Preface |
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xv | |
Authors |
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xix | |
1 Introduction |
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1 | (12) |
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1.1 Different Types of Scientific Study |
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1 | (2) |
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1.2 Relating Sample Results to More General Populations |
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3 | (1) |
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1.3 Constructing Models to Represent Reality |
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4 | (3) |
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7 | (1) |
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1.5 Estimating the Parameters of Linear Models |
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8 | (1) |
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1.6 Summarizing the Importance of Model Terms |
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9 | (2) |
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1.7 The Scope of This Book |
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11 | (2) |
2 A Review of Basic Statistics |
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13 | (30) |
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2.1 Summary Statistics and Notation for Sample Data |
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13 | (3) |
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2.2 Statistical Distributions for Populations |
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16 | (12) |
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17 | (5) |
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22 | (2) |
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2.2.3 The Normal Distribution |
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24 | (2) |
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2.2.4 Distributions Derived from Functions of Normal Random Variables |
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26 | (2) |
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2.3 From Sample Data to Conclusions about the Population |
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28 | (2) |
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2.3.1 Estimating Population Parameters Using Summary Statistics |
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28 | (1) |
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2.3.2 Asking Questions about the Data: Hypothesis Testing |
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29 | (1) |
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2.4 Simple Tests for Population Means |
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30 | (6) |
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2.4.1 Assessing the Mean Response: The One-Sample t-Test |
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30 | (2) |
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2.4.2 Comparing Mean Responses: The Two-Sample t-Test |
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32 | (4) |
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2.5 Assessing the Association between Variables |
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36 | (3) |
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2.6 Presenting Numerical Results |
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39 | (2) |
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41 | (2) |
3 Principles for Designing Experiments |
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43 | (26) |
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43 | (9) |
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46 | (2) |
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48 | (3) |
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51 | (1) |
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3.2 Forms of Experimental Structure |
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52 | (5) |
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3.3 Common Forms of Design for Experiments |
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57 | (5) |
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3.3.1 The Completely Randomized Design |
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57 | (1) |
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3.3.2 The Randomized Complete Block Design |
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58 | (1) |
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3.3.3 The Latin Square Design |
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59 | (1) |
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3.3.4 The Split-Plot Design |
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60 | (1) |
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3.3.5 The Balanced Incomplete Block Design |
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61 | (1) |
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3.3.6 Generating a Randomized Design |
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62 | (1) |
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62 | (7) |
4 Models for a Single Factor |
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69 | (24) |
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69 | (4) |
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4.2 Estimating the Model Parameters |
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73 | (1) |
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4.3 Summarizing the Importance of Model Terms |
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74 | (10) |
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4.3.1 Calculating Sums of Squares |
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76 | (4) |
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4.3.2 Calculating Degrees of Freedom and Mean Squares |
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80 | (1) |
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4.3.3 Calculating Variance Ratios as Test Statistics |
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81 | (1) |
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4.3.4 The Summary ANOVA Table |
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82 | (2) |
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4.4 Evaluating the Response to Treatments |
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84 | (4) |
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4.4.1 Prediction of Treatment Means |
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84 | (1) |
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4.4.2 Comparison of Treatment Means |
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85 | (3) |
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4.5 Alternative Forms of the Model |
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88 | (2) |
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90 | (3) |
5 Checking Model Assumptions |
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93 | (20) |
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5.1 Estimating Deviations |
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93 | (3) |
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94 | (1) |
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5.1.2 Standardized Residuals |
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95 | (1) |
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5.2 Using Graphical Tools to Diagnose Problems |
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96 | (8) |
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5.2.1 Assessing Homogeneity of Variances |
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96 | (2) |
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5.2.2 Assessing Independence |
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98 | (3) |
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5.2.3 Assessing Normality |
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101 | (1) |
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5.2.4 Using Permutation Tests Where Assumptions Fail |
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102 | (1) |
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5.2.5 The Impact of Sample Size |
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103 | (1) |
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5.3 Using Formal Tests to Diagnose Problems |
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104 | (4) |
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5.4 Identifying Inconsistent Observations |
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108 | (2) |
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110 | (3) |
6 Transformations of the Response |
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113 | (16) |
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6.1 Why Do We Need to Transform the Response? |
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113 | (1) |
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6.2 Some Useful Transformations |
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114 | (8) |
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114 | (5) |
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119 | (1) |
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120 | (1) |
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6.2.4 Other Transformations |
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121 | (1) |
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6.3 Interpreting the Results after Transformation |
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122 | (1) |
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6.4 Interpretation for Log-Transformed Responses |
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123 | (3) |
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126 | (1) |
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127 | (2) |
7 Models with a Simple Blocking Structure |
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129 | (20) |
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130 | (2) |
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7.2 Estimating the Model Parameters |
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132 | (2) |
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7.3 Summarizing the Importance of Model Terms |
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134 | (6) |
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7.4 Evaluating the Response to Treatments |
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140 | (1) |
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7.5 Incorporating Strata: The Multi-Stratum Analysis of Variance |
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141 | (5) |
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146 | (3) |
8 Extracting Information about Treatments |
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149 | (60) |
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8.1 From Scientific Questions to the Treatment Structure |
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150 | (2) |
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8.2 A Crossed Treatment Structure with Two Factors |
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152 | (12) |
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8.2.1 Models for a Crossed Treatment Structure with Two Factors |
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153 | (2) |
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8.2.2 Estimating the Model Parameters |
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155 | (3) |
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8.2.3 Assessing the Importance of Individual Model Terms |
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158 | (2) |
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8.2.4 Evaluating the Response to Treatments: Predictions from the Fitted Model |
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160 | (2) |
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8.2.5 The Advantages of Factorial Structure |
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162 | (1) |
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8.2.6 Understanding Different Parameterizations |
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163 | (1) |
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8.3 Crossed Treatment Structures with Three or More Factors |
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164 | (9) |
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8.3.1 Assessing the Importance of Individual Model Terms |
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166 | (5) |
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8.3.2 Evaluating the Response to Treatments: Predictions from the Fitted Model |
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171 | (2) |
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8.4 Models for Nested Treatment Structures |
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173 | (6) |
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8.5 Adding Controls or Standards to a Set of Treatments |
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179 | (3) |
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8.6 Investigating Specific Treatment Comparisons |
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182 | (8) |
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8.7 Modelling Patterns for Quantitative Treatments |
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190 | (5) |
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8.8 Making Treatment Comparisons from Predicted Means |
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195 | (11) |
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8.8.1 The Bonferroni Correction |
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196 | (1) |
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8.8.2 The False Discovery Rate |
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197 | (1) |
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8.8.3 All Pairwise Comparisons |
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198 | (3) |
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8.8.3.1 The LSD and Fisher's Protected LSD |
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198 | (1) |
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8.8.3.2 Multiple Range Tests |
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199 | (1) |
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8.8.3.3 Tukey's Simultaneous Confidence Intervals |
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200 | (1) |
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8.8.4 Comparison of Treatments against a Control |
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201 | (1) |
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8.8.5 Evaluation of a Set of Pre-Planned Comparisons |
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201 | (4) |
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205 | (1) |
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206 | (3) |
9 Models with More Complex Blocking Structure |
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209 | (32) |
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9.1 The Latin Square Design |
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209 | (11) |
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211 | (1) |
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9.1.2 Estimating the Model Parameters |
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211 | (1) |
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9.1.3 Assessing the Importance of Individual Model Terms |
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212 | (3) |
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9.1.4 Evaluating the Response to Treatments: Predictions from the Fitted Model |
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215 | (2) |
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9.1.5 Constraints and Extensions of the Latin Square Design |
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217 | (3) |
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9.2 The Split-Plot Design |
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220 | (12) |
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222 | (1) |
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9.2.2 Assessing the Importance of Individual Model Terms |
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223 | (2) |
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9.2.3 Evaluating the Response to Treatments: Predictions from the Fitted Model |
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225 | (3) |
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9.2.4 Drawbacks and Variations of the Split-Plot Design |
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228 | (4) |
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9.3 The Balanced Incomplete Block Design |
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232 | (6) |
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235 | (1) |
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9.3.2 Assessing the Importance of Individual Model Terms |
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236 | (1) |
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9.3.3 Drawbacks and Variations of the Balanced Incomplete Block Design |
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237 | (1) |
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238 | (3) |
10 Replication and Power |
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241 | (16) |
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10.1 Simple Methods for Determining Replication |
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242 | (3) |
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10.1.1 Calculations Based on the LSD |
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242 | (1) |
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10.1.2 Calculations Based on the Coefficient of Variation |
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243 | (1) |
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10.1.3 Unequal Replication and Models with Blocking |
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244 | (1) |
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10.2 Estimating the Background Variation |
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245 | (1) |
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10.3 Assessing the Power of a Design |
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245 | (4) |
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10.4 Constructing a Design for a Particular Experiment |
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249 | (4) |
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10.5 A Different Hypothesis: Testing for Equivalence |
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253 | (3) |
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256 | (1) |
11 Dealing with Non-Orthogonality |
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257 | (30) |
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11.1 The Benefits of Orthogonality |
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257 | (2) |
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11.2 Fitting Models with Non-Orthogonal Terms |
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259 | (13) |
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11.2.1 Parameterizing Models for Two Non-Orthogonal Factors |
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259 | (6) |
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11.2.2 Assessing the Importance of Non-Orthogonal Terms: The Sequential ANOVA Table |
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265 | (4) |
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11.2.3 Calculating the Impact of Model Terms |
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269 | (1) |
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11.2.4 Selecting the Best Model |
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270 | (1) |
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11.2.5 Evaluating the Response to Treatments: Predictions from the Fitted Model |
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270 | (2) |
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11.3 Designs with Planned Non-Orthogonality |
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272 | (2) |
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11.3.1 Fractional Factorial Designs |
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273 | (1) |
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11.3.2 Factorial Designs with Confounding |
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274 | (1) |
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11.4 The Consequences of Missing Data |
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274 | (3) |
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11.5 Incorporating the Effects of Unplanned Factors |
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277 | (3) |
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11.6 Analysis Approaches for Non-Orthogonal Designs |
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280 | (4) |
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11.6.1 A Simple Approach: The Intra-Block Analysis |
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281 | (3) |
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284 | (3) |
12 Models for a Single Variate: Simple Linear Regression |
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287 | (38) |
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288 | (4) |
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12.2 Estimating the Model Parameters |
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292 | (4) |
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12.3 Assessing the Importance of the Model |
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296 | (3) |
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12.4 Properties of the Model Parameters |
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299 | (2) |
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12.5 Using the Fitted Model to Predict Responses |
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301 | (4) |
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12.6 Summarizing the Fit of the Model |
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305 | (1) |
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12.7 Consequences of Uncertainty in the Explanatory Variate |
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306 | (2) |
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12.8 Using Replication to Test Goodness of Fit |
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308 | (5) |
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12.9 Variations on the Model |
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313 | (8) |
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12.9.1 Centering and Scaling the Explanatory Variate |
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313 | (1) |
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12.9.2 Regression through the Origin |
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314 | (6) |
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320 | (1) |
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321 | (4) |
13 Checking Model Fit |
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325 | (20) |
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13.1 Checking the Form of the Model |
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325 | (3) |
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13.2 More Ways of Estimating Deviations |
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328 | (2) |
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13.3 Using Graphical Tools to Check Assumptions |
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330 | (2) |
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13.4 Looking for Influential Observations |
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332 | (6) |
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13.4.1 Measuring Potential Influence: Leverage |
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333 | (2) |
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13.4.2 The Relationship between Residuals and Leverages |
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335 | (1) |
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13.4.3 Measuring the Actual Influence of Individual Observations |
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336 | (2) |
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13.5 Assessing the Predictive Ability of a Model: Cross-Validation |
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338 | (4) |
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342 | (3) |
14 Models for Several Variates: Multiple Linear Regression |
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345 | (36) |
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14.1 Visualizing Relationships between Variates |
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345 | (2) |
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347 | (3) |
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14.3 Estimating the Model Parameters |
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350 | (2) |
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14.4 Assessing the Importance of Individual Explanatory Variates |
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352 | (6) |
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14.4.1 Adding Terms into the Model: Sequential ANOVA and Incremental Sums of Squares |
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353 | (3) |
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14.4.2 The Impact of Removing Model Terms: Marginal Sums of Squares |
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356 | (2) |
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14.5 Properties of the Model Parameters and Predicting Responses |
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358 | (1) |
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14.6 Investigating Model Misspecification |
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359 | (2) |
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14.7 Dealing with Correlation among Explanatory Variates |
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361 | (4) |
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14.8 Summarizing the Fit of the Model |
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365 | (1) |
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14.9 Selecting the Best Model |
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366 | (12) |
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14.9.1 Strategies for Sequential Variable Selection |
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369 | (7) |
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14.9.2 Problems with Procedures for the Selection of Subsets of Variables |
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376 | (1) |
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14.9.3 Using Cross-Validation as a Tool for Model Selection |
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377 | (1) |
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14.9.4 Some Final Remarks on Procedures for Selecting Models |
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378 | (1) |
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378 | (3) |
15 Models for Variates and Factors |
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381 | (46) |
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15.1 Incorporating Groups into the Simple Linear Regression Model |
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382 | (17) |
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15.1.1 An Overview of Possible Models |
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383 | (5) |
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15.1.2 Defining and Choosing between the Models |
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388 | (8) |
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15.1.2.1 Single Line Model |
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388 | (1) |
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15.1.2.2 Parallel Lines Model |
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388 | (2) |
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15.1.2.3 Separate Lines Model |
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390 | (1) |
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15.1.2.4 Choosing between the Models: The Sequential ANOVA Table |
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391 | (5) |
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15.1.3 An Alternative Sequence of Models |
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396 | (2) |
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15.1.4 Constraining the Intercepts |
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398 | (1) |
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15.2 Incorporating Groups into the Multiple Linear Regression Model |
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399 | (7) |
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15.3 Regression in Designed Experiments |
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406 | (3) |
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15.4 Analysis of Covariance: A Special Case of Regression with Groups |
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409 | (5) |
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15.5 Complex Models with Factors and Variates |
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414 | (3) |
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15.5.1 Selecting the Predictive Model |
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414 | (3) |
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15.5.2 Evaluating the Response: Predictions from the Fitted Model |
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417 | (1) |
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15.6 The Connection between Factors and Variates |
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417 | (6) |
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15.6.1 Rewriting the Model in Matrix Notation |
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421 | (2) |
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423 | (4) |
16 Incorporating Structure: Linear Mixed Models |
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427 | (24) |
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16.1 Incorporating Structure |
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427 | (1) |
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16.2 An Introduction to Linear Mixed Models |
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428 | (2) |
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16.3 Selecting the Best Fixed Model |
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430 | (2) |
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16.4 Interpreting the Random Model |
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432 | (3) |
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16.4.1 The Connection between the Linear Mixed Model and Multi-Stratum ANOVA |
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434 | (1) |
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16.5 What about Random Effects? |
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435 | (1) |
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16.6 Predicting Responses |
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436 | (1) |
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437 | (1) |
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438 | (6) |
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16.9 Some Pitfalls and Dangers |
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444 | (1) |
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16.10 Extending the Model |
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445 | (2) |
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447 | (4) |
17 Models for Curved Relationships |
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451 | (28) |
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17.1 Fitting Curved Functions by Transformation |
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451 | (12) |
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17.1.1 Simple Transformations of an Explanatory Variate |
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451 | (5) |
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456 | (4) |
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17.1.3 Trigonometric Models for Periodic Patterns |
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460 | (3) |
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17.2 Curved Surfaces as Functions of Two or More Variates |
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463 | (9) |
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17.3 Fitting Models Including Non-Linear Parameters |
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472 | (4) |
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476 | (3) |
18 Models for Non-Normal Responses: Generalized Linear Models |
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479 | (38) |
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18.1 Introduction to Generalized Linear Models |
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480 | (1) |
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18.2 Analysis of Proportions Based on Counts: Binomial Responses |
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481 | (21) |
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18.2.1 Understanding and Defining the Model |
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483 | (4) |
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18.2.2 Assessing the Importance of the Model and Individual Terms: The Analysis of Deviance |
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487 | (7) |
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18.2.2.1 Interpreting the ANODEV with No Over-Dispersion |
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489 | (1) |
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18.2.2.2 Interpreting the ANODEV with Over-Dispersion |
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490 | (3) |
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18.2.2.3 The Sequential ANODEV Table |
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493 | (1) |
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18.2.3 Checking the Model Fit and Assumptions |
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494 | (2) |
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18.2.4 Properties of the Model Parameters |
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496 | (2) |
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18.2.5 Evaluating the Response to Explanatory Variables: Prediction |
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498 | (2) |
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18.2.6 Aggregating Binomial Responses |
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500 | (1) |
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18.2.7 The Special Case of Binary Data |
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501 | (1) |
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18.2.8 Other Issues with Binomial Responses |
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501 | (1) |
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18.3 Analysis of Count Data: Poisson Responses |
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502 | (10) |
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18.3.1 Understanding and Defining the Model |
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503 | (3) |
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18.3.2 Analysis of the Model |
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506 | (3) |
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18.3.3 Analysing Poisson Responses with Several Explanatory Variables |
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509 | (3) |
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18.3.4 Other Issues with Poisson Responses |
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512 | (1) |
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18.4 Other Types of GLM and Extensions |
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512 | (1) |
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513 | (4) |
19 Practical Design and Data Analysis for Real Studies |
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517 | (28) |
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19.1 Designing Real Studies |
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518 | (17) |
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19.1.1 Aims, Objectives and Choice of Explanatory Structure |
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518 | (1) |
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19.1.2 Resources, Experimental Units and Constraints |
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519 | (1) |
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19.1.3 Matching the Treatments to the Resources |
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520 | (1) |
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19.1.4 Designs for Series of Studies and for Studies with Multiple Phases |
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521 | (2) |
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19.1.5 Design Case Studies |
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523 | (12) |
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19.2 Choosing the Best Analysis Approach |
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535 | (3) |
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19.2.1 Analysis of Designed Experiments |
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536 | (1) |
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19.2.2 Analysis of Observational Studies |
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537 | (1) |
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19.2.3 Different Types of Data |
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538 | (1) |
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19.3 Presentation of Statistics in Reports, Theses and Papers |
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538 | (5) |
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19.3.1 Statistical Information in the Materials and Methods |
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539 | (1) |
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19.3.2 Presentation of Results |
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540 | (3) |
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543 | (2) |
References |
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545 | (6) |
Appendix A: Data Tables |
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551 | (8) |
Appendix B: Quantiles of Statistical Distributions |
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559 | (4) |
Appendix C: Statistical and Mathematical Results |
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563 | (6) |
Index |
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569 | |