Using Multivariate Statistics

Transcription

1 SEVENTH EDITION Using Multivariate Statistics Barbara G. Tabachnick California State University, Northridge Linda S. Fidell California State University, Northridge 330 Hudson Street, NY NY A01_TABA0541_07_ALC_FM.indd 1

2 Portfolio Manager: Tanimaa Mehra Content Producer: Kani Kapoor Portfolio Manager Assistant: Anna Austin Product Marketer: Jessica Quazza Art/Designer: Integra Software Services Pvt. Ltd. Fu ll-service Project Manager: Integra Software Services Pvt. Ltd. Compositor: Integra Software Services Pvt. Ltd. Printer/Binder: LSC Communications, Inc. Cover Printer: Phoenix Color/Hagerstown Cover Design: Lumina Datamatics, Inc. Cover Art: Shutterstock Acknowledgments of third party content appear on pages within the text, which constitutes an extension of this copyright page. Copyright 2019, 2013, 2007 by Pearson Education, Inc. or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit PEARSON and ALWAYS LEARNING are exclusive trademarks owned by Pearson Education, Inc. or its affiliates, in the U.S., and/or other countries. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees or distributors. Many of the designations by manufacturers and seller to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed in initial caps or all caps. Library of Congress Cataloging-in-Publication Data Names: Tabachnick, Barbara G., author. Fidell, Linda S., author. Title: Using multivariate statistics/barbara G. Tabachnick, California State University, Northridge, Linda S. Fidell, California State University, Northridge. Description: Seventh edition. Boston: Pearson, [2019] Chapter 14, by Jodie B. Ullman. Identifiers: LCCN ISBN ISBN Subjects: LCSH: Multivariate analysis. Statistics. Classification: LCC QA278.T DDC 519.5/35 dc23 LC record available at Books a la Carte ISBN-10: ISBN-13: A01_TABA0541_07_ALC_FM.indd 2

3 Contents Preface xiv 1 Introduction Multivariate Statistics: Why? The Domain of Multivariate Statistics: Numbers of IVs and DVs Experimental and Nonexperimental Research Computers and Multivariate Statistics Garbage In, Roses Out? Some Useful Definitions Continuous, Discrete, and Dichotomous Data Samples and Populations Descriptive and Inferential Statistics Orthogonality: Standard and Sequential Analyses Linear Combinations of Variables Number and Nature of Variables to Include Statistical Power Data Appropriate for Multivariate Statistics The Data Matrix The Correlation Matrix The Variance Covariance Matrix The Sum-of-Squares and Cross-Products Matrix Residuals Organization of the Book 14 2 A Guide to Statistical Techniques: Using the Book Research Questions and Associated Techniques Degree of Relationship Among Variables Bivariate r Multiple R Sequential R Canonical R Multiway Frequency Analysis Multilevel Modeling Significance of Group Differences One-Way ANOVA and t Test One-Way ANCOVA Factorial ANOVA Factorial ANCOVA Hotelling s T One-Way MANOVA One-Way MANCOVA Factorial MANOVA Factorial MANCOVA Profile Analysis of Repeated Measures Prediction of Group Membership One-Way Discriminant Analysis Sequential One-Way Discriminant Analysis Multiway Frequency Analysis (Logit) Logistic Regression Sequential Logistic Regression Factorial Discriminant Analysis Sequential Factorial Discriminant Analysis Structure Principal Components Factor Analysis Structural Equation Modeling Time Course of Events Survival/Failure Analysis Time-Series Analysis Some Further Comparisons A Decision Tree Technique Chapters Preliminary Check of the Data 28 3 Review of Univariate and Bivariate Statistics Hypothesis Testing One-Sample z Test as Prototype Power Extensions of the Model Controversy Surrounding Significance Testing Analysis of Variance One-Way Between-Subjects ANOVA Factorial Between-Subjects ANOVA Within-Subjects ANOVA Mixed Between-Within-Subjects ANOVA Design Complexity Nesting Latin-Square Designs Unequal n and Nonorthogonality Fixed and Random Effects Specific Comparisons Weighting Coefficients for Comparisons Orthogonality of Weighting Coefficients Obtained F for Comparisons Critical F for Planned Comparisons Critical F for Post Hoc Comparisons Parameter Estimation Effect Size 47 iii A01_TABA0541_07_ALC_FM.indd 3

4 iv Contents 3.5 Bivariate Statistics: Correlation and Regression Correlation Regression Chi-Square Analysis 50 4 Cleaning Up Your Act: Screening Data Prior to Analysis Important Issues in Data Screening Accuracy of Data File Honest Correlations Inflated Correlation Deflated Correlation Missing Data Deleting Cases or Variables Estimating Missing Data Using a Missing Data Correlation Matrix Treating Missing Data as Data Repeating Analyses with and without Missing Data Choosing Among Methods for Dealing with Missing Data Outliers Detecting Univariate and Multivariate Outliers Describing Outliers Reducing the Influence of Outliers Outliers in a Solution Normality, Linearity, and Homoscedasticity Normality Linearity Homoscedasticity, Homogeneity of Variance, and Homogeneity of Variance Covariance Matrices Common Data Transformations Multicollinearity and Singularity A Checklist and Some Practical Recommendations Complete Examples of Data Screening Screening Ungrouped Data Accuracy of Input, Missing Data, Distributions, and Univariate Outliers Linearity and Homoscedasticity Transformation Detecting Multivariate Outliers Variables Causing Cases to Be Outliers Multicollinearity Screening Grouped Data Accuracy of Input, Missing Data, Distributions, Homogeneity of Variance, and Univariate Outliers Linearity Multivariate Outliers Variables Causing Cases to Be Outliers Multicollinearity 97 5 Multiple Regression General Purpose and Description Kinds of Research Questions Degree of Relationship Importance of IVs Adding IVs Changing IVs Contingencies Among IVs Comparing Sets of IVs Predicting DV Scores for Members of a New Sample Parameter Estimates Limitations to Regression Analyses Theoretical Issues Practical Issues Ratio of Cases to IVs Absence of Outliers Among the IVs and on the DV Absence of Multicollinearity and Singularity Normality, Linearity, and Homoscedasticity of Residuals Independence of Errors Absence of Outliers in the Solution Fundamental Equations for Multiple Regression General Linear Equations Matrix Equations Computer Analyses of Small-Sample Example Major Types of Multiple Regression Standard Multiple Regression Sequential Multiple Regression Statistical (Stepwise) Regression Choosing Among Regression Strategies Some Important Issues Importance of IVs Standard Multiple Regression Sequential or Statistical Regression Commonality Analysis Relative Importance Analysis Statistical Inference Test for Multiple R Test of Regression Components Test of Added Subset of IVs Confidence Limits Comparing Two Sets of Predictors Adjustment of R Suppressor Variables Regression Approach to ANOVA Centering When Interactions and Powers of IVs Are Included Mediation in Causal Sequence 137 A01_TABA0541_07_ALC_FM.indd 4

5 Contents v 5.7 Complete Examples of Regression Analysis Evaluation of Assumptions Ratio of Cases to IVs Normality, Linearity, Homoscedasticity, and Independence of Residuals Outliers Multicollinearity and Singularity Standard Multiple Regression Sequential Regression Example of Standard Multiple Regression with Missing Values Multiply Imputed Comparison of Programs IBM SPSS Package SAS System SYSTAT System Analysis of Covariance General Purpose and Description Kinds of Research Questions Main Effects of IVs Interactions Among IVs Specific Comparisons and Trend Analysis Effects of Covariates Effect Size Parameter Estimates Limitations to Analysis of Covariance Theoretical Issues Practical Issues Unequal Sample Sizes, Missing Data, and Ratio of Cases to IVs Absence of Outliers Absence of Multicollinearity and Singularity Normality of Sampling Distributions Homogeneity of Variance Linearity Homogeneity of Regression Reliability of Covariates Fundamental Equations for Analysis of Covariance Sums of Squares and Cross-Products Significance Test and Effect Size Computer Analyses of Small-Sample Example Some Important Issues Choosing Covariates Evaluation of Covariates Test for Homogeneity of Regression Design Complexity Within-Subjects and Mixed Within-Between Designs Unequal Sample Sizes Specific Comparisons and Trend Analysis Effect Size Alternatives to ANCOVA Complete Example of Analysis of Covariance Evaluation of Assumptions Unequal n and Missing Data Normality Linearity Outliers Multicollinearity and Singularity Homogeneity of Variance Homogeneity of Regression Reliability of Covariates Analysis of Covariance Main Analysis Evaluation of Covariates Homogeneity of Regression Run Comparison of Programs IBM SPSS Package SAS System SYSTAT System Multivariate Analysis of Variance and Covariance General Purpose and Description Kinds of Research Questions Main Effects of IVs Interactions Among IVs Importance of DVs Parameter Estimates Specific Comparisons and Trend Analysis Effect Size Effects of Covariates Repeated-Measures Analysis of Variance Limitations to Multivariate Analysis of Variance and Covariance Theoretical Issues Practical Issues Unequal Sample Sizes, Missing Data, and Power Multivariate Normality Absence of Outliers Homogeneity of Variance Covariance Matrices Linearity Homogeneity of Regression Reliability of Covariates Absence of Multicollinearity and Singularity Fundamental Equations for Multivariate Analysis of Variance and Covariance Multivariate Analysis of Variance 212 A01_TABA0541_07_ALC_FM.indd 5

6 vi Contents Computer Analyses of Small-Sample Example Multivariate Analysis of Covariance Some Important Issues MANOVA Versus ANOVAs Criteria for Statistical Inference Assessing DVs Univariate F Roy Bargmann Stepdown Analysis Using Discriminant Analysis Choosing Among Strategies for Assessing DVs Specific Comparisons and Trend Analysis Design Complexity Within-Subjects and Between- Within Designs Unequal Sample Sizes Complete Examples of Multivariate Analysis of Variance and Covariance Evaluation of Assumptions Unequal Sample Sizes and Missing Data Multivariate Normality Linearity Outliers Homogeneity of Variance Covariance Matrices Homogeneity of Regression Reliability of Covariates Multicollinearity and Singularity Multivariate Analysis of Variance Multivariate Analysis of Covariance Assessing Covariates Assessing DVs Comparison of Programs IBM SPSS Package SAS System SYSTAT System Profile Analysis: The Multivariate Approach to Repeated Measures General Purpose and Description Kinds of Research Questions Parallelism of Profiles Overall Difference Among Groups Flatness of Profiles Contrasts Following Profile Analysis Parameter Estimates Effect Size Limitations to Profile Analysis Theoretical Issues Practical Issues Sample Size, Missing Data, and Power Multivariate Normality Absence of Outliers Homogeneity of Variance Covariance Matrices Linearity Absence of Multicollinearity and Singularity Fundamental Equations for Profile Analysis Differences in Levels Parallelism Flatness Computer Analyses of Small-Sample Example Some Important Issues Univariate Versus Multivariate Approach to Repeated Measures Contrasts in Profile Analysis Parallelism and Flatness Significant, Levels Not Significant (Simple-Effects Analysis) Parallelism and Levels Significant, Flatness Not Significant (Simple-Effects Analysis) Parallelism, Levels, and Flatness Significant (Interaction Contrasts) Only Parallelism Significant Doubly Multivariate Designs Classifying Profiles Imputation of Missing Values Complete Examples of Profile Analysis Profile Analysis of Subscales of the WISC Evaluation of Assumptions Profile Analysis Doubly Multivariate Analysis of Reaction Time Evaluation of Assumptions Doubly Multivariate Analysis of Slope and Intercept Comparison of Programs IBM SPSS Package SAS System SYSTAT System Discriminant Analysis General Purpose and Description Kinds of Research Questions Significance of Prediction Number of Significant Discriminant Functions Dimensions of Discrimination Classification Functions Adequacy of Classification Effect Size Importance of Predictor Variables Significance of Prediction with Covariates Estimation of Group Means 304 A01_TABA0541_07_ALC_FM.indd 6

7 Contents vii 9.3 Limitations to Discriminant Analysis Theoretical Issues Practical Issues Unequal Sample Sizes, Missing Data, and Power Multivariate Normality Absence of Outliers Homogeneity of Variance Covariance Matrices Linearity Absence of Multicollinearity and Singularity Fundamental Equations for Discriminant Analysis Derivation and Test of Discriminant Functions Classification Computer Analyses of Small-Sample Example Types of Discriminant Analyses Direct Discriminant Analysis Sequential Discriminant Analysis Stepwise (Statistical) Discriminant Analysis Some Important Issues Statistical Inference Criteria for Overall Statistical Significance Stepping Methods Number of Discriminant Functions Interpreting Discriminant Functions Discriminant Function Plots Structure Matrix of Loadings Evaluating Predictor Variables Effect Size Design Complexity: Factorial Designs Use of Classification Procedures Cross-Validation and New Cases Jackknifed Classification Evaluating Improvement in Classification Complete Example of Discriminant Analysis Evaluation of Assumptions Unequal Sample Sizes and Missing Data Multivariate Normality Linearity Outliers Homogeneity of Variance Covariance Matrices Multicollinearity and Singularity Direct Discriminant Analysis Comparison of Programs IBM SPSS Package SAS System SYSTAT System Logistic Regression General Purpose and Description Kinds of Research Questions Prediction of Group Membership or Outcome Importance of Predictors Interactions Among Predictors Parameter Estimates Classification of Cases Significance of Prediction with Covariates Effect Size Limitations to Logistic Regression Analysis Theoretical Issues Practical Issues Ratio of Cases to Variables Adequacy of Expected Frequencies and Power Linearity in the Logit Absence of Multicollinearity Absence of Outliers in the Solution Independence of Errors Fundamental Equations for Logistic Regression Testing and Interpreting Coefficients Goodness of Fit Comparing Models Interpretation and Analysis of Residuals Computer Analyses of Small-Sample Example Types of Logistic Regression Direct Logistic Regression Sequential Logistic Regression Statistical (Stepwise) Logistic Regression Probit and Other Analyses Some Important Issues Statistical Inference Assessing Goodness of Fit of Models Tests of Individual Predictors Effect Sizes Effect Size for a Model Effect Sizes for Predictors Interpretation of Coefficients Using Odds Coding Outcome and Predictor Categories Number and Type of Outcome Categories Classification of Cases Hierarchical and Nonhierarchical Analysis 372 A01_TABA0541_07_ALC_FM.indd 7

8 viii Contents Importance of Predictors Logistic Regression for Matched Groups Complete Examples of Logistic Regression Evaluation of Limitations Ratio of Cases to Variables and Missing Data Multicollinearity Outliers in the Solution Direct Logistic Regression with Two-Category Outcome and Continuous Predictors Limitation: Linearity in the Logit Direct Logistic Regression with Two-Category Outcome Sequential Logistic Regression with Three Categories of Outcome Limitations of Multinomial Logistic Regression Sequential Multinomial Logistic Regression Comparison of Programs IBM SPSS Package SAS System SYSTAT System Survival/Failure Analysis General Purpose and Description Kinds of Research Questions Proportions Surviving at Various Times Group Differences in Survival Survival Time with Covariates Treatment Effects Importance of Covariates Parameter Estimates Contingencies Among Covariates Effect Size and Power Limitations to Survival Analysis Theoretical Issues Practical Issues Sample Size and Missing Data Normality of Sampling Distributions, Linearity, and Homoscedasticity Absence of Outliers Differences Between Withdrawn and Remaining Cases Change in Survival Conditions over Time Proportionality of Hazards Absence of Multicollinearity Fundamental Equations for Survival Analysis Life Tables Standard Error of Cumulative Proportion Surviving Hazard and Density Functions Plot of Life Tables Test for Group Differences Computer Analyses of Small-Sample Example Types of Survival Analyses Actuarial and Product-Limit Life Tables and Survivor Functions Prediction of Group Survival Times from Covariates Direct, Sequential, and Statistical Analysis Cox Proportional-Hazards Model Accelerated Failure-Time Models Choosing a Method Some Important Issues Proportionality of Hazards Censored Data Right-Censored Data Other Forms of Censoring Effect Size and Power Statistical Criteria Test Statistics for Group Differences in Survival Functions Test Statistics for Prediction from Covariates Predicting Survival Rate Regression Coefficients (Parameter Estimates) Hazard Ratios Expected Survival Rates Complete Example of Survival Analysis Evaluation of Assumptions Accuracy of Input, Adequacy of Sample Size, Missing Data, and Distributions Outliers Differences Between Withdrawn and Remaining Cases Change in Survival Experience over Time Proportionality of Hazards Multicollinearity Cox Regression Survival Analysis Effect of Drug Treatment Evaluation of Other Covariates Comparison of Programs SAS System IBM SPSS Package SYSTAT System Canonical Correlation General Purpose and Description Kinds of Research Questions 448 A01_TABA0541_07_ALC_FM.indd 8

9 Contents ix Number of Canonical Variate Pairs Interpretation of Canonical Variates Importance of Canonical Variates and Predictors Canonical Variate Scores Limitations Theoretical Limitations Practical Issues Ratio of Cases to IVs Normality, Linearity, and Homoscedasticity Missing Data Absence of Outliers Absence of Multicollinearity and Singularity Fundamental Equations for Canonical Correlation Eigenvalues and Eigenvectors Matrix Equations Proportions of Variance Extracted Computer Analyses of Small-Sample Example Some Important Issues Importance of Canonical Variates Interpretation of Canonical Variates Complete Example of Canonical Correlation Evaluation of Assumptions Missing Data Normality, Linearity, and Homoscedasticity Outliers Multicollinearity and Singularity Canonical Correlation Comparison of Programs SAS System IBM SPSS Package SYSTAT System Principal Components and Factor Analysis General Purpose and Description Kinds of Research Questions Number of Factors Nature of Factors Importance of Solutions and Factors Testing Theory in FA Estimating Scores on Factors Limitations Theoretical Issues Practical Issues Sample Size and Missing Data Normality Linearity Absence of Outliers Among Cases Absence of Multicollinearity and Singularity Factorability of R Absence of Outliers Among Variables Fundamental Equations for Factor Analysis Extraction Orthogonal Rotation Communalities, Variance, and Covariance Factor Scores Oblique Rotation Computer Analyses of Small-Sample Example Major Types of Factor Analyses Factor Extraction Techniques PCA Versus FA Principal Components Principal Factors Image Factor Extraction Maximum Likelihood Factor Extraction Unweighted Least Squares Factoring Generalized (Weighted) Least Squares Factoring Alpha Factoring Rotation Orthogonal Rotation Oblique Rotation Geometric Interpretation Some Practical Recommendations Some Important Issues Estimates of Communalities Adequacy of Extraction and Number of Factors Adequacy of Rotation and Simple Structure Importance and Internal Consistency of Factors Interpretation of Factors Factor Scores Comparisons Among Solutions and Groups Complete Example of FA Evaluation of Limitations Sample Size and Missing Data Normality Linearity Outliers Multicollinearity and Singularity Factorability of R Outliers Among Variables Principal Factors Extraction with Varimax Rotation 515 A01_TABA0541_07_ALC_FM.indd 9

10 x Contents 13.8 Comparison of Programs IBM SPSS Package SAS System SYSTAT System Structural Equation Modeling by Jodie B. Ullman General Purpose and Description Kinds of Research Questions Adequacy of the Model Testing Theory Amount of Variance in the Variables Accounted for by the Factors Reliability of the Indicators Parameter Estimates Intervening Variables Group Differences Longitudinal Differences Multilevel Modeling Latent Class Analysis Limitations to Structural Equation Modeling Theoretical Issues Practical Issues Sample Size and Missing Data Multivariate Normality and Outliers Linearity Absence of Multicollinearity and Singularity Residuals Fundamental Equations for Structural Equations Modeling Covariance Algebra Model Hypotheses Model Specification Model Estimation Model Evaluation Computer Analysis of Small-Sample Example Some Important Issues Model Identification Estimation Techniques Estimation Methods and Sample Size Estimation Methods and Nonnormality Estimation Methods and Dependence Some Recommendations for Choice of Estimation Method Assessing the Fit of the Model Comparative Fit Indices Absolute Fit Index Indices of Proportion of Variance Accounted Degree of Parsimony Fit Indices Residual-Based Fit Indices Choosing Among Fit Indices Model Modification Chi-Square Difference Test Lagrange Multiplier (LM) Test Wald Test Some Caveats and Hints on Model Modification Reliability and Proportion of Variance Discrete and Ordinal Data Multiple Group Models Mean and Covariance Structure Models Complete Examples of Structural Equation Modeling Analysis Confirmatory Factor Analysis of the WISC Model Specification for CFA Evaluation of Assumptions for CFA CFA Model Estimation and Preliminary Evaluation Model Modification SEM of Health Data SEM Model Specification Evaluation of Assumptions for SEM SEM Model Estimation and Preliminary Evaluation Model Modification Comparison of Programs EQS LISREL AMOS SAS System Multilevel Linear Modeling General Purpose and Description Kinds of Research Questions Group Differences in Means Group Differences in Slopes Cross-Level Interactions Meta-Analysis Relative Strength of Predictors at Various Levels Individual and Group Structure Effect Size Path Analysis at Individual and Group Levels Analysis of Longitudinal Data Multilevel Logistic Regression Multiple Response Analysis 618 A01_TABA0541_07_ALC_FM.indd 10

11 Contents xi 15.3 Limitations to Multilevel Linear Modeling Theoretical Issues Practical Issues Sample Size, Unequal-n, and Missing Data Independence of Errors Absence of Multicollinearity and Singularity Fundamental Equations Intercepts-Only Model The Intercepts-Only Model: Level-1 Equation The Intercepts-Only Model: Level-2 Equation Computer Analyses of Intercepts-Only Model Model with a First-Level Predictor Level-1 Equation for a Model with a Level-1 Predictor Level-2 Equations for a Model with a Level-1 Predictor Computer Analysis of a Model with a Level-1 Predictor Model with Predictors at First and Second Levels Level-1 Equation for Model with Predictors at Both Levels Level-2 Equations for Model with Predictors at Both Levels Computer Analyses of Model with Predictors at First and Second Levels Types of MLM Repeated Measures Higher-Order MLM Latent Variables Nonnormal Outcome Variables Multiple Response Models Some Important Issues Intraclass Correlation Centering Predictors and Changes in Their Interpretations Interactions Random and Fixed Intercepts and Slopes Statistical Inference Assessing Models Tests of Individual Effects Effect Size Estimation Techniques and Convergence Problems Exploratory Model Building Complete Example of MLM Evaluation of Assumptions Sample Sizes, Missing Data, and Distributions Outliers Multicollinearity and Singularity Independence of Errors: Intraclass Correlations Multilevel Modeling Comparison of Programs SAS System IBM SPSS Package HLM Program MLwiN Program SYSTAT System Multiway Frequency Analysis General Purpose and Description Kinds of Research Questions Associations Among Variables Effect on a Dependent Variable Parameter Estimates Importance of Effects Effect Size Specific Comparisons and Trend Analysis Limitations to Multiway Frequency Analysis Theoretical Issues Practical Issues Independence Ratio of Cases to Variables Adequacy of Expected Frequencies Absence of Outliers in the Solution Fundamental Equations for Multiway Frequency Analysis Screening for Effects Total Effect First-Order Effects Second-Order Effects Third-Order Effect Modeling Evaluation and Interpretation Residuals Parameter Estimates Computer Analyses of Small-Sample Example Some Important Issues Hierarchical and Nonhierarchical Models Statistical Criteria Tests of Models Tests of Individual Effects Strategies for Choosing a Model IBM SPSS HILOGLINEAR (Hierarchical) 697 A01_TABA0541_07_ALC_FM.indd 11

12 xii Contents IBM SPSS GENLOG (General Log-Linear) SAS CATMOD and IBM SPSS LOGLINEAR (General Log-Linear) Complete Example of Multiway Frequency Analysis Evaluation of Assumptions: Adequacy of Expected Frequencies Hierarchical Log-Linear Analysis Preliminary Model Screening Stepwise Model Selection Adequacy of Fit Interpretation of the Selected Model Comparison of Programs IBM SPSS Package SAS System SYSTAT System Time-Series Analysis General Purpose and Description Kinds of Research Questions Pattern of Autocorrelation Seasonal Cycles and Trends Forecasting Effect of an Intervention Comparing Time Series Time Series with Covariates Effect Size and Power Assumptions of Time-Series Analysis Theoretical Issues Practical Issues Normality of Distributions of Residuals Homogeneity of Variance and Zero Mean of Residuals Independence of Residuals Absence of Outliers Sample Size and Missing Data Fundamental Equations for Time-Series ARIMA Models Identification of ARIMA (p, d, q) Models Trend Components, d: Making the Process Stationary Auto-Regressive Components Moving Average Components Mixed Models ACFs and PACFs Estimating Model Parameters Diagnosing a Model Computer Analysis of Small-Sample Time-Series Example Types of Time-Series Analyses Models with Seasonal Components Models with Interventions Abrupt, Permanent Effects Abrupt, Temporary Effects Gradual, Permanent Effects Models with Multiple Interventions Adding Continuous Variables Some Important Issues Patterns of ACFs and PACFs Effect Size Forecasting Statistical Methods for Comparing Two Models Complete Examples of Time-Series Analysis Time-Series Analysis of Introduction of Seat Belt Law Evaluation of Assumptions Baseline Model Identification and Estimation Baseline Model Diagnosis Intervention Analysis Time-Series Analysis of Introduction of a Dashboard to an Educational Computer Game Evaluation of Assumptions Baseline Model Identification and Diagnosis Intervention Analysis Comparison of Programs IBM SPSS Package SAS System SYSTAT System An Overview of the General Linear Model Linearity and the General Linear Model Bivariate to Multivariate Statistics and Overview of Techniques Bivariate Form Simple Multivariate Form Full Multivariate Form Alternative Research Strategies 782 Appendix A A Skimpy Introduction to Matrix Algebra 783 A.1 The Trace of a Matrix 784 A.2 Addition or Subtraction of a Constant to a Matrix 784 A.3 Multiplication or Division of a Matrix by a Constant 784 A.4 Addition and Subtraction of Two Matrices 785 A.5 Multiplication, Transposes, and Square Roots of Matrice 785 A01_TABA0541_07_ALC_FM.indd 12

13 Contents xiii A.6 Matrix Division (Inverses and Determinants) 786 A.7 Eigenvalues and Eigenvectors: Procedures for Consolidating Variance from a Matrix 788 Appendix B Research Designs for Complete Examples 791 B.1 Women s Health and Drug Study 791 B.2 Sexual Attraction Study 793 B.3 Learning Disabilities Data Bank 794 B.4 Reaction Time to Identify Figures 794 B.5 Field Studies of Noise-Induced Sleep Disturbance 795 B.6 Clinical Trial for Primary Biliary Cirrhosis 795 B.7 Impact of Seat Belt Law 795 B.8 The Selene Online Educational Game 796 Appendix C Statistical Tables 797 C.1 Normal Curve Areas 798 C.2 Critical Values of the t Distribution for a =.05 and.01, Two-Tailed Test 799 C.3 Critical Values of the F Distribution 800 C.4 Critical Values of Chi Square (x 2 ) 804 C.5 Critical Values for Squares Multiple Correlation (R 2 ) in Forward Stepwise Selection: a = C.6 Critical Values for F MAX (S 2 MAX/S 2 MIN) Distribution for a =.05 and References 808 Index 815 A01_TABA0541_07_ALC_FM.indd 13

14 Preface Some good things seem to go on forever: friendship and updating this book. It is difficult to believe that the first edition manuscript was typewritten, with real cutting and pasting. The publisher required a paper manuscript with numbered pages that was almost our downfall. We could write a book on multivariate statistics, but we couldn t get the same number of pages (about 1200, double-spaced) twice in a row. SPSS was in release 9.0, and the other program we demonstrated was BMDP. There were a mere 11 chapters, of which 6 of them were describing techniques. Multilevel and structural equation modeling were not yet ready for prime time. Logistic regression and survival analysis were not yet popular. Material new to this edition includes a redo of all SAS examples, with a pretty new output format and replacement of interactive analyses that are no longer available. We ve also re-run the IBM SPSS examples to show the new output format. We ve tried to update the references in all chapters, including only classic citations if they date prior to New work on relative importance has been incorporated in multiple regression, canonical correlation, and logistic regression analysis complete with demonstrations. Multiple imputation procedures for dealing with missing data have been updated, and we ve added a new time-series example, taking advantage of an IBM SPSS expert modeler that replaces previous tea-leaf reading aspects of the analysis. Our goals in writing the book remain the same as in all previous editions to present complex statistical procedures in a way that is maximally useful and accessible to researchers who are not necessarily statisticians. We strive to be short on theory but long on conceptual understanding. The statistical packages have become increasingly easy to use, making it all the more critical to make sure that they are applied with a good understanding of what they can and cannot do. But above all else what does it all mean? We have not changed the basic format underlying all of the technique chapters, now 14 of them. We start with an overview of the technique, followed by the types of research questions the techniques are designed to answer. We then provide the cautionary tale what you need to worry about and how to deal with those worries. Then come the fundamental equations underlying the technique, which some readers truly enjoy working through (we know because they helpfully point out any errors and/or inconsistencies they find); but other readers discover they can skim (or skip) the section without any loss to their ability to conduct meaningful analysis of their research. The fundamental equations are in the context of a small, made-up, usually silly data set for which computer analyses are provided usually IBM SPSS and SAS. Next, we delve into issues surrounding the technique (such as different types of the analysis, follow-up procedures to the main analysis, and effect size, if it is not amply covered elsewhere). Finally, we provide one or two full-bore analyses of an actual real-life data set together with a Results section appropriate for a journal. Data sets for these examples are available at in IBM SPSS, SAS, and ASCII formats. We end each technique chapter with a comparison of features available in IBM SPSS, SAS, SYSTAT and sometimes other specialized programs. SYSTAT is a statistical package that we reluctantly had to drop a few editions ago for lack of space. We apologize in advance for the heft of the book; it is not our intention to line the coffers of chiropractors, physical therapists, acupuncturists, and the like, but there s really just so much to say. As to our friendship, it s still going strong despite living in different cities. Art has taken the place of creating belly dance costumes for both of us, but we remain silly in outlook, although serious in our analysis of research. The lineup of people to thank grows with each edition, far too extensive to list: students, reviewers, editors, and readers who send us corrections and point out areas of confusion. As always, we take full responsibility for remaining errors and lack of clarity. xiv Barbara G. Tabachnick Linda S. Fidell A01_TABA0541_07_ALC_FM.indd 14