# The Analysis of Variance: Fixed, Random and Mixed Models

## Hardeo Sahai; Mohammed I. Ageel

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The analysis of variance (ANOYA) models have become one of the most widely used tools of modern statistics for analyzing multifactor data. The ANOYA models provide versatile statistical tools for studying the relationship between a dependent variable and one or more independent variables. The ANOYA mod- els are employed to determine whether different variables interact and which factors or factor combinations are most important. They are appealing because they provide a conceptually simple technique for investigating statistical rela- tionships among different independent variables known as factors. Currently there are several texts and monographs available on the sub- ject. However, some of them such as those of Scheffe (1959) and Fisher and McDonald (1978), are written for mathematically advanced readers, requiring a good background in calculus, matrix algebra, and statistical theory; whereas others such as Guenther (1964), Huitson (1971), and Dunn and Clark (1987), although they assume only a background in elementary algebra and statistics, treat the subject somewhat scantily and provide only a superficial discussion of the random and mixed effects analysis of variance.

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Recensione:

"Intended as a text for an introductory analysis of variance service course, it requires little more than a precalculus understanding of probability, estimation, and hypothesis testing…. The student and practitioner will benefit from a well-balanced mixture of statistical theory, formulas, and explanations and the great care exercised by the authors in discussing properties and analysis of fixed, random, and mixed models in parallel…. The book employs several devices to aid readability…. In summary, the text is…a valuable source for the practitioner and nonstatistics major… It is well organized and well written at a difficulty level that precisely meets the target audience’s needs."   —JASA

Contenuti:

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.

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## 1.Analysis of Variance: Fixed, Random and Mixed Models

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