Applied Statistics: From Bivariate Through Multivariate Techniques

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Dedication Title Copyright Brief Contents Detailed Contents Preface Acknowledgments About the Author Chapter 1. Review of Basic Concepts 1.1 Introduction 1.2 A Simple Example of a Research Problem 1.3 Discrepancies Between Real and Ideal Research Situations 1.4 Samples and Populations 1.5 Descriptive Versus Inferential Uses of Statistics 1.6 Levels of Measurement and Types of Variables 1.7 The Normal Distribution 1.8 Research Design 1.8.1 Experimental Design 1.8.2 Quasi-Experimental Design 1.8.3 Nonexperimental Research Design 1.8.4 Between-Subjects Versus Within-Subjects or Repeated Measures 1.9 Combinations of These Design Elements 1.10 Parametric Versus Nonparametric Statistics 1.11 Additional Implicit Assumptions 1.12 Selection of an Appropriate Bivariate Analysis 1.13 Summary Comprehension Questions Chapter 2. Basic Statistics, Sampling Error, and Confidence Intervals 2.1 Introduction 2.2 Research Example: Description of a Sample of HR Scores 2.3 Sample Mean (M) 2.4 Sum of Squared Deviations (SS) and Sample Variance (s2) 2.5 Degrees of Freedom (df) for a Sample Variance 2.6 Why Is There Variance? 2.7 Sample Standard Deviation (s) 2.8 Assessment of Location of a Single X Score Relative to a Distribution of Scores 2.9 A Shift in Level of Analysis: The Distribution of Values of M Across Many Samples From the Same Population 2.10 An Index of Amount of Sampling Error: The Standard Error of the Mean (σM) 2.11 Effect of Sample Size (N) on the Magnitude of the Standard Error (σM) 2.12 Sample Estimate of the Standard Error of the Mean (SEM) 2.13 The Family of t Distributions 2.14 Confidence Intervals 2.14.1 The General Form of a CI 2.14.2 Setting Up a CI for M When σ Is Known 2.14.3 Setting Up a CI for M When the Value of σ Is Not Known 2.14.4 Reporting CIs 2.15 Summary Appendix on SPSS Comprehension Questions Chapter 3. Statistical Significance Testing 3.1 The Logic of Null Hypothesis Significance Testing (NHST) 3.2 Type I Versus Type II Error 3.3 Formal NHST Procedures: The z Test for a Null Hypothesis About One Population Mean 3.3.1 Obtaining a Random Sample From the Population of Interest 3.3.2 Formulating a Null Hypothesis (H0) for the One-Sample z Test 3.3.3 Formulating an Alternative Hypothesis (H1) 3.3.4 Choosing a Nominal Alpha Level 3.3.5 Determining the Range of z Scores Used to Reject H0 3.3.6 Determining the Range of Values of M Used to Reject H0 3.3.7 Reporting an “Exact” p Value 3.4 Common Research Practices Inconsistent With Assumptions and Rules for NHST 3.4.1 Use of Convenience Samples 3.4.2 Modification of Decision Rules After the Initial Decision 3.4.3 Conducting Large Numbers of Significance Tests 3.4.4 Impact of Violations of Assumptions on Risk of Type I Error 3.5 Strategies to Limit Risk of Type I Error 3.5.1 Use of Random and Representative Samples 3.5.2 Adherence to the Rules for NHST 3.5.3 Limit the Number of Significance Tests 3.5.4 Bonferroni-Corrected Per-Comparison Alpha Levels 3.5.5 Replication of Outcome in New Samples 3.5.6 Cross-Validation 3.6 Interpretation of Results 3.6.1 Interpretation of Null Results 3.6.2 Interpretation of Statistically Significant Results 3.7 When Is a t Test Used Instead of a z Test? 3.8 Effect Size 3.8.1 Evaluation of “Practical” (vs. Statistical) Significance 3.8.2 Formal Effect-Size Index: Cohen’s d 3.9 Statistical Power Analysis 3.10 Numerical Results for a One-Sample t Test Obtained From SPSS 3.11 Guidelines for Reporting Results 3.12 Summary 3.12.1 Logical Problems With NHST 3.12.2 Other Applications of the t Ratio 3.12.3 What Does It Mean to Say “p < .05”? Comprehension Questions Chapter 4. Preliminary Data Screening 4.1 Introduction: Problems in Real Data 4.2 Quality Control During Data Collection 4.3 Example of an SPSS Data Worksheet 4.4 Identification of Errors and Inconsistencies 4.5 Missing Values 4.6 Empirical Example of Data Screening for Individual Variables 4.6.1 Frequency Distribution Tables 4.6.2 Removal of Impossible or Extreme Scores 4.6.3 Bar Chart for a Categorical Variable 4.6.4 Histogram for a Quantitative Variable 4.7 Identification and Handling of Outliers 4.8 Screening Data for Bivariate Analyses 4.8.1 Bivariate Data Screening for Two Categorical Variables 4.8.2 Bivariate Data Screening for One Categorical and One Quantitative Variable 4.8.3 Bivariate Data Screening for Two Quantitative Variables 4.9 Nonlinear Relations 4.10 Data Transformations 4.11 Verifying That Remedies Had the Desired Effects 4.12 Multivariate Data Screening 4.13 Reporting Preliminary Data Screening 4.14 Summary and Checklist for Data Screening 4.15 Final Notes Comprehension Questions Chapter 5. Comparing Group Means Using the Independent Samples t Test 5.1 Research Situations Where the Independent Samples t Test Is Used 5.2 A Hypothetical Research Example 5.3 Assumptions About the Distribution of Scores on the Quantitative Dependent Variable 5.3.1 Quantitative, Approximately Normally Distributed 5.3.2 Equal Variances of Scores Across Groups (the Homogeneity of Variance Assumption) 5.3.3 Independent Observations Both Between and Within Groups 5.3.4 Robustness to Violations of Assumptions 5.4 Preliminary Data Screening 5.5 Issues in Designing a Study 5.6 Formulas for the Independent Samples t Test 5.6.1 The Pooled Variances t Test 5.6.2 Computation of the Separate Variances t Test and Its Adjusted df 5.6.3 Evaluation of Statistical Significance of a t Ratio 5.6.4 Confidence Interval Around M1 – M2 5.7 Conceptual Basis: Factors That Affect the Size of the t Ratio 5.7.1 Design Decisions That Affect the Difference Between Group Means, M1 – M2 5.7.2 Design Decisions That Affect Pooled Within-Group Variance, s2p 5.7.3 Design Decisions About Sample Sizes, n1 and n2 5.7.4 Summary: Factors That Influence the Size of t 5.8 Effect-Size Indexes for t 5.8.1 Eta Squared (η2) 5.8.2 Cohen’s d 5.8.3 Point Biserial r (rpb) 5.9 Statistical Power and Decisions About Sample Size for the Independent Samples t Test 5.10 Describing the Nature of the Outcome 5.11 SPSS Output and Model Results Section 5.12 Summary Comprehension Questions Chapter 6. One-Way Between-Subjects Analysis of Variance 6.1 Research Situations Where One-Way Between-Subjects Analysis of Variance (ANOVA) Is Used 6.2 Hypothetical Research Example 6.3 Assumptions About Scores on the Dependent Variable for One-Way Between-S ANOVA 6.4 Issues in Planning a Study 6.5 Data Screening 6.6 Partition of Scores Into Components 6.7 Computations for the One-Way Between-S ANOVA 6.7.1 Comparison Between the Independent Samples t Test and One-Way Between-S ANOVA 6.7.2 Summarizing Information About Distances Between Group Means: Computing MSbetween 6.7.3 Summarizing Information About Variability of Scores Within Groups: Computing MSwithin 6.7.4 The F Ratio: Comparing MSbetween With MSwithin 6.7.5 Patterns of Scores Related to the Magnitudes of MSbetween and MSwithin 6.7.6 Expected Value of F When H0 Is True 6.7.7 Confidence Intervals (CIs) for Group Means 6.8 Effect-Size Index for One-Way Between-S ANOVA 6.9 Statistical Power Analysis for One-Way Between-S ANOVA 6.10 Nature of Differences Among Group Means 6.10.1 Planned Contrasts 6.10.2 Post Hoc or “Protected” Tests 6.11 SPSS Output and Model Results 6.12 Summary Comprehension Questions Chapter 7. Bivariate Pearson Correlation 7.1 Research Situations Where Pearson’s r Is Used 7.2 Hypothetical Research Example 7.3 Assumptions for Pearson’s r 7.4 Preliminary Data Screening 7.5 Design Issues in Planning Correlation Research 7.6 Computation of Pearson’s r 7.7 Statistical Significance Tests for Pearson’s r 7.7.1 Testing the Hypothesis That ρXY = 0 7.7.2 Testing Other Hypotheses About ρXY 7.7.3 Assessing Differences Between Correlations 7.7.4 Reporting Many Correlations: Need to Control Inflated Risk of Type I Error 7.7.4.1 Limiting the Number of Correlations 7.7.4.2 Cross-Validation of Correlations 7.7.4.3 Bonferroni Procedure: A More Conservative Alpha Level for Tests of Individual Correlations 7.8 Setting Up CIs for Correlations 7.9 Factors That Influence the Magnitude and Sign of Pearson’s r 7.9.1 Pattern of Data Points in the X, Y Scatter Plot 7.9.2 Biased Sample Selection: Restricted Range or Extreme Groups 7.9.3 Correlations for Samples That Combine Groups 7.9.4 Control of Extraneous Variables 7.9.5 Disproportionate Influence by Bivariate Outliers 7.9.6 Shapes of Distributions of X and Y 7.9.7 Curvilinear Relations 7.9.8 Transformations of Data 7.9.9 Attenuation of Correlation Due to Unreliability of Measurement 7.9.10 Part-Whole Correlations 7.9.11 Aggregated Data 7.10 Pearson’s r and r2 as Effect-Size Indexes 7.11 Statistical Power and Sample Size for Correlation Studies 7.12 Interpretation of Outcomes for Pearson’s r 7.12.1 “Correlation Does Not Necessarily Imply Causation” (So What Does It Imply?) 7.12.2 Interpretation of Significant Pearson’s r Values 7.12.3 Interpretation of a Nonsignificant Pearson’s r Value 7.13 SPSS Output and Model Results Write-Up 7.14 Summary Comprehension Questions Chapter 8. Alternative Correlation Coefficients 8.1 Correlations for Different Types of Variables 8.2 Two Research Examples 8.3 Correlations for Rank or Ordinal Scores 8.4 Correlations for True Dichotomies 8.4.1 Point Biserial r (rpb) 8.4.2 Phi Coefficient (Φ) 8.5 Correlations for Artificially Dichotomized Variables 8.5.1 Biserial r (rb) 8.5.2 Tetrachoric r (rtet) 8.6 Assumptions and Data Screening for Dichotomous Variables 8.7 Analysis of Data: Dog Ownership and Survival After a Heart Attack 8.8 Chi-Square Test of Association (Computational Methods for Tables of Any Size) 8.9 Other Measures of Association for Contingency Tables 8.10 SPSS Output and Model Results Write-Up 8.11 Summary Comprehension Questions Chapter 9. Bivariate Regression 9.1 Research Situations Where Bivariate Regression Is Used 9.2 A Research Example: Prediction of Salary From Years of Job Experience 9.3 Assumptions and Data Screening 9.4 Issues in Planning a Bivariate Regression Study 9.5 Formulas for Bivariate Regression 9.6 Statistical Significance Tests for Bivariate Regression 9.7 Setting Up Confidence Intervals Around Regression Coefficients 9.8 Factors That Influence the Magnitude and Sign of b 9.8.1 Factors That Affect the Size of the b Coefficient 9.8.2 Comparison of Coefficients for Different Predictors or for Different Groups 9.9 Effect Size/Partition of Variance in Bivariate Regression 9.10 Statistical Power 9.11 Raw Score Versus Standard Score Versions of the Regression Equation 9.12 Removing the Influence of X From the Y Variable by Looking at Residuals From Bivariate Regression 9.13 Empirical Example Using SPSS 9.13.1 Information to Report From a Bivariate Regression 9.14 Summary Comprehension Questions Chapter 10. Adding a Third Variable: Preliminary Exploratory Analyses 10.1 Three-Variable Research Situations 10.2 First Research Example 10.3 Exploratory Statistical Analyses for Three-Variable Research Situations 10.4 Separate Analysis of the X1, Y Relationship for Each Level of the Control Variable X2 10.5 Partial Correlation Between X1 and Y, Controlling for X2 10.6 Understanding Partial Correlation as the Use of Bivariate Regression to Remove Variance Predictable by X2 From Both X1 and Y 10.7 Computation of Partial r From Bivariate Pearson Correlations 10.8 Intuitive Approach to Understanding Partial r 10.9 Significance Tests, Confidence Intervals, and Statistical Power for Partial Correlations 10.9.1 Statistical Significance of Partial r 10.9.2 Confidence Intervals for Partial r 10.9.3 Effect Size, Statistical Power, and Sample Size Guidelines for Partial r 10.10 Interpretation of Various Outcomes for rY1.2 and rY1 10.11 Two-Variable Causal Models 10.12 Three-Variable Models: Some Possible Patterns of Association Among X1, Y, and X2 10.12.1 X1 and Y Are Not Related Whether You Control for X2 or Not 10.12.2 X2 Is Irrelevant to the X1, Y Relationship 10.12.3 When You Control for X2, the X1, Y Correlation Drops to 0 or Close to 0 10.12.3.1 Completely Spurious Correlation 10.12.3.2 Completely Mediated Association Between X1 and Y 10.12.4 When You Control for X, the Correlation Between X2 and Y1 Becomes Smaller (but Does Not Drop to 0 and Does Not Change Sign) 10.12.4.1 X2 Partly Accounts for the X1, Y Association, or X1 and X2 Are Correlated Predictors of Y 10.12.4.2 X2 Partly Mediates the X1, Y Relationship 10.12.5 Suppression: When You Control for X2, the X1, Y Correlation Becomes Larger Than r1Y or Becomes Opposite in Sign Relative to r1Y 10.12.5.1 Suppression of Error Variance in a Predictor Variable 10.12.5.2 Sign of X1 as a Predictor of Y Reverses When Controlling for X2 10.12.5.3 Predictor Variables With Opposite Signs 10.12.6 “None of the Above” 10.13 Mediation Versus Moderation 10.13.1 Preliminary Analysis to Identify Possible Moderation 10.13.2 Preliminary Analysis to Detect Possible Mediation 10.13.3 Experimental Tests for Mediation Models 10.14 Model Results 10.15 Summary Comprehension Questions Chapter 11. Multiple Regression With Two Predictor Variables 11.1 Research Situations Involving Regression With Two Predictor Variables 11.2 Hypothetical Research Example 11.3 Graphic Representation of Regression Plane 11.4 Semipartial (or “Part”) Correlation 11.5 Graphic Representation of Partition of Variance in Regression With Two Predictors 11.6 Assumptions for Regression With Two Predictors 11.7 Formulas for Regression Coefficients, Significance Tests, and Confidence Intervals 11.7.1 Formulas for Standard Score Beta Coefficients 11.7.2 Formulas for Raw Score (b) Coefficients 11.7.3 Formula for Multiple R and Multiple R2 11.7.4 Test of Significance for Overall Regression: Overall F Test for H0: R = 0 11.7.5 Test of Significance for Each Individual Predictor: t Test for H0: bi = 0 11.7.6 Confidence Interval for Each b Slope Coefficient 11.8 SPSS Regression Results 11.9 Conceptual Basis: Factors That Affect the Magnitude and Sign of β and b Coefficients in Multiple Regression With Two Predictors 11.10 Tracing Rules for Causal Model Path Diagrams 11.11 Comparison of Equations for β, b, pr, and sr 11.12 Nature of Predictive Relationships 11.13 Effect-Size Information in Regression With Two Predictors 11.13.1 Effect Size for Overall Model 11.13.2 Effect Size for Individual Predictor Variables 11.14 Statistical Power 11.15 Issues in Planning a Study 11.15.1 Sample Size 11.15.2 Selection of Predictor Variables 11.15.3 Multicollinearity Among Predictors 11.15.4 Range of Scores 11.16 Results 11.17 Summary Comprehension Questions Chapter 12. Dummy Predictor Variables in Multiple Regression 12.1 Research Situations Where Dummy Predictor Variables Can Be Used 12.2 Empirical Example 12.3 Screening for Violations of Assumptions 12.4 Issues in Planning a Study 12.5 Parameter Estimates and Significance Tests for Regressions With Dummy Variables 12.6 Group Mean Comparisons Using One-Way Between-S ANOVA 12.6.1 Gender Differences in Mean Salary 12.6.2 College Differences in Mean Salary 12.7 Three Methods of Coding for Dummy Variables 12.7.1 Regression With Dummy-Coded Dummy Predictor Variables 12.7.1.1 Two-Group Example With a Dummy-Coded Dummy Variable 12.7.1.2 Multiple-Group Example With Dummy-Coded Dummy Variables 12.7.2 Regression With Effect-Coded Dummy Predictor Variables 12.7.2.1 Two-Group Example With an Effect-Coded Dummy Variable 12.7.2.2 Multiple-Group Example With Effect-Coded Dummy Variables 12.7.3 Orthogonal Coding of Dummy Predictor Variables 12.8 Regression Models That Include Both Dummy and Quantitative Predictor Variables 12.9 Effect Size and Statistical Power 12.10 Nature of the Relationship and/or Follow-Up Tests 12.11 Results 12.12 Summary Comprehension Questions Chapter 13. Factorial Analysis of Variance 13.1 Research Situations and Research Questions 13.1.1 First Null Hypothesis: Test of Main Effect for Factor A 13.1.2 Second Null Hypothesis: Test of Main Effect for Factor B 13.1.3 Third Null Hypothesis: Test of the A × B Interaction 13.2 Screening for Violations of Assumptions 13.3 Issues in Planning a Study 13.4 Empirical Example: Description of Hypothetical Data 13.5 Computations for Between-S Factorial ANOVA 13.5.1 Notation for Sample Statistics That Estimate Score Components in Factorial ANOVA 13.5.2 Notation for Theoretical Effect Terms (or Unknown Population Parameters) in Factorial ANOVA 13.5.3 Formulas for Sums of Squares and Degrees of Freedom 13.6 Conceptual Basis: Factors That Affect the Size of Sums of Squares and F Ratios in Factorial ANOVA 13.6.1 Distances Between Group Means (Magnitude of the α and β Effects) 13.6.2 Number of Scores (n) Within Each Group or Cell 13.6.3 Variability of Scores Within Groups or Cells (Magnitude of MSwithin) 13.7 Effect-Size Estimates for Factorial ANOVA 13.8 Statistical Power 13.9 Nature of the Relationships, Follow-Up Tests, and Information to Include in the Results 13.9.1 Nature of a Two-Way Interaction 13.9.2 Nature of Main Effect Differences 13.10 Factorial ANOVA Using the SPSS GLM Procedure 13.10.1 Further Discussion of Results: Comparison of the Factorial ANOVA (in Figures 13.7 and 13.8) With the One-Way ANOVA (in Figure 13.1) 13.11 Summary Appendix: Nonorthogonal Factorial ANOVA (ANOVA With Unbalanced Numbers of Cases in the Cells or Groups) Comprehension Questions Chapter 14. Multiple Regression With More Than Two Predictors 14.1 Research Questions 14.2 Empirical Example 14.3 Screening for Violations of Assumptions 14.4 Issues in Planning a Study 14.5 Computation of Regression Coefficients With k Predictor Variables 14.6 Methods of Entry for Predictor Variables 14.6.1 Standard or Simultaneous Method of Entry 14.6.2 Sequential or Hierarchical (User-Determined) Method of Entry 14.6.3 Statistical (Data-Driven) Order of Entry 14.7 Variance Partitioning in Regression for Standard or Simultaneous Regression Versus Regressions That Involve a Series of Steps 14.8 Significance Test for an Overall Regression Model 14.9 Significance Tests for Individual Predictors in Multiple Regression 14.10 Effect Size 14.10.1 Effect Size for Overall Regression (Multiple R) 14.10.2 Effect Sizes for Individual Predictor Variables (sr2) 14.11 Changes in F and R as Additional Predictors Are Added to a Model in Sequential or Statistical Regression 14.12 Statistical Power 14.13 Nature of the Relationship Between Each X Predictor and Y (Controlling for Other Predictors) 14.14 Assessment of Multivariate Outliers in Regression 14.15 SPSS Example and Results 14.15.1 SPSS Screen Shots, Output, and Results for Standard Regression 14.15.2 SPSS Screen Shots, Output, and Results for Sequential Regression 14.15.3 SPSS Screen Shots, Output, and Results for Statistical Regression 14.16 Summary Appendix 14.A: A Review of Matrix Algebra Notation and Operations and Application of Matrix Algebra to Estimation of Slope Coefficients for Regression With More Than k Predictor Variables Appendix 14.B: Tables for the Wilkinson and Dallal (1981) Test of Significance of Multiple R2 in Method = Forward Statistical Regression Comprehension Questions Chapter 15. Moderation: Tests for Interaction in Multiple Regression 15.1 Moderation Versus Mediation 15.2 Situations in Which Researchers Test Interactions 15.2.1 Factorial ANOVA Designs 15.2.2 Regression Analyses That Include Interaction Terms 15.3 When Should Interaction Terms Be Included in Regression Analysis? 15.4 Types of Predictor Variables Included in Interactions 15.4.1 Interaction Between Two Categorical Predictor Variables 15.4.2 Interaction Between a Quantitative and a Categorical Predictor Variable 15.4.3 Interaction Between Two Quantitative Predictor Variables 15.5 Assumptions and Preliminary Data Screening 15.6 Issues in Designing a Study 15.7 Sample Size and Statistical Power in Tests of Moderation or Interaction 15.8 Effect Size for Interaction 15.9 Additional Issues in Analysis 15.10 Preliminary Example: One Categorical and One Quantitative Predictor Variable With No Significant Interaction 15.11 Example 1: Significant Interaction Between One Categorical and One Quantitative Predictor Variable 15.12 Graphing Regression Lines for Subgroups 15.13 Interaction With a Categorical Predictor With More Than Two Categories 15.14 Results Section for Interaction Involving One Categorical and One Quantitative Predictor Variable 15.15 Example 2: Interaction Between Two Quantitative Predictors 15.16 Results for Example 2: Interaction Between Two Quantitative Predictors 15.17 Graphing the Interaction for Selected Values of Two Quantitative Predictors 15.18 Results Section for Example 2: Interaction of Two Quantitative Predictors 15.19 Additional Issues and Summary Comprehension Questions Chapter 16. Mediation 16.1 Definition of Mediation 16.1.1 Path Model Notation 16.1.2 Circumstances When Mediation May Be a Reasonable Hypothesis 16.2 A Hypothetical Research Example Involving One Mediating Variable 16.3 Limitations of Causal Models 16.3.1 Reasons Why Some Path Coefficients May Be Not Statistically Significant 16.3.2 Possible Interpretations for a Statistically Significant Path 16.4 Questions in a Mediation Analysis 16.5 Issues in Designing a Mediation Analysis Study 16.5.1 Type and Measurement of Variables in Mediation Analysis 16.5.2 Temporal Precedence or Sequence of Variables in Mediation Studies 16.5.3 Time Lags Between Variables 16.6 Assumptions in Mediation Analysis and Preliminary Data Screening 16.7 Path Coefficient Estimation 16.8 Conceptual Issues: Assessment of Direct Versus Indirect Paths 16.8.1 The Mediated or Indirect Path: ab 16.8.2 Mediated and Direct Path as Partition of Total Effect 16.8.3 Magnitude of Mediated Effect 16.9 Evaluating Statistical Significance 16.9.1 Causal-Steps Approach 16.9.2 Joint Significance Test 16.9.3 Sobel Test of H0: ab = 0 16.9.4 Bootstrapped Confidence Interval for ab 16.10 Effect-Size Information 16.11 Sample Size and Statistical Power 16.12 Additional Examples of Mediation Models 16.12.1 Tests of Multiple Mediating Variables 16.12.2 Multiple-Step Mediated Paths 16.12.3 Mediated Moderation and Moderated Mediation 16.13 Use of Structural Equation Modeling Programs to Test Mediation Models 16.13.1 Comparison of Regression and SEM Tests of Mediation 16.13.2 Steps in Running Amos 16.13.3 Opening the Amos Graphics Program 16.13.4 Amos Tools 16.13.5 First Steps Toward Drawing and Labeling an Amos Path Model 16.13.6 Adding Additional Variables and Paths to the Amos Path Diagram 16.13.7 Adding Error Terms for Dependent Variables 16.13.8 Correcting Mistakes and Printing the Path Model 16.13.9 Opening a Data File From Amos 16.13.10 Specification of Analysis Method and Request for Output 16.13.11 Running the Amos Analysis and Examining Preliminary Results 16.13.12 Unstandardized Path Coefficients on Path Diagram 16.13.13 Examining Text Output From Amos 16.13.14 Locating and Interpreting Output for Bootstrapped CI for the ab Indirect Effect 16.13.15 Why Use Amos/SEM Rather Than OLS Regression? 16.14 Results Section 16.15 Summary Comprehension Questions Chapter 17. Analysis of Covariance 17.1 Research Situations and Research Questions 17.2 Empirical Example 17.3 Screening for Violations of Assumptions 17.4 Variance Partitioning in ANCOVA 17.5 Issues in Planning a Study 17.6 Formulas for ANCOVA 17.7 Computation of Adjusted Effects and Adjusted Y* Means 17.8 Conceptual Basis: Factors That Affect the Magnitude of SSAadj and SSresidual and the Pattern of Adjusted Group Means 17.9 Effect Size 17.10 Statistical Power 17.11 Nature of the Relationship and Follow-Up Tests: Information to Include in the Results Section 17.12 SPSS Analysis and Model Results 17.13 Additional Discussion of ANCOVA Results 17.14 Summary Appendix: Alternative Methods for the Analysis of Pretest/Posttest Data Comprehension Questions Chapter 18. Discriminant Analysis 18.1 Research Situations and Research Questions 18.2 Introduction of an Empirical Example 18.3 Screening for Violations of Assumptions 18.4 Issues in Planning a Study 18.5 Equations for Discriminant Analysis 18.6 Conceptual Basis: Factors That Affect the Magnitude of Wilks’s Λ 18.7 Effect Size 18.8 Statistical Power and Sample Size Recommendations 18.9 Follow-Up Tests to Assess What Pattern of Scores Best Differentiates Groups 18.10 Results 18.11 One-Way ANOVA on Scores on Discriminant Functions 18.12 Summary Appendix: Eigenvalue/Eigenvector Problem Comprehension Questions Chapter 19. Multivariate Analysis of Variance 19.1 Research Situations and Research Questions 19.2 Introduction of the Initial Research Example: A One-Way MANOVA 19.3 Why Include Multiple Outcome Measures? 19.4 Equivalence of MANOVA and DA 19.5 The General Linear Model 19.6 Assumptions and Data Screening 19.7 Issues in Planning a Study 19.8 Conceptual Basis of MANOVA and Some Formulas for MANOVA 19.9 Multivariate Test Statistics 19.10 Factors That Influence the Magnitude of Wilks’s Λ 19.11 Effect Size for MANOVA 19.12 Statistical Power and Sample Size Decisions 19.13 SPSS Output for a One-Way MANOVA: Career Group Data From Chapter 18 19.14 A 2 × 3 Factorial MANOVA of the Career Group Data 19.14.1 Potential Follow-Up Tests to Assess the Nature of Significant Main Effects 19.14.2 Possible Follow-Up Tests to Assess the Nature of the Interaction 19.14.3 Further Discussion of Problems With This 2 × 3 Factorial MANOVA 19.15 A Significant Interaction in a 3 × 6 MANOVA 19.16 Comparison of Univariate and Multivariate Follow-Up Analyses for MANOVA 19.17 Summary Comprehension Questions Chapter 20. Principal Components and Factor Analysis 20.1 Research Situations 20.2 Path Model for Factor Analysis 20.3 Factor Analysis as a Method of Data Reduction 20.4 Introduction of an Empirical Example 20.5 Screening for Violations of Assumptions 20.6 Issues in Planning a Factor-Analytic Study 20.7 Computation of Loadings 20.8 Steps in the Computation of Principal Components or Factor Analysis 20.8.1 Computation of the Correlation Matrix R 20.8.2 Computation of the Initial Loading Matrix A 20.8.3 Limiting the Number of Components or Factors 20.8.4 Rotation of Factors 20.8.5 Naming or Labeling Components or Factors 20.9 Analysis 1: Principal Components Analysis of Three Items Retaining All Three Components 20.9.1 Communality for Each Item Based on All Three Components 20.9.2 Variance Reproduced by Each of the Three Components 20.9.3 Reproduction of Correlations From Loadings on All Three Components 20.10 Analysis 2: Principal Component Analysis of Three Items Retaining Only the First Component 20.10.1 Communality for Each Item Based on One Component 20.10.2 Variance Reproduced by the First Component 20.10.3 Partial Reproduction of Correlations From Loadings on Only One Component 20.11 Principal Components Versus Principal Axis Factoring 20.12 Analysis 3: PAF of Nine Items, Two Factors Retained, No Rotation 20.12.1 Communality for Each Item Based on Two Retained Factors 20.12.2 Variance Reproduced by Two Retained Factors 20.12.3 Partial Reproduction of Correlations From Loadings on Only Two Factors 20.13 Geometric Representation of Correlations Between Variables and Correlations Between Components or Factors 20.13.1 Factor Rotation 20.14 The Two Sets of Multiple Regressions 20.14.1 Construction of Factor Scores (Such as Score on F1) From z Scores 20.14.2 Prediction of Standard Scores on Variables (zxi) From Factors (F1, F2, …, F9) 20.15 Analysis 4: PAF With Varimax Rotation 20.15.1 Variance Reproduced by Each Factor at Three Stages in the Analysis 20.15.2 Rotated Factor Loadings 20.15.3 Example of a Reverse-Scored Item 20.16 Questions to Address in the Interpretation of Factor Analysis 20.16.1 How Many Factors or Components or Latent Variables Are Needed to Account for (or Reconstruct) the Pattern of Correlations Among the Measured Variables? 20.16.2 How “Important” Are the Factors or Components? How Much Variance Does Each Factor or Component Explain? 20.16.3 What, if Anything, Do the Retained Factors or Components Mean? Can We Label or Name Our Factors? 20.16.4 How Adequately Do the Retained Components or Factors Reproduce the Structure in the Original Data—That Is, the Correlation Matrix? 20.17 Results Section for Analysis 4: PAF With Varimax Rotation 20.18 Factor Scores Versus Unit-Weighted Composites 20.19 Summary of Issues in Factor Analysis 20.20 Optional: Brief Introduction to Concepts in Structural Equation Modeling Appendix: The Matrix Algebra of Factor Analysis Comprehension Questions Chapter 21. Reliability, Validity, and Multiple-Item Scales 21.1 Assessment of Measurement Quality 21.1.1 Reliability 21.1.2 Validity 21.1.3 Sensitivity 21.1.4 Bias 21.2 Cost and Invasiveness of Measurements 21.2.1 Cost 21.2.2 Invasiveness 21.2.3 Reactivity of Measurement 21.3 Empirical Examples of Reliability Assessment 21.3.1 Definition of Reliability 21.3.2 Test-Retest Reliability Assessment for a Quantitative Variable 21.3.3 Interobserver Reliability Assessment for Scores on a Categorical Variable 21.4 Concepts From Classical Measurement Theory 21.4.1 Reliability as Partition of Variance 21.4.2 Attenuation of Correlations Due to Unreliability of Measurement 21.5 Use of Multiple-Item Measures to Improve Measurement Reliability 21.6 Computation of Summated Scales 21.6.1 Assumption: All Items Measure Same Construct and Are Scored in Same Direction 21.6.2 Initial (Raw) Scores Assigned to Individual Responses 21.6.3 Variable Naming, Particularly for Reverse-Worded Questions 21.6.4 Factor Analysis to Assess Dimensionality of a Set of Items 21.6.5 Recoding Scores for Reverse-Worded Items 21.6.6 Summing Scores Across Items to Compute Total Score: Handling Missing Data 21.6.7 Sums of (Unit-Weighted) Item Scores Versus Saved Factor Scores 21.6.7.1 Simple Unit-Weighted Sum of Raw Scores 21.6.7.2 Simple Unit-Weighted Sum of z Scores 21.6.7.3 Saved Factor Scores or Other Optimally Weighted Linear Composites 21.6.7.4 Correlation Between Sums of Items Versus Factor Scores 21.6.7.5 Choice Among Methods of Scoring 21.7 Assessment of Internal Homogeneity for Multiple-Item Measures: Cronbach’s Alpha Reliability Coefficient 21.7.1 Cronbach’s Alpha: Conceptual Basis 21.7.2 Empirical Example: Cronbach’s Alpha for Five Selected CES-D Scale Items 21.7.3 Improving Cronbach’s Alpha by Dropping a “Poor” Item 21.7.4 Improving Cronbach’s Alpha by Increasing the Number of Items 21.7.5 Other Methods of Reliability Assessment for Multiple-Item Measures 21.7.5.1 Split-Half Reliability 21.7.5.2 Parallel Forms Reliability 21.8 Validity Assessment 21.8.1 Content and Face Validity 21.8.2 Criterion-Oriented Validity 21.8.2.1 Convergent Validity 21.8.2.2 Discriminant Validity 21.8.2.3 Concurrent Validity 21.8.2.4 Predictive Validity 21.8.3 Construct Validity: Summary 21.9 Typical Scale Development Process 21.9.1 Generating and Modifying the Pool of Items or Measures 21.9.2 Administer Survey to Participants 21.9.3 Factor Analyze Items to Assess the Number and Nature of Latent Variables or Constructs 21.9.4 Development of Summated Scales 21.9.5 Assess Scale Reliability 21.9.6 Assess Scale Validity 21.9.7 Iterative Process 21.9.8 Create the Final Scale 21.10 Modern Measurement Theory 21.11 Reporting Reliability Assessment 21.12 Summary Appendix: The CES-D Scale Comprehension Questions Chapter 22. Analysis of Repeated Measures 22.1 Introduction 22.2 Empirical Example: Experiment to Assess Effect of Stress on Heart Rate 22.2.1 Analysis of Data From the Stress/HR Study as a Between-S or Independent Samples Design 22.2.2 Independent Samples t Test for the Stress/HR Data 22.2.3 One-Way Between-S ANOVA for the Stress/HR Data 22.3 Discussion of Sources of Within-Group Error in Between-S Versus Within-S Data 22.4 The Conceptual Basis for the Paired Samples t Test and One-Way Repeated Measures ANOVA 22.5 Computation of a Paired Samples t Test to Compare Mean HR Between Baseline and Pain Conditions 22.6 SPSS Example: Analysis of Stress/HR Data Using a Paired Samples t Test 22.7 Comparison Between Independent Samples t Test and Paired Samples t Test 22.8 SPSS Example: Analysis of Stress/HR Data Using a Univariate One-Way Repeated Measures ANOVA 22.9 Using the SPSS GLM Procedure for Repeated Measures ANOVA 22.10 Screening for Violations of Assumptions in Univariate Repeated Measures 22.11 The Greenhouse-Geisser ε and Huynh-Feldt ε Correction Factors 22.12 MANOVA Approach to Analysis of Repeated Measures Data 22.13 Effect Size 22.14 Statistical Power 22.15 Planned Contrasts 22.16 Results 22.17 Design Problems in Repeated Measures Studies 22.18 More Complex Designs 22.19 Alternative Analyses for Pretest and Posttest Scores 22.20 Summary Comprehension Questions Chapter 23. Binary Logistic Regression 23.1 Research Situations 23.1.1 Types of Variables 23.1.2 Research Questions 23.1.3 Assumptions Required for Linear Regression Versus Binary Logistic Regression 23.2 Simple Empirical Example: Dog Ownership and Odds of Death 23.3 Conceptual Basis for Binary Logistic Regression Analysis 23.3.1 Why Ordinary Linear Regression Is Inadequate 23.3.2 Modifying the Method of Analysis to Handle These Problems 23.4 Definition and Interpretation of Odds 23.5 A New Type of Dependent Variable: The Logit 23.6 Terms Involved in Binary Logistic Regression Analysis 23.6.1 Estimation of Coefficients for a Binary Logistic Regression Model 23.6.2 Assessment of Overall Goodness of Fit for a Binary Logistic Regression Model 23.6.3 Alternative Assessments of Overall Goodness of Fit 23.6.4 Information About Predictive Usefulness of Individual Predictor Variables 23.6.5 Evaluating Accuracy of Group Classification 23.7 Analysis of Data for First Empirical Example: Dog Ownership/Death Study 23.7.1 SPSS Menu Selections and Dialog Windows 23.7.2 SPSS Output 23.7.2.1 Null Model 23.7.2.2 Full Model 23.7.3 Results for the Dog Ownership/Death Study 23.8 Issues in Planning and Conducting a Study 23.8.1 Preliminary Data Screening 23.8.2 Design Decisions 23.8.3 Coding Scores on Binary Variables 23.9 More Complex Models 23.10 Binary Logistic Regression for Second Empirical Analysis: Drug Dose and Gender as Predictors of Odds of Death 23.11 Comparison of Discriminant Analysis to Binary Logistic Regression 23.12 Summary Comprehension Questions Appendix A: Proportions of Area Under a Standard Normal Curve Appendix B: Critical Values for t Distribution Appendix C: Critical Values of F Appendix D: Critical Values of Chi-Square Appendix E: Critical Values of the Pearson Correlation Coefficient Appendix F: Critical Values of the Studentized Range Statistic Appendix G: Transformation of r (Pearson Correlation) to Fisher Z Glossary References Index