Biostatistics for Animal Science Uae长ra and Universi o Misour-Columbi USA CABI Publishing
Biostatistics for Animal Science Miroslav Kaps University of Zagreb, Croatia and William R. Lamberson University of Missouri-Columbia, USA CABI Publishing
Table of Contents PREFACE CHAPTER 1 PRESENTING AND SUMMARIZING DATA .1 DATA AND VARIABLES TION OF QUALITATIVE DATA GRAPHICAL PRESENTA ION OF QUANTITATIVE DATA 3 ofa Hist 1.4 NUMERICAL METHODS FOR PRESENTING DATA. 14 143 144 Measures of the Shape of a Distribution. 1.4.5 Measures of Relative Position. EXERCISES XAMPLE CHAPTER 2 PROBABILITY. 15 RULES ABOUT PROBABILITIES OF SIMPLE EVENTS 17 2.2.3 Combinations 18 324 Partition Rule COMPOUND EVENTS 24 BAYES THEOREM. EXERCISES. 25 CHAPTER 3 RANDOM VARIABLES AND THEIR DISTRIBUTIONS 26 31 PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES EXPECTATIONS AND VARIANCES OF RANDOM VARIABLES 32 3.2.1 Expectation and Variance of a Discrete Random Variable ut10n. 03 324 3.2.5 Poisson Distribution .2.6 Multinomial Distribution. 35 33 FOR CONTINUOUS 3.3.1 67 332 Normal Distribution 37 3.3.3 Multivariate Normal Distribution. eaDistrbution 灯锅
v Table of Contents PREFACE XII CHAPTER 1 PRESENTING AND SUMMARIZING DATA .1 1.1 DATA AND VARIABLES.1 1.2 GRAPHICAL PRESENTATION OF QUALITATIVE DATA .2 1.3 GRAPHICAL PRESENTATION OF QUANTITATIVE DATA .3 1.3.1 Construction of a Histogram .3 1.4 NUMERICAL METHODS FOR PRESENTING DATA.6 1.4.1 Symbolic Notation .6 1.4.2 Measures of Central Tendency.7 1.4.3 Measures of Variability.8 1.4.4 Measures of the Shape of a Distribution .9 1.4.5 Measures of Relative Position.11 1.5 SAS EXAMPLE .12 EXERCISES .13 CHAPTER 2 PROBABILITY .15 2.1 RULES ABOUT PROBABILITIES OF SIMPLE EVENTS .15 2.2 COUNTING RULES .16 2.2.1 Multiplicative Rule.17 2.2.2 Permutations.17 2.2.3 Combinations .18 2.2.4 Partition Rule .18 2.2.5 Tree Diagram .18 2.3 COMPOUND EVENTS.19 2.4 BAYES THEOREM .23 EXERCISES .25 CHAPTER 3 RANDOM VARIABLES AND THEIR DISTRIBUTIONS .26 3.1 EXPECTATIONS AND VARIANCES OF RANDOM VARIABLES .26 3.2 PROBABILITY DISTRIBUTIONS FOR DISCRETE RANDOM VARIABLES .28 3.2.1 Expectation and Variance of a Discrete Random Variable .29 3.2.2 Bernoulli Distribution .30 3.2.3 Binomial Distribution.31 3.2.4 Hyper-geometric Distribution .33 3.2.5 Poisson Distribution.34 3.2.6 Multinomial Distribution.35 3.3 PROBABILITY DISTRIBUTIONS FOR CONTINUOUS RANDOM VARIABLES.36 3.3.1 Uniform Distribution.37 3.3.2 Normal Distribution .37 3.3.3 Multivariate Normal Distribution.45 3.3.4 Chi-square Distribution.47 3.3.5 Student t Distribution .48
vi Biostatistics for Animal Science CHAPTER 4 POPULATION AND SAMPLE 53 4.1 FUNCTIONS OF RANDOM VARIABLES AND SAMPLING DISTRIBUTIO 4.1 entra tions Other than Normal DEGREES OF FREEDOM. 55 CHAPTER 5 ESTIMATION OF PARAMETERS 56 51 POINT ESTIMATION 5.3 INTERVA TIM HOOD ESTIMATION F PARAMETERS OF A NORMAL POPULATION m Like ood E ea 54.3 terval nterval Es mation of th /ariance. 6 EXERCISE 64 CHAPTER 6 HYPOTHESIS TESTING 65 6.1 HYPOTHESIS TEST OF A POPULATION MEAN 6 69 sided 6.13 othesis Test Can Be One 71 6 TWO POPULATION MEANS 17 621 73 62 sand Equal Var nces 65 d Unequal Variances 75 Depender 75 63S 76 626 ofTwo Population Means 70 63 HYPOTHESIS TEST OF A POPULATION PROPORTION. 81 HYPOTHESIS TEST OF THE DIFFERENCE BETWEEN PROPORTIONS EROM TWO POPIL ATIONS 82 6.5 CHI-SQUARE TEST OF THE DIFFERENCE BETWEEN OBSERVED AND EXpecTed FREOUENCIES 84 6.5.1 SAS Example for Testing the Difference between Observed and Expected 85 6.6 HYPOTHESIS TEST OF DIFFERENCES AMONG PROPORTIONS FROM SEVERAL POPULATIONS .86 6.6.1 SAS Example for Testing Differences among Proportions from Several Populations 88 6.7 HYPOTHESIS TEST OF POPULATION VARIANCE. 90 6.8 HYPOTHESIS TEST OF THE DIFFERENCE OF TWO POPULATION VARIANCES. 6.9 HYPOTHESIS TESTS USING CONFIDENCE INTERVALS 91
vi Biostatistics for Animal Science 3.3.6 F Distribution.50 EXERCISES .51 CHAPTER 4 POPULATION AND SAMPLE .53 4.1 FUNCTIONS OF RANDOM VARIABLES AND SAMPLING DISTRIBUTIONS .53 4.1.1 Central Limit Theorem.54 4.1.2 Statistics with Distributions Other than Normal .54 4.2 DEGREES OF FREEDOM.55 CHAPTER 5 ESTIMATION OF PARAMETERS.56 5.1 POINT ESTIMATION .56 5.2 MAXIMUM LIKELIHOOD ESTIMATION .57 5.3 INTERVAL ESTIMATION .58 5.4 ESTIMATION OF PARAMETERS OF A NORMAL POPULATION.60 5.4.1 Maximum Likelihood Estimation .60 5.4.2 Interval Estimation of the Mean.61 5.4.3 Interval Estimation of the Variance.62 EXERCISES .64 CHAPTER 6 HYPOTHESIS TESTING.65 6.1 HYPOTHESIS TEST OF A POPULATION MEAN.66 6.1.1 P value.69 6.1.2 A Hypothesis Test Can Be One- or Two-sided.70 6.1.3 Hypothesis Test of a Population Mean for a Small Sample.71 6.2 HYPOTHESIS TEST OF THE DIFFERENCE BETWEEN TWO POPULATION MEANS.72 6.2.1 Large Samples.72 6.2.2 Small Samples and Equal Variances.74 6.2.3 Small Samples and Unequal Variances.75 6.2.4 Dependent Samples.75 6.2.5 Nonparametric Test.76 6.2.6 SAS Examples for Hypotheses Tests of Two Population Means.79 6.3 HYPOTHESIS TEST OF A POPULATION PROPORTION.81 6.4 HYPOTHESIS TEST OF THE DIFFERENCE BETWEEN PROPORTIONS FROM TWO POPULATIONS.82 6.5 CHI-SQUARE TEST OF THE DIFFERENCE BETWEEN OBSERVED AND EXPECTED FREQUENCIES .84 6.5.1 SAS Example for Testing the Difference between Observed and Expected Frequencies .85 6.6 HYPOTHESIS TEST OF DIFFERENCES AMONG PROPORTIONS FROM SEVERAL POPULATIONS .86 6.6.1 SAS Example for Testing Differences among Proportions from Several Populations.88 6.7 HYPOTHESIS TEST OF POPULATION VARIANCE.90 6.8 HYPOTHESIS TEST OF THE DIFFERENCE OF TWO POPULATION VARIANCES.90 6.9 HYPOTHESIS TESTS USING CONFIDENCE INTERVALS.91
Contents vii 6.10 STATISTICAL AND PRACTICAL SIGNIFICANCE. 92 6.11 TYPES OF ERRORS IN INFERENCES AND POWER OF TEST 2 6.11.1 SAS Examples for the Power of Test. 6.12 SAMPLE SIZE 103 6.12.1 SAS Examples for Sample Size. 104 EXERCISES. 107 CHAPTER 7 SIMPLE LINEAR REGRESSION 109 7.1 THE SIMPLE REGRESSION MODEL 109 7.2 ESTIMATION OF THE REGRESSION PARAMETERS-LEAST SQUARES ESTIMATION. 113 73 MAXIMUM LIKELIHOOD ESTIMATION .116 7.4 RESIDUALS AND THEIR PROPERTIES. .117 EXPECTATIONS AND VARIANCES OF THE PARAMETER ESTIMATORS. .119 7.6 STUDENT TTEST IN TESTING HYPOTHESES ABOUT THE PARAMETERS. 120 CONFIDENCE INTERVALS OF THE PARAMETERS. 121 MEAN AND PREDICTION CONFIDENCE INTERVALS OF THE RESPONSE VARIABLE .122 7.9 PARTITIONING TOTAL VARIABILITY. 124 7.9.1 Relationships among Sums of Squares. 126 7.9.2 Theoretical Distribution of Sum of Squares. 127 7.10 TEST OF HYPOTHESES-FTEST. 12 7.11 LIKELIHOOD RATIO TEST. 130 7.12 COEFFICIENT OF DETERMINATION. 132 7.12.1 Shortcut Calculation of Sums of Squares and the Coefficient of Determination. 133 7.13 MATRIX APPROACH TO SIMPLE LINEAR REGRESSION. 134 7.13.1 The Simple Regression Model. 134 7.132 Estimation of Parameters. 7.13.3 Maximum Likelihood Estimation 7.14 SAS EXAMPLE FOR SIMPLE LINEAR REGRESSION 139 7.15 PW上ROF IESIS 7.15.1 SAS Examples for Calculating the Power of Test. 142 EXERCISES. 144 CHAPTER 8 CORRELATION. 146 81 ESTIMATION OF THE COEFFICIENT OF CORRELATION AND TESTS OF HYPOIHES 147 8.2 CAL RELATIONSHIP BETWEEN THE SAMPLE COEFFICIENT OF CORRELATION AND THE COEFFICIENT OF DETERMINATION. 149 8.2.1 SAS for Correlation. 151 83.1 SAS Example for Rank Correlation EXERCISES. 153
Contents vii 6.10 STATISTICAL AND PRACTICAL SIGNIFICANCE.92 6.11 TYPES OF ERRORS IN INFERENCES AND POWER OF TEST .92 6.11.1 SAS Examples for the Power of Test.99 6.12 SAMPLE SIZE .103 6.12.1 SAS Examples for Sample Size .104 EXERCISES .107 CHAPTER 7 SIMPLE LINEAR REGRESSION .109 7.1 THE SIMPLE REGRESSION MODEL.109 7.2 ESTIMATION OF THE REGRESSION PARAMETERS – LEAST SQUARES ESTIMATION .113 7.3 MAXIMUM LIKELIHOOD ESTIMATION .116 7.4 RESIDUALS AND THEIR PROPERTIES.117 7.5 EXPECTATIONS AND VARIANCES OF THE PARAMETER ESTIMATORS.119 7.6 STUDENT T TEST IN TESTING HYPOTHESES ABOUT THE PARAMETERS.120 7.7 CONFIDENCE INTERVALS OF THE PARAMETERS .121 7.8 MEAN AND PREDICTION CONFIDENCE INTERVALS OF THE RESPONSE VARIABLE .122 7.9 PARTITIONING TOTAL VARIABILITY.124 7.9.1 Relationships among Sums of Squares .126 7.9.2 Theoretical Distribution of Sum of Squares.127 7.10 TEST OF HYPOTHESES - F TEST .128 7.11 LIKELIHOOD RATIO TEST.130 7.12 COEFFICIENT OF DETERMINATION .132 7.12.1 Shortcut Calculation of Sums of Squares and the Coefficient of Determination.133 7.13 MATRIX APPROACH TO SIMPLE LINEAR REGRESSION .134 7.13.1 The Simple Regression Model .134 7.13.2 Estimation of Parameters .135 7.13.3 Maximum Likelihood Estimation .138 7.14 SAS EXAMPLE FOR SIMPLE LINEAR REGRESSION .139 7.15 POWER OF TESTS.140 7.15.1 SAS Examples for Calculating the Power of Test .142 EXERCISES .144 CHAPTER 8 CORRELATION.146 8.1 ESTIMATION OF THE COEFFICIENT OF CORRELATION AND TESTS OF HYPOTHESES .147 8.2 NUMERICAL RELATIONSHIP BETWEEN THE SAMPLE COEFFICIENT OF CORRELATION AND THE COEFFICIENT OF DETERMINATION.149 8.2.1 SAS Example for Correlation .150 8.3 RANK CORRELATION .151 8.3.1 SAS Example for Rank Correlation .152 EXERCISES .153
viii Biostatistics for Animal Science CHAPTER 9 MULTIPLE LINEAR REGRESSION 154 9.1 TWO INDEPENDENT VARIABLES 15 9 913 dent r test 160 9 LIK SAS EXAMPLE FOR MULTI LE RE RE 68 9 POWER OF MULTIPL ESSION 5.1 96 Power PROBLEMS WIH REGRESSION 061 nalys 173 9.6.2 963 Mu 17 9.6.4 Detectin g Problems with Regression 17 97 CHOOSING THE BEST MODEL 181 71 SAS E xample for r Model Selection 183 CHAPTER 10 CURVILINEAR REGRESSION 185 10.1 POLYNOMIAL REGRESSION 1011 SAS Ex ale for ou adratic Regression 102 NonLInEAR REGRESSION 100 1021 SASEx ple for Nonlinear Regression 192 10.3 SEGMENTED REGRESSION 194 10.3.1 SAS Examples for Segme nted Regression. 10Q 10311 gress with Two Simple Regressions. 198 10312 le for S ented re on with platea 200 CHAPTER 11 ONE-WAY ANALYSIS OF VARIANCE 204 Il THE FIXED EFFECTS ONE-WAY MODEL 206 11 11 Partitioning Total variability 20R 11.1.2 Hypothesis Test-FTest )10 1112 214 11.1.4 Maximum Likelihood Estimation 21d 11.1.5 Likelihood Ratio Test. 215 11 1 6 Multinle Comparisons among groun Means 217 11161 Least Significance Difference (LSD). 217 11.1.6.2 Tukey Test. 218 11.1.63 Con asts 220 11.1.64 nal contrasts 221 11165 Scheffe Test. 22 11.1.7 Test of Homogeneity of Variance. 225 11.1.8 SAS Example for the Fixed Effects One-way Model. 226 11.1.9 Power of the Fixed Effects One-way Model. 228 11.1.9.1 SAS Example for Calculating Power. 230 11.2 THE RANDOM EFFECTS ONE-WAY MODEL 231 11.2.1 Hypothesis Test. 233
viii Biostatistics for Animal Science CHAPTER 9 MULTIPLE LINEAR REGRESSION.154 9.1 TWO INDEPENDENT VARIABLES .155 9.1.1 Estimation of Parameters .156 9.1.2 Student t test in Testing Hypotheses .159 9.1.3 Partitioning Total Variability and Tests of Hypotheses .160 9.2 PARTIAL AND SEQUENTIAL SUMS OF SQUARES .162 9.3 TESTING MODEL FIT USING A LIKELIHOOD RATIO TEST.166 9.4 SAS EXAMPLE FOR MULTIPLE REGRESSION.168 9.5 POWER OF MULTIPLE REGRESSION .170 9.5.1 SAS Example for Calculating Power .171 9.6 PROBLEMS WITH REGRESSION.172 9.6.1 Analysis of Residuals.173 9.6.2 Extreme Observations .174 9.6.3 Multicollinearity.177 9.6.4 SAS Example for Detecting Problems with Regression .178 9.7 CHOOSING THE BEST MODEL .181 9.7.1 SAS Example for Model Selection .183 CHAPTER 10 CURVILINEAR REGRESSION .185 10.1 POLYNOMIAL REGRESSION.185 10.1.1 SAS Example for Quadratic Regression .189 10.2 NONLINEAR REGRESSION.190 10.2.1 SAS Example for Nonlinear Regression.192 10.3 SEGMENTED REGRESSION.194 10.3.1 SAS Examples for Segmented Regression.198 10.3.1.1 SAS Example for Segmented Regression with Two Simple Regressions.198 10.3.1.2 SAS Example for Segmented Regression with Plateau .200 CHAPTER 11 ONE-WAY ANALYSIS OF VARIANCE .204 11.1 THE FIXED EFFECTS ONE-WAY MODEL .206 11.1.1 Partitioning Total Variability .208 11.1.2 Hypothesis Test - F Test .210 11.1.3 Estimation of Group Means .214 11.1.4 Maximum Likelihood Estimation .214 11.1.5 Likelihood Ratio Test.215 11.1.6 Multiple Comparisons among Group Means .217 11.1.6.1 Least Significance Difference (LSD).217 11.1.6.2 Tukey Test.218 11.1.6.3 Contrasts .220 11.1.6.4 Orthogonal contrasts .221 11.1.6.5 Scheffe Test.223 11.1.7 Test of Homogeneity of Variance .225 11.1.8 SAS Example for the Fixed Effects One-way Model .226 11.1.9 Power of the Fixed Effects One-way Model.228 11.1.9.1 SAS Example for Calculating Power .230 11.2 THE RANDOM EFFECTS ONE-WAY MODEL .231 11.2.1 Hypothesis Test.233