Using Multivariate Statistics

Transcription

1 / K FIFTH EDITION 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Using Multivariate Statistics Barbara G. Tabachnick California State University, Northridge Linda S. Fidell California State University, Northridge Boston New York San Francisco Mexico City Montreal Toronto London Madrid Munich Paris Hong Kong Singapore Tokyo Cape Town Sydney

2 Preface xxvii _L 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 Residuais Organization of the Book 16 A Guide to Statistical Techniques: Using the Book Research Questions and Associated Techniques Degree of Relationship among Variables L.l.l Bivariate r Multiple R Sequential/? Canonical/J L.1.5 Multiway Frequency Analy sis Multilevel Modeling iii

3 2.1.2 Significance of Group Differences 19 * 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 Sequential One-Way Discriminant 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 32 f ' Review of Univariate and Bivariate Statistics Hypothesis Testing One-Sample z Test as Prototype Power Extensions of the Model Controversy Surrounding Significance Testing Analysis ofvariance One-Way Between-Subjects ANOVA Factorial Between-Subjects ANOVA Within-Subjects ANOVA Mixed Between-Within-Subjects ANOVA 46

4 3.2.5 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 EffectSize Bivariate Statistics: Correlation and Regression Correlation Regression Chi-Square Analysis 58 Cleaning Up Your Act: Screening Data Prior to Analysis Important Issues in Data Screening Accuracy of Data File Honest Correlations Inflated Correlation Deflated Correlation MissingData 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 85

5 i 0 VI CONTENTS 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 Ouüiers 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 114 Multiple Regression General Purpose and Description Kinds of Research Questions Degree of Relationship ImportanceofIVs AddingIVs 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, Homoscedasticity of Residuais 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 134

6 VÜ 5.5 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 Statistical Inference Test for Multiple R Test of Regression Components Test of Added Subset of IVs Confidence Limits around B and Multiple R Comparing Two Sets of Predictors Adjustmentof/? Suppressor Variables Regression Approach to ANOVA Centering when Interactions and Powers of IVs Are Included Mediation in Causal Sequences Complete Examples of Regression Analysis Evaluation of Assumptions Ratio of Cases to IVs Normality, Linearity, Homoscedasticity, and Independence of Residuais Outliers Multicollinearity and Singularity Standard Multiple Regression Sequential Regression Example of Standard Multiple Regression with Missing Values Multiply Imputed Comparison of Programs SPSSPackage SAS System SYSTAT System 194 Analysis of Covariance General Purpose and Description Kinds of Research Questions Main Effectsof IVs Interactions among IVs Specific Comparisons and Trend Analysis Effectsof Covariates 199

7 Vlll CONTENTS Effect Size Parameter Estimates Limitations to Analysis of Covariance Theoretical Issues Practical Issues Unequal Sample Sizes, Missing Data, and Ratio of Cases toivs Absence of Outliers Absence of Multicollinearity and Singularity Normality of Sampling Distributions Homogeneity of Variance Linearity Homogeneity of Regression ReliabilityofCovariates 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 SPSSPackage 240

8 i CONTENTS IX SAS System SYSTAT System 240 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 Computer Analyses of Small-Sample Example Multivariate Analysis of Covariance Some Important Issues MANOVA vs. ANOVAs Criteria for Statistical Inference AssessingDVs 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 276

9 7.6 Complete Examples of Multivariate Analysis of Variance and Covariance Evaluation of Assumptions Unequal Sample Sizes and Missing Data Multivariate Normality Linearity Outliers Homogeneity ofvariance-covariancematrices Homogeneity of Regression Reliability of Covariates Multicollinearity and Singularity Multivariate Analysis of Variance Multivariate Analysis of Covariance Assessing Covariates Assessing DVs Comparison of Programs SPSSPackage SAS System SYSTAT System 310 O Profile Analysis: The Multivariate Approach to Repeated Measures General Purpose and Description Kinds of Research Questions Parallelismof 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 ofvariance-covariancematrices Linearity Absence of Multicollinearity and Singularity Fundamental Equations for Profile Analysis Differences in Levels Parallelism 318

10 xi Flatness Computer Analyses of Small-Sample Example Some Important Issues Univariate vs. 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 SPSSPackage SAS System SYSTAT System 374 Discriminant Analysis General Purpose and Description Kinds of Research Questions Significance ofprediction 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 Limitations to Discriminant Analysis Theoretical Issues 381

11 Xll CONTENTS Practical Issues Unequal Sample Sizes, Missing Data, and Power Multivariate Normality Absence of Outliers Homogeneity ofvariance-covariancematrices Linearity Absence of Multicollinearity and Singularity Fundamental Equations for Discriminant Analysis Derivation and Test of Discriminant Functions Classification Computer Analyses of Small-Sample Example Typesof Discriminant Function 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 ofloadings 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 ofvariance-covariancematrices Multicollinearity and Singularity Direct Discriminant Analysis Comparison of Programs SPSSPackage SAS System SYSTAT System 436

12 i. CONTENTS XÜi AU Logistic Regression General Purpose and Description Kinds of Research Questions Prediction of Group Membership or Outcome ImportanceofPredictors 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 Typesof 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 oflndividual Variables Effect Size for a Model Interpretation of Coefficients Using Odds Coding Outcome and Predictor Categories Number and Type of Outcome Categories Classification of Cases Hierarchical and Nonhierarchical Analysis 468

13 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 Comparisons of Programs SPSSPackage SAS System SYSTAT System 504 1J. 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 510

14 XV 11.4 Fundamental Equations for Survival Analysis LifeTables Standard Error of Cumulative Proportion Surviving Hazard and Density Functions PlotofLifeTables 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 CensoredData Right-Censored Data OtherFormsofCensoring 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) OddsRatios 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 ofdrugtreatment Evaluation of Other Covariates Comparison of Programs SAS System SPSSPackage SYSTAT System 566

15 XVi CONTENTS Canonical Correlation General Purpose and Description Kinds of Research Questions Number of Canonical Variate Pairs Interpretation of Canonical Variates Importance of Canonical Variates 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 SPSSPackage SYSTAT System 606 JL«5 Principal Components and Factor Analysis General Purpose and Description Kinds of Research Questions Number offactors 610

16 XVÜ Natureof 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 FactorabilityofR 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 PCAvs. 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 Geometrie 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 651

17 XV111 CONTENTS 13.7 Complete Example offa Evaluation of Limitations Sample Size and Missing Data Normality Linearity Outliers Multicollinearity and Singularity Outliers among Variables Principal Factors Extraction with Varimax Rotation Comparison of Programs SPSSPackage SAS System SYSTAT System 675 A4 Structural Equation Modeling 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 Limitations to Structural Equation Modeling Theoretical Issues Practical Issues Sample Size and Missing Data Multivariate Normality and Absence of 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 696

18 X 14.5 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 780 A5 Multilevel Linear Modeling General Purpose and Description 781

19 XX CONTENTS 15.2 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 Path Analysis at Individual and Group Levels Analysis of Longitudinal Data Multilevel Logistic Regression Multiple Response Analysis Limitations to Multilevel Linear Modeling Theoretical Issues Practical Issues Sample Size, Unequal-n, and Missing Data IndependenceofErrors 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 Analysis 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 TypesofMLM Repeated Measures Higher-OrderMLM 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 826

20 XXI 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 SPSSPackage HLM Program MLwiN Program SYSTAT System 857 Au 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 871

21 XXII CONTENTS 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 SPSS HILOGLINEAR (Hierarchical) SPSS GENLOG (General Log-Linear) SAS CATMOD and SPSS LOGLINEAR (General Log-Linear) Complete Example of Multiway Frequency Analysis Evaluation of As sumptions: 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 SPSSPackage 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 Füll Multivariate Form Alternative Research Strategies 918 Time-Series Analysis (available online at General Purpose and Description 18-1

22 XX 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 ZeroMeanof Residuals Independence of Residuals Absence of Outliers Fundamental Equations for Time-Series ARIMA Models Identification ARIMA (p, d, q) Models Trend Components, d: Making the Process Stationary Auto-Regressive Components Moving Average Components Mixed Models ACFsandPACFs Estimating Model Parameters Diagnosing a Model Computer Analysis of Small-Sample Time-Series Example Typesof 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 Patternsof ACFsandPACFs Effect Size Forecasting Statistical Methods for Comparing Two Models Complete Example of a Time-Series Analysis Evaluation of Assumptions Normality of Sampling Distributions Homogeneity of Variance Outliers 18-48

23 XXiv CONTENTS Baseline Model Identification and Estimation Baseline Model Diagnosis Intervention Analysis Model Diagnosis Model Interpretation Comparison of Programs SPSSPackage SAS System SYSTAT System Appendix A A Skimpy Introduction to Matrix Algebra 924 A.l The Trace of a Matrix 925 A.2 Addition or Subtraction of a Constant to a Matrix 925 A.3 Multiplication or Division of a Matrix by a Constant 925 A.4 Addition and Subtraction oftwo Matrices 926 A.5 Multiplication, Transposes, and Square Roots of Matrices 927 A.6 Matrix "Division" (Inverses and Determinante) 929 A.7 Eigenvalues and Eigenvectors: Procedures for Consolidating Variance from a Matrix 930 Appendix 15 Research Designs for Complete Examples 934 B.l Women's Health and Drug Study 934 B.2 Sexual Attraction Study 935 B.3 Learning Disabilities Data Bank 938 B.4 Reaction Time to Identify Figures 939 B.5 Field Studies of Noise-Induced Sleep Disturbance 939 B.6 Clinical Trial for Primary Biliary Cirrhosis 940 B.7 ImpactofSeatBeltLaw 940 Appendix v^ Statistical Tables 941 C.l Normal Curve Areas 942 C.2 Critical Values of the t Distribution for a =.05 and.01, Two-Tailed Test 943

24 XXV C.3 Critical Values of the F Distribution 944 C.4 Critical Values of Chi Square (/ 2 ) 949 C.5 Critical Values for Squared Multiple Correlation (R 2 ) in Forward Stepwise Selection 950 C.6 Critical Values for F MAX (S^AX /S^IN ) Distribution for a =.05 and References 953 Index 963