The Analysis of Variance

1. Introduction.- 1.0 Preview.- 1.1 Historical Developments.- 1.2 Analysis of Variance Models.- 1.3 Concept of Fixed and Random Effects.- 1.4 Finite and Infinite Populations.- 1.5 General and Generalized Linear Models.- 1.6 Scope of the Book.- 2. One-Way Classification.- 2.0 Preview.- 2.1 Mathematical Model.- 2.2 Assumptions of the Model.- 2.3 Partition of the Total Sum of Squares.- 2.4 The Concept of Degrees of Freedom.- 2.5 Mean Squares and Their Expectations.- 2.6 Sampling Distribution of Mean Squares.- 2.7 Test of Hypothesis: The Analysis of VarianceFTest.- Model I (Fixed Effects).- Model II (Random Effects).- 2.8 Analysis of Variance Table.- 2.9 Point Estimation: Estimation of Treatment Effects and Variance Components.- 2.10 Confidence Intervals for Variance Components.- 2.11 Computational Formulae and Procedure.- 2.12 Analysis of Variance for Unequal Number of Observations.- 2.13 Worked Examples for Model I.- 2.14 Worked Examples for Model II.- 2.15 Use of Statistical Computing Packages.- 2.16 Worked Examples Using Statistical Packages.- 2.17 Power of the Analysis of VarianceFTest.- Model I (Fixed Effects).- Model II (Random Effects).- 2.18 Power and Determination of Sample Size.- Sample Size Determination Using Smallest Detectable Difference.- 2.19 Inference About the Difference Between Treatment Means:.- Multiple Comparisons 64 Linear Combination of Means, Contrast and.- Orthogonal Contrasts.- Test of Hypothesis Involving a Contrast.- The Use of Multiple Comparisons.- Tukey’s method.- Scheffé’s method.- Interpretation of Tukey’s and Scheffé’s methods.- Comparison of Tukey’s and Scheffé’s methods.- Other Multiple Comparison Methods.- Least significant difference test.- Bonferroni’s test.- Dunn-?idák’s test.- Newman-Keuls’s test.- Duncan’s multiple range test.- Dunnett’s test.- Multiple Comparisons for Unequal Sample Sizes and Variances.- Unequal sample sizes.- Unequal population variances.- 2.20 Effects of Departures from Assumptions Underlying the Analysis of Variance Model.- Departures from Normality.- Departures from Equal Variances.- Departures from Independence of Error Terms.- 2.21 Tests for Departures from Assumptions of the Model.- Tests for Normality.- Chi-square goodness-of-fit test.- Test for skewness.- Test for kurtosis.- Other Tests for Normality.- Shapiro-Wilk’s W test.- Shapiro-Francia’s test.- D’Agostino’sDtest.- Tests for Homoscedasticity.- Bartlett’s test.- Hartley’s test.- Cochran’s test.- Comments on Bartlett’s, Hartley’s and Cochran’s tests.- Other tests of homoscedasticity.- 2.22 Corrections for Departures from Assumptions of the Model..- Transformations to Correct Lack of Normality.- Logarithmic transformation.- Square-root transformation.- Arcsine transformation.- Transformations to Correct Lack of Homoscedasticity.- Logarithmic transformation.- Square-root transformation.- Reciprocal transformation.- Arcsine transformation.- Square transformation.- Power transformation.- Exercises.- 3. Two-Way Crossed Classification Without Interaction.- 3.0 Preview.- 3.1 Mathematical Model.- 3.2 Assumptions of the Model.- 3.3 Partition of the Total Sum of Squares.- 3.4 Mean Squares and Their Expectations.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 3.5 Sampling Distribution of Mean Squares.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 3.6 Tests of Hypotheses: The Analysis of VarianceFTests.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 3.7 Point Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 3.8 Interval Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 3.9 Computational Formulae and Procedure.- 3.10 Missing Observations.- 3.11 Power of the Analysis of VarianceFTests.- 3.12 Multiple Comparison Methods.- 3.13 Worked Example for Model I.- 3.14 Worked Example for Model II.- 3.15 Worked Example for Model III.- 3.16 Worked Example for Missing Value Analysis.- 3.17 Use of Statistical Computing Packages.- 3.18 Worked Examples Using Statistical Packages.- 3.19 Effects of Violations of Assumptions of the Model.- Exercises.- 4. Two-Way Crossed Classification With Interaction.- 4.0 Preview.- 4.1 Mathematical Model.- 4.2 Assumptions of the Model.- 4.3 Partition of the Total Sum of Squares.- 4.4 Mean Squares and Their Expectations.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 4.5 Sampling Distribution of Mean Squares.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 4.6 Tests of Hypotheses: The Analysis of VarianceFTests.- Model I (Fixed Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factor A effects.- Model II (Random Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factorAeffects.- Model III (Mixed Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factor A effects.- Summary of Models and Tests.- 4.7 Point Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 4.8 Interval Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 4.9 Computational Formulae and Procedure.- 4.10 Analysis of Variance with Unequal Sample Sizes Per Cell.- Fixed Effects Analysis.- Proportional frequencies.- General case of unequal frequencies.- Random Effects Analysis.- Proportional frequencies.- General case of unequal frequencies.- Mixed Effects Analysis.- 4.11 Power of the Analysis of VarianceFTests.- Model I (Fixed Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factor A effects.- Model II (Random Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factor A effects.- Model III (Mixed Effects).- Test forABinteractions.- Test for factorBeffects.- Test for factor A effects.- 4.12 Multiple Comparison Methods.- 4.13 Worked Example for Model I.- 4.14 Worked Example for Model I: Unequal Sample Sizes Per Cell.- 4.15 Worked Example for Model II.- 4.16 Worked Example for Model III.- 4.17 Use of Statistical Computing Packages.- 4.18 Worked Examples Using Statistical Packages.- 4.19 The Meaning and Interpretation of Interaction.- 4.20 Interaction With One Observation Per Cell.- 4.21 Alternate Mixed Models.- 4.22 Effects of Violations of Assumptions of the Model.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- Exercises.- 5. Three-Way and Higher-Order Crossed Classifications.- 5.0 Preview.- 5.1 Mathematical Model.- 5.2 Assumptions of the Model.- 5.3 Partition of the Total Sum of Squares.- 5.4 Mean Squares and Their Expectations.- 5.5 Tests of Hypotheses: The Analysis of VarianceFTests...- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 5.6 Point and Interval Estimation.- 5.7 Computational Formulae and Procedure.- 5.8 Power of the Analysis of VarianceFTests.- 5.9 Multiple Comparison Methods.- 5.10 Three-Way Classification with One Observation Per Cell..- 5.11 Four-Way Crossed Classification.- 5.12 Higher-Order Crossed Classifications.- 5.13 Unequal Sample Sizes in Three-and Higher-Order Classifications.- 5.14 Worked Example for Model I.- 5.15 Worked Example for Model II.- 5.16 Worked Example for Model III.- 5.17 Use of Statistical Computing Packages.- 5.18 Worked Examples Using Statistical Packages.- Exercises.- 6. Two-Way Nested (Hierarchical) Classification.- 6.0 Preview.- 6.1 Mathematical Model.- 6.2 Assumptions of the Model.- 6.3 Analysis of Variance.- 6.4 Tests of Hypotheses: The Analysis of VarianceFTests.- 6.5 Point Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 6.6 Interval Estimation.- Model I (Fixed Effects).- Model II (Random Effects).- Model III (Mixed Effects).- 6.7 Computational Formulae and Procedure.- 6.8 Power of the Analysis of VarianceFTests.- 6.9 Multiple Comparison Methods.- 6.10 Unequal Numbers in the Subclasses.- Tests of Hypotheses.- Point and Interval Estimation.- 6.11 Worked Example for Model I.- 6.12 Worked Example for Model II.- 6.13 Worked Example for Model II: Unequal Numbers in the Subclasses.- 6.14 Worked Example for Model III.- 6.15 Use of Statistical Computing Packages.- 6.16 Worked Examples Using Statistical Packages.- Exercises.- 7. Three-Way and Higher-Order Nested Classifications.- 7.0 Preview.- 7.1 Mathematical Model.- 7.2 Analysis of Variance.- 7.3 Tests of Hypotheses and Estimation.- 7.4 Unequal Numbers in the Subclasses.- 7.5 Four-Way Nested Classification.- 7.6 General q-Way Nested Classification.- 7.7 Worked Example for Model II.- 7.8 Worked Example for Model II: Unequal Numbers in the Subclasses.- 7.9 Worked Example for Model III.- 7.10 Use of Statistical Computing Packages.- 7.11 Worked Examples Using Statistical Packages.- Exercises.- 8. Partially Nested Classifications.- 8.0 Preview.- 8.1 Mathematical Model.- 8.2 Analysis of Variance.- 8.3 Computational Formulae and Procedure.- 8.4 A Four-Factor Partially Nested Classification.- 8.5 Worked Example for Model II.- 8.6 Worked Example for Model III.- 8.7 Use of Statistical Computing Packages.- 8.8 Worked Example Using Statistical Packages.- Exercises.- 9. Finite Population and Other Models.- 9.0 Preview.- 9.1 One-Way Finite Population Model.- 9.2 Two-Way Crossed Finite Population Model.- Tests of Hypotheses.- FTests.- Point Estimation.- Interval Estimation.- 9.3 Three-Way Crossed Finite Population Model.- 9.4 Four-Way Crossed Finite Population Model.- 9.5 Nested Finite Population Models.- 9.6 Unbalanced Finite Population Models.- 9.7 Worked Example for a Finite Population Model.- 9.8 Other Models.- 9.9 Use of Statistical Computing Packages.- Exercises.- 10. Some Simple Experimental Designs.- 10.0 Preview.- 10.1 Principles of Experimental Design.- Replication.- Randomization.- Control.- 10.2 Completely Randomized Design.- Model and Analysis.- Worked Example.- 10.3 Randomized Block Design.- Model and Analysis.- Both blocks and treatments fixed.- Both blocks and treatments random.- Blocks random and treatments fixed.- Blocks fixed and treatments random.- Missing Observations.- Relative Efficiency of the Design.- Replications.- Worked Example.- 10.4 Latin Square Design.- Model and Analysis.- Point and Interval Estimation.- Power of theFTest.- Multiple Comparisons.- Computational Formulae.- Missing Observations.- Tests for Interaction.- Relative Efficiency of the Design.- Replications.- Worked Example.- 10.5 Graeco-Latin Square Design.- Model and Analysis.- Worked Example.- 10.6 Split-Plot Design.- Model and Analysis.- Worked Example.- 10.7 Other Designs.- Incomplete Block Designs.- Lattice Designs.- Youden Squares.- Cross-Over Designs.- Repeated Measures Designs.- Hyper-Graeco-Latin and Hyper Squares.- Magic and Super Magic Latin Squares.- Split-Split-Plot Design.- 2PDesign and Fractional Replications.- 10.8 Use of Statistical Computing Packages.- Exercises.- 11. Analysis of Variance Using Statistical Computing Packages.- 11.0 Preview.- 11.1 Analysis of Variance Using SAS.- 11.2 Analysis of Variance Using SPSS.- 11.3 Analysis of Variance Using BMDP.- 11.4 Use of Statistical Packages for Computing Power.- 11.5 Use of Statistical Packages for Multiple.- Comparison Procedures.- 11.6 Use of Statistical Packages for Tests of Homoscedasticity.- 11.7 Use of Statistical Packages for Tests of Normality.- Appendices.- B Chi-Square Distribution.- E Noncentral Chi-Square Distribution.- I Studentized Range Distribution.- J Studentized Maximum Modulus Distribution.- K Satterthwaite Procedure and Its Application to.- Analysis of Variance.- L Components of Variance.- M Intraclass Correlation.- N Analysis of Covariance.- Q Expected Value and Variance.- R Covariance and Correlation.- S Rules for Determining the Analysis of Variance Model.- T Rules for Calculating Sums of Squares and Degrees of Freedom.- U Rules for Finding Expected Mean Squares.- V Samples and Sampling Distribution.- W Methods of Statistical Inference.- X Some Selected Latin Squares.- Y Some Selected Graeco-Latin Squares.- Z PROC MIXED Outputs for Some Selected.- Worked Examples.- Statistical Tables and Charts.- Tables.- I Cumulative Standard Normal Distribution.- II Percentage Points of the Standard Normal Distribution.- IV Critical Values of the Chi-Square Distribution.- X Critical Values of the Studentized Range Distribution.- XI Critical Values of the Dunnett’s Test.- XII Critical Values of the Duncan’s Multiple Range Test.- XIV Critical Values of the Dunn-Sidák’s Multiple Comparison Test.- XV Critical Values of the Studentized Maximum Modulus Distribution.- XVI Critical Values of the Studentized Augmented Range
Distribution.- XVIII Coefficients of Order Statistics for the Shapiro-Wilk’s W Test for Normality.- XIX Critical Values of the Shapiro-Wilk’s W Test for Normality.- XXI Critical Values of the Bartlett’s Test for Homogeneity of Variances.- XXIII Critical Values of the Cochran’s C Test for Homogeneity of Variances.- XXIV Random Numbers.- Charts.- IV Curves of Constant Power for Determination of Sample Size in a One-Way Analysis of Variance (Fixed.- Effects Model): Feldt-Mahmoud Charts.- References.- Author Index.