
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2 SSRES s = = MSRES = 115. 826 n
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Confidence intervals follow the cla
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spread yi about y A) y y * * * * *
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SS REG ( SSxy ) = SS xx 2 Chapter 7
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E(MSREG) = σ 2 + β1 2 SSxx Chapte
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The likelihood function when H0 is
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2 R = SS SS REG TOT Chapter 7 Simpl
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= (X'X) -1 X'y A maximum likelihood
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DATA a; alpha=0.05; n=6; b=7.52941;
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Chapter 8 Correlation 147 of correl
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has a t-distribution with (n - 2) d
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Chapter 8 Correlation 151 under H0:
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Exercises Chapter 8 Correlation 153
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The assumptions of the model are: 1
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where: Chapter 9 Multiple Linear Re
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The residual mean square, which is
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The null and alternative hypotheses
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Chapter 9 Multiple Linear Regressio
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yi = the weight of bull i x1i = the
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SAS program: DATA bulls; INPUT weig
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9.6.1 Analysis of Residuals Chapter
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y * * 1 * * * * * * * * * 2 * * * 4
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where: Chapter 9 Multiple Linear Re
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SAS program: DATA bull; INPUT weigh
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where: SSRES = residual mean square
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Chapter 10 Curvilinear Regression I
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Chapter 10 Curvilinear Regression 1
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The variance estimate is s 2 = 210.
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SAS program: DATA a; INPUT age weig
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Weight (g) 1000 800 600 400 200 0 0
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Chapter 11 One-way Analysis of Vari
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From the assumptions it follows: Ch
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By defining: SS SS SS TOT TRT RES =
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1) Total sum: Σi Σj yij = (270 +
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2 L( µ , σ | y) = The log likelih
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And as shown previously: Thus: 2
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The hypotheses can be tested using
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ANOVA table: Chapter 11 One-way Ana
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F 3 = 2890 2 296. 67 = 4. 871 Chapt
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SAS output: Chapter 11 One-way Anal
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The effect of sow 1 is: ˆ 1 τ = 0
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The log likelihood that is to be ma
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Solution for Random Effects Chapter
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11.3.1.2 Estimating Parameters Chap
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The sums of squares needed for test
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The Xr'Xr matrix and its inverse ar
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11.3.2 The Random Effects Model 11.
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11.3.2.4 Restricted Maximum Likelih
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Chapter 12 Concepts of Experimental
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12.5 Required Number of Replication
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SAS output: alfa df1 df2 n power 0.
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Chapter 13 Blocking 273 environment
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1) Total sum: Σi Σj yij 2) Correc
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Chapter 13 Blocking 277 Example: Th
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13.1.3 SAS Example for Block Design
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Animal No. (Treatment) Blocks I II
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Mean square for treatments: Mean sq
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= s y − y MS ij . i' j' . RES ⎛
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1347. 90 F = = 7. 67 175. 83 F test
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Dependent Variable: d_gain i/j 1 2
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Chapter 13 Blocking 291 the adjuste
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Exercises Chapter 13 Blocking 293 1
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µ = the overall mean τi = the fix
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BLOCK II The hypotheses are: Chapte
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The results are shown in the ANOVA
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Chapter 14 Change-over Designs 301
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3) Total (corrected) sum of squares
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3) Total (corrected) sum of squares
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Least Squares Means Adjustment for
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Chapter 15 Factorial Experiments A
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1) Total sum: Σi Σj Σk yijk 2) C
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Chapter 15 Factorial Experiments 31
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5) Sum of squares for vitamin II: (
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Chapter 15 Factorial Experiments 32
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Chapter 16 Hierarchical or Nested D
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6) Sum of squares within factor B (
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3) Total sum of squares: Chapter 16
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SAS output of the NESTED procedure:
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Chapter 17 More about Blocking If t
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Chapter 17 More about Blocking 333
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SAS program: DATA steer; INPUT pen
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SAS output of the GLM procedure: De
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Chapter 17 More about Blocking 339
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The ANOVA table is: Source Degrees
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The model for this design is: where
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Chapter 18 Split-plot Design 345 su
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Least Squares Means Chapter 18 Spli
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Source Degrees of freedom Factor A
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Effects Means Estimators Standard e
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Chapter 18 Split-plot Design 353 Ex
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Chapter 19 Analysis of Covariance A
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DATA gain; INPUT treatment $ initia
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Chapter 19 Analysis of Covariance 3
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For males (M) the model is: E(yi) =
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Chapter 19 Analysis of Covariance 3
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Chapter 20 Repeated Measures Experi
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Chapter 20 Repeated Measures 367 Ex
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8 9 1 1.3 8 10 1 1.3 8 11 1 1.3 8 1
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SAS output: Covariance Parameter Es
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where: Chapter 20 Repeated Measures
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Chapter 20 Repeated Measures 375 Ex
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σ ⎡ σ b σ ⎡ ⎤ 0 b0b1 [ 1 t
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⎡1 ⎢ ⎢ 1 R ˆ = ⎢1 ⎢ ⎣1
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⎡1 ⎢ ˆR ⎢ 1 = ⎢1 ⎢ ⎣1
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Chapter 20 Repeated Measures 383
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Chapter 21 Analysis of Numerical Tr
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21.2.1 SAS Example for Polynomial C
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Chapter 22 Discrete Dependent Varia
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where: pi = the proportion with mas
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Analysis Of Parameter Estimates Cha
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SAS program: DATA a; INPUT n y farm
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Farm The model is: where: Total no.
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LR Statistics For Type 1 Analysis C
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Solutions of Exercises 1.1. Mean =
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Appendix A: Vectors and Matrices A
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⎡a A + B = ⎢ ⎢ a ⎢⎣ a A +
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a = (X'X) -1 X'y Appendix A: Vector
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Critical Values of Student t Distri
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Critical Values of Chi-square Distr
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Critical Values of F Distributions,
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Critical Value of F Distributions,
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Critical Values of the Studentized
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References 437 Gianola, D. and Hamm
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Subject Index accuracy, 266 Akaike
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y hypothesis testing, 56 by paramet
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hypothesis test, 186 t test, 186 ra
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of regression estimators, 119, 159
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