DANIEL J. DENIS, PhD, is Associate Professor of Quantitative Psychology at the University of Montana where he teaches courses in univariate and multivariate statistics. He has published a number of articles in peer-reviewed journals and has served as consultant to researchers and practitioners in a variety of fields.

Preface

1 Preliminary Considerations

1.1 The Philosophical Bases of Knowledge: Rationalistic versus

Empiricist Pursuits

1.2 What is a “Model”?

1.3 Social Sciences versus Hard Sciences

1.4 Is Complexity a Good Depiction of Reality? Are Multivariate

Methods Useful?

1.5 Causality

1.6 The Nature of Mathematics: Mathematics as a Representation

of Concepts

1.7 As a Scientist, How Much Mathematics Do You Need

to Know?

1.8 Statistics and Relativity

1.9 Experimental versus Statistical Control

1.10 Statistical versus Physical Effects

1.11 Understanding What “Applied Statistics” Means

Review Exercises

2 Introductory Statistics

2.1 Densities and Distributions

2.1.2 Binomial Distributions

2.1.3 Normal Approximation

2.1.4 Joint Probability Densities: Bivariate and Multivariate

Distributions

2.2 Chi-Square Distributions and Goodness-of-Fit Test

2.2.1 Power for Chi-Square Test of Independence

2.3 Sensitivity and Specificity

2.4 Scales of Measurement: Nominal, Ordinal, and Interval, Ratio

2.4.1 Nominal Scale

2.4.2 Ordinal Scale

2.4.3 Interval Scale

2.4.4 Ratio Scale

2.5 Mathematical Variables versus Random Variables

2.6 Moments and Expectations

2.7 Estimation and Estimators

2.8 Variance

2.9 Degrees of Freedom

2.10 Skewness and Kurtosis

2.11 Sampling Distributions

2.11.1 Sampling Distribution of the Mean

2.12 Central Limit Theorem

2.13 Confidence Intervals

2.14 Maximum Likelihood

2.15 Akaike’s Information Criteria

2.16 Covariance and Correlation

2.17 Psychometric Validity, Reliability: A Common Use of Correlation Coefficients

2.18 Covariance and Correlation Matrices

2.19 Other Correlation Coefficients

2.20 Student’s t Distribution

2.20.1 t-Tests for One Sample

2.20.2 t-Tests for Two Samples

2.21 Statistical Power

2.21.1 Power Estimation Using R and G∗Power

2.21.2 Estimating Sample Size and Power for Independent

Samples t-Test

2.22 Paired Samples t-Test: Statistical Test for Matched Pairs

(Elementary Blocking) Designs

2.23 Blocking with Several Conditions

2.24 Composite Variables: Linear Combinations

2.25 Models in Matrix Form

2.26 Graphical Approaches

2.26.1 Box-and-Whisker Plots

2.27 What Makes a p-Value Small? A Critical Overview and Simple

Demonstration of Null Hypothesis Significance Testing

2.27.1 Null Hypothesis Significance Testing: A History

of Criticism

2.27.2 The Makeup of a p-Value: A Brief Recap and Summary

2.27.3 The Issue of Standardized Testing: Are Students in

Your School Achieving More Than the National Average?

2.27.4 Other Test Statistics

2.27.5 The Solution

2.27.6 Statistical Distance: Cohen’s d

2.27.7 Why and Where the Significance Test Still Makes Sense

2.28 Chapter Summary and Highlights

Review Exercises

3 Analysis of Variance: Fixed Effects Models

3.1 What is Analysis of Variance? Fixed versus Random Effects

3.1.1 Small Sample Example: Achievement as a

Function of Teacher

3.2 How Analysis of Variance Works: A Big Picture Overview

3.2.1 Is the Observed Difference Likely? ANOVA as a

Comparison (Ratio) of Variances

3.3 Logic and Theory of ANOVA: A Deeper Look

3.3.1 Independent Samples t-tests versus Analysis of Variance

3.3.2 The ANOVA Model: Explaining Variation

3.3.3 Breaking Down a Deviation

3.3.4 Naming the Deviations

3.3.5 The Sums of Squares of ANOVA

3.4 From Sums of Squares to Unbiased Variance Estimators:

Dividing by Degrees of Freedom

3.5 Expected Mean Squares for One-Way Fixed Effects Model:

Deriving the F-Ratio

3.6 The Null Hypothesis in ANOVA

3.7 Fixed Effects ANOVA: Model Assumptions

3.8 A Word on Experimental Design and Randomization

3.9 A Preview of the Concept of Nesting

3.10 Balanced versus Unbalanced Data in ANOVA Models

3.11 Measures of Association and Effect Size in ANOVA:

Measures of Variance Explained

3.11.1 Eta-Squared

3.11.2 Omega-Squared

3.12 The F-Test and the Independent Samples t-Test

3.13 Contrasts and Post-Hocs

3.13.1 Independence of Contrasts

3.13.2 Independent Samples t-Test as a Linear Contrast

3.14 Post-Hoc Tests

3.14.1 Newman–Keuls and Tukey HSD

3.14.2 Tukey HSD

3.14.3 Scheffé Test

3.14.4 Contrast versus Post-Hoc? Which Should I Be Doing?

3.15 Sample Size and Power for ANOVA: Estimation with R and

G∗Power

3.15.1 Power for ANOVA in R and G∗Power

3.16 Fixed Effects One-Way Analysis of Variance in R:

Mathematics Achievement as a Function of Teacher

3.17 Analysis of Variance Via R’s lm

3.18 Kruskal–Wallis Test in R and the Motivation Behind Nonparametric Tests

3.19 ANOVA in SPSS: Achievement as a Function of Teacher

3.20 Chapter Summary and Highlights

Review Exercises

4 Factorial Analysis of Variance: Modeling Interactions

4.1 What is Factorial Analysis of Variance?

4.2 Theory of Factorial ANOVA: A Deeper Look

4.2.1 Deriving the Model for Two-Way Factorial ANOVA

4.2.2 Cell Effects

4.2.3 Interaction Effects

4.2.4 A Model for the Two-Way Fixed Effects ANOVA

4.3 Comparing One-Way ANOVA to Two-Way ANOVA: Cell

Effects in Factorial ANOVA versus Sample Effects

in One-Way ANOVA

4.4 Partitioning the Sums of Squares for Factorial ANOVA:

The Case of Two Factors

4.4.1 SS Total: A Measure of Total Variation

4.4.2 Model Assumptions: Two-Way Factorial Model

4.4.3 Expected Mean Squares for Factorial Design

4.5 Interpreting Main Effects in the Presence of Interactions

4.6 Effect Size Measures

4.7 Three-Way, Four-Way, and Higher-Order Models

4.8 Simple Main Effects

4.9 Nested Designs

4.9.1 Varieties of Nesting: Nesting of Levels versus Subjects

4.10 Achievement as a Function of Teacher and Textbook: Example of

Factorial ANOVA in R

4.10.1 Simple Main Effects for Achievement Data: Breaking Down

Interaction Effects

4.11 Interaction Contrasts

4.12 Chapter Summary and Highlights

Review Exercises

5 Introduction to Random Effects and Mixed Models

5.1 What is Random Effects Analysis of Variance?

5.2 Theory of Random Effects Models

5.3 Estimation in Random Effects Models

5.3.1 Transitioning from Fixed Effects to Random Effects

5.3.2 Expected Mean Squares for MS Between and MS Within

5.4 Defining Null Hypotheses in Random Effects Models

5.4.1 F-Ratio for Testing 

5.5 Comparing Null Hypotheses in Fixed versus Random Effects Models:

The Importance of Assumptions

5.6 Estimating Variance Components in Random Effects Models:

ANOVA, ML, REML Estimators

5.6.1 ANOVA Estimators of Variance Components

5.6.2 Maximum Likelihood and Restricted Maximum

Likelihood

5.7 Is Achievement a Function of Teacher? One-Way Random Effects

Model in R

5.7.1 Proportion of Variance Accounted for by Teacher

5.8 R Analysis Using REML

5.9 Analysis in SPSS: Obtaining Variance Components

5.10 Factorial Random Effects: A Two-Way Model

5.11 Fixed Effects versus Random Effects: A Way of Conceptualizing Their

Differences

5.12 Conceptualizing the Two-Way Random Effects Model: The Makeup of

a Randomly Chosen Observation

5.13 Sums of Squares and Expected Mean Squares for Random Effects: The

Contaminating Influence of Interaction Effects

5.13.1 Testing Null Hypotheses

5.14 You Get What You Go in with: The Importance of Model Assumptions

and Model Selection

5.15 Mixed Model Analysis of Variance: Incorporating Fixed and Random

Effects

5.15.1 Mixed Model in R

5.16 Mixed Models in Matrices

5.17 Multilevel Modeling as a Special Case of the Mixed Model:

Incorporating Nesting and Clustering

5.18 Chapter Summary and Highlights

Review Exercises

6 Randomized Blocks and Repeated Measures

6.1 What Is a Randomized Block Design?

6.2 Randomized Block Designs: Subjects Nested Within Blocks

6.3 Theory of Randomized Block Designs

6.3.1 Nonadditive Randomized Block Design

6.3.2 Additive Randomized Block Design

6.4 Tukey Test for Nonadditivity

6.5 Assumptions for the Variance–Covariance Matrix

6.6 Intraclass Correlation

6.7 Repeated Measures Models: A Special Case of Randomized Block

Designs

6.8 Independent versus Paired Samples t-Test

6.9 The Subject Factor: Fixed or Random Effect?

6.10 Model for One-Way Repeated Measures Design

6.10.1 Expected Mean Squares for Repeated Measures Models

6.11 Analysis Using R: One-Way Repeated Measures: Learning as a

Function of Trial

6.12 Analysis Using SPSS: One-Way Repeated Measures: Learning as a

Function of Trial

6.12.1 Which Results Should Be Interpreted?

6.13 SPSS: Two-Way Repeated Measures Analysis of Variance: Mixed

Design: One Between Factor, One Within Factor

6.13.1 Another Look at the Between-Subjects Factor

6.14 Chapter Summary and Highlights

Review Exercises

7 Linear Regression

7.1 Brief History of Regression

7.2 Regression Analysis and Science: Experimental versus Correlational

Distinctions

7.3 A Motivating Example: Can Offspring Height Be Predicted?

7.4 Theory of Regression Analysis: A Deeper Look

7.5 Multilevel Yearnings

7.6 The Least-Squares Line

7.7 Making Predictions Without Regression

7.8 More About 

7.9 Model Assumptions for Linear Regression

7.9.1 Model Specification

7.9.2 Measurement Error

7.10 Estimation of Model Parameters in Regression

7.10.1 Ordinary Least-Squares

7.11 Null Hypotheses for Regression

7.12 Significance Tests and Confidence Intervals for Model

Parameters

7.13 Other Formulations of the Regression Model

7.14 The Regression Model in Matrices: Allowing for More Complex

Multivariable Models

7.15 Ordinary Least-Squares in Matrices

7.16 Analysis of Variance for Regression

7.17 Measures of Model Fit for Regression: How Well Does the Linear

Equation Fit?

7.18 Adjusted 

7.19 What “Explained Variance” Means: And More Importantly,

What It Does Not Mean

7.20 Values Fit by Regression

7.21 Least-Squares Regression in R: Using Matrix Operations

7.22 Linear Regression Using R

7.23 Regression Diagnostics: A Check on Model Assumptions

7.23.1 Understanding How Outliers Influence a Regression Model

7.23.2 Examining Outliers and Residuals

7.24 Regression in SPSS: Predicting Quantitative from Verbal

7.25 Power Analysis for Linear Regression in R

7.26 Chapter Summary and Highlights

Review Exercises

8 Multiple Linear Regression

8.1 Theory of Partial Correlation

8.2 Semipartial Correlations

8.3 Multiple Regression

8.4 Some Perspective on Regression Coefficients: “Experimental

Coefficients”?

8.5 Multiple Regression Model in Matrices

8.6 Estimation of Parameters

8.7 Conceptualizing Multiple R

8.8 Interpreting Regression Coefficients: Correlated Versus Uncorrelated Predictors

8.9 Anderson’s Iris Data: Predicting Sepal Length from Petal

Length and Petal Width

8.10 Fitting Other Functional Forms: A Brief Look at Polynomial

Regression

8.11 Measures of Collinearity in Regression: Variance Inflation Factor

and Tolerance

8.12 R-Squared as a Function of Partial and Semipartial Correlations:

The Stepping Stones to Forward and Stepwise Regression

8.13 Model-Building Strategies: Simultaneous, Hierarchichal,

Forward, and Stepwise

8.13.1 Simultaneous, Hierarchical, and Forward

8.13.2 Stepwise Regression

8.13.3 Selection Procedures in R

8.13.4 Which Regression Procedure Should Be Used? Concluding

Comments and Recommendations Regarding

Model-Building

8.14 Power Analysis for Multiple Regression

8.15 Introduction to Statistical Mediation: Concepts

and Controversy

8.15.1 Statistical versus True Mediation: Some Philosophical

Pitfalls in the Interpretation of Mediation Analysis

8.16 Brief Survey of Ridge and Lasso Regression: Penalized Regression Models and the Concept of Shrinkage

8.17 Chapter Summary and Highlights

Review Exercises

9 Interactions in Multiple Linear Regression: Dichotomous,

Polytomous, and Continuous Moderators

9.1 The Additive Regression Model with Two Predictors

9.2 Why the Interaction is the Product Term  : Drawing

an Analogy to Factorial ANOVA

9.3 A Motivating Example of Interaction in Regression: Crossing a

Continuous Predictor with a Dichotomous Predictor

9.4 Analysis of Covariance

9.5 Continuous Moderators

9.6 Summing Up the Idea of Interactions in Regression

9.7 Do Moderators Really “Moderate” Anything? Some Philosophical

Considerations

9.8 Interpreting Model Coefficients in the Context of Moderators

9.9 Mean-Centering Predictors: Improving the Interpretability

of Simple Slopes

9.10 Multilevel Regression: Another Special Case of the Mixed

Model

9.11 Chapter Summary and Highlights

Review Exercises

10 Logistic Regression and the Generalized Linear Model

10.1 Nonlinear Models

10.2 Generalized Linear Models

10.2.1 The Logic of the Generalized Linear Model: How the Link

Function Transforms Nonlinear Response Variables

10.3 Canonical Links

10.3.1 Canonical Link for Gaussian Variable

10.4 Distributions and Generalized Linear Models

10.4.1 Logistic Models

10.4.2 Poisson Models

10.5 Dispersion Parameters and Deviance

10.6 Logistic Regression: A Generalized Linear Model for Binary

Responses

10.6.1 Model for Single Predictor

10.7 Exponential and Logarithmic Functions

10.7.1 Logarithms

10.7.2 The Natural Logarithm

10.8 Odds and the Logit

10.9 Putting It All Together: The Logistic Regression Model

10.9.1 Interpreting the Logit: A Survey of Logistic Regression

Output

10.10 Logistic Regression in R: Challenger O-ring Data

10.11 Challenger Analysis in SPSS

10.11.1 Predictions of New Cases

10.12 Sample Size, Effect Size, and Power

10.13 Further Directions

10.14 Chapter Summary and Highlights

Review Exercises

11 Multivariate Analysis of Variance

11.1 A Motivating Example: Quantitative and Verbal Ability

as a Variate

11.2 Constructing the Composite

11.3 Theory of MANOVA

11.4 Is the Linear Combination Meaningful?

11.5 Multivariate Hypotheses

11.6 Assumptions of MANOVA

11.7 Hotelling’s  : The Case of Generalizing from Univariate

to Multivariate

11.8 The Covariance Matrix 

11.9 From Sums of Squares and Cross-Products to Variances and

Covariances

11.10 Hypothesis and Error Matrices of MANOVA

11.11 Multivariate Test Statistics

11.11.1 Pillai’s Trace

11.11.2 Lawley–Hotelling’s Trace

11.12 Equality of Variance–Covariance Matrices

11.13 Multivariate Contrasts

11.14 MANOVA in R and SPSS

11.14.1 Univariate Analyses

11.15 MANOVA of Fisher’s Iris Data

11.16 Power Analysis and Sample Size for MANOVA

11.17 Multivariate Analysis of Covariance and Multivariate Models:

A Bird’s Eye View of Linear Models

11.18 Chapter Summary and Highlights

Review Exercises

12 Discriminant Analysis

12.1 What is Discriminant Analysis? The Big Picture

on the Iris Data

12.2 Theory of Discriminant Analysis

12.2.1 Discriminant Analysis for Two Populations

12.3 LDA in R and SPSS

12.4 Discriminant Analysis for Several Populations

12.4.1 Theory for Several Populations

12.5 Discriminating Species of Iris: Discriminant Analyses

for Three Populations

12.6 A Note on Classification and Error Rates

12.7 Discriminant Analysis and Beyond

12.8 Canonical Correlation

12.9 Motivating Example for Canonical Correlation: Hotelling’s 1936

Data

12.10 Canonical Correlation as a General Linear Model

12.11 Theory of Canonical Correlation

12.12 Canonical Correlation of Hotelling’s Data

12.13 Canonical Correlation on the Iris Data: Extracting Canonical

Correlation from Regression, MANOVA, LDA

12.14 Chapter Summary and Highlights

Review Exercises

13 Principal Components Analysis

13.1 History of Principal Components Analysis

13.2 Hotelling 1933

13.3 Theory of Principal Components Analysis

13.3.1 The Theorem of Principal Components Analysis

13.4 Eigenvalues as Variance

13.5 Principal Components as Linear Combinations

13.6 Extracting the First Component

13.6.1 Sample Variance of a Linear Combination

13.7 Extracting the Second Component

13.8 Extracting Third and Remaining Components

13.9 The Eigenvalue as the Variance of a Linear Combination

Relative to Its Length

13.10 Demonstrating Principal Components Analysis: Pearson’s 1901

Illustration

13.11 Scree Plots

13.12 Principal Components versus Least-Squares Regression Lines

13.13 Covariance versus Correlation Matrices: Principal Components

and Scaling

13.14 Principal Components Analysis Using SPSS

13.15 Chapter Summary and Highlights

Review Exercises

14 Factor Analysis

14.1 History of Factor Analysis

14.2 Factor Analysis: At a Glance

14.3 Exploratory vs. Confirmatory Factor Analysis

14.4 Theory of Factor Analysis: The Exploratory Factor-Analytic

Model

14.5 The Common Factor-Analytic Model

14.6 Assumptions of the Factor-Analytic Model

14.7 Why Model Assumptions are Important

14.8 The Factor Model as an Implication for the Covariance

Matrix 

14.9 Again, Why is   so Important a Result?

14.10 The Major Critique Against Factor Analysis: Indeterminacy

and the Nonuniqueness of Solutions

14.11 Has Your Factor Analysis Been Successful?

14.12 Estimation of Parameters in Exploratory Factor Analysis

14.13 Principal Factor

14.14 Maximum Likelihood

14.15 The Concepts (and Criticisms) of Factor Rotation

14.16 Varimax and Quartimax Rotation

14.17 Should Factors Be Rotated? Is That Not “Cheating?”

14.18 Sample Size for Factor Analysis

14.19 Principal Components Analysis versus Factor Analysis:

Two Key Differences

14.19.1 Hypothesized Model and Underlying Theoretical

Assumptions

14.19.2 Solutions Are Not Invariant in Factor Analysis

14.20 Principal Factor in SPSS: Principal Axis Factoring

14.21 Bartlett Test of Sphericity and Kaiser–Meyer–Olkin Measure

of Sampling Adequacy (MSA)

14.23 Factor Analysis in R: Holzinger and Swineford (1939)

14.23 Cluster Analysis

14.24 What Is Cluster Analysis? The Big Picture

14.25 Measuring Proximity

14.26 Hierarchical Clustering Approaches

14.27 Nonhierarchical Clustering Approaches

14.28 K-Means Cluster Analysis in R

14.29 Guidelines and Warnings About Cluster Analysis

14.30 A Brief Look at Multidimensional Scaling

14.31 Chapter Summary and Highlights

Review Exercises

15 Path Analysis and Structural Equation Modeling

15.1 Path Analysis: A Motivating Example—Predicting IQ Across

Generations

15.2 Path Analysis and “Causal Modeling”

15.3 Early Post-Wright Path Analysis: Predicting Child’s IQ

(Burks, 1928)

15.4 Decomposing Path Coefficients

15.5 Path Coefficients and Wright’s Contribution

15.6 Path Analysis in R: A Quick Overview—Modeling

Galton’s Data

15.7 Confirmatory Factor Analysis: The Measurement Model

15.7.1. Confirmatory Factor Analysis as a Means of Evaluating Construct Validity and Assessing Psychometric Qualities

15.8 Structural Equation Models

15.9 Direct, Indirect, and Total Effects

15.10 Theory of Statistical Modeling: A Deeper Look into Covariance

Structures and General Modeling

15.11 The Discrepancy Function and Chi-Square

15.13 Identification

15.14 Disturbance Variables

15.15 Measures and Indicators of Model Fit

15.16 Overall Measures of Model Fit

15.16.1 Root Mean Square Residual and Standardized Root

Mean Square Residual

15.16.2 Root Mean Square Error of Approximation

15.17 Model Comparison Measures: Incremental Fit Indices

15.18 Which Indicator of Model Fit Is Best?

15.19 Structural Equation Model in R

15.20 How All Variables Are Latent: A Suggestion for Resolving

the Manifest–Latent Distinction

15.21 The Structural Equation Model as a General Model:

Some Concluding Thoughts on Statistics and Science

15.22 Chapter Summary and Highlights

Review Exercises

References

Index

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