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Applied Statistics II

Applied Statistics II
Multivariable and Multivariate Techniques

Third Edition

January 2020 | 720 pages | SAGE Publications, Inc

Rebecca M. Warner’s bestselling Applied Statistics: From Bivariate Through Multivariate Techniques has been split into two volumes for ease of use over a two-course sequence. This new multivariate statistics text, Applied Statistics II: Multivariable and Multivariate Techniques, Third Edition is based on chapters from the second half of original book, but with much additional material. This text now provides a distinctive bridge between earlier courses and advanced topics through extensive discussion of statistical control (adding a third variable), a new chapter on the "new statistics", a new chapter on outliers and missing values, and a final chapter that provides an introduction to structural equation modeling. This text provides a solid introduction to concepts such as statistical control, mediation, moderation, and path modeling necessary to students taking intermediate and advanced statistics courses across the social sciences. Examples are provided in SPSS with datasets available on an accompanying website. A companion study guide reproducing the exercises and examples in R will also be available.

1. The New Statistics
Required Background

What is the “New Statistics”?

Common Misinterpretations of p Values

Problems with NHST Logic The Replication Crises

Common Misuses of NHST

The Replication Crisis

Some Proposed Remedies for NHST Problems

Review of Confidence Intervals

Effect Size

Brief Introduction to Meta-Analysis

Recommendations for Better Research and Analysis


2. Advanced Data Screening: Outliers and Missing Values

Variable Names and File Management

Sources of Bias

Screening Sample Data

Possible Remedy for Skewness: Nonlinear Data Transformations

Identification of Outliers

Handling Outliers

Testing Linearity Assumptions

Evaluation of Other Assumptions Specific to Analyses

Describing Amount of Missing Data

How Missing Data Arise

Patterns in Missing Data

Empirical Example: Detecting Type a Missingness

Possible Remedies for Missing Data

Empirical Example: Multiple Imputation to Replace Missing Values

Data Screening Checklist

Reporting Guidelines


Appendix 2 A Brief Note About Zero Inflated Binomial or Poisson Regression

3. Statistical Control: What Can Happen When You Add a Third Variable?
What is Statistical Control?

First Research Example: Controlling for a Categorical X2 Variable

Assumptions for Partial Correlation Between X1 and Y, Controlling for X2

Notation for Partial Correlation

Computing Partial Correlation: Use of Bivariate Regressions to Remove Variance Predictable by X2 from Both X1 and Y

Partial Correlation Makes No Sense if There is An X1 x X2 Interaction

Computation of Partial r From Bivariate Pearson Correlations

Significance Tests, Confidence Intervals, and Statistical Power for Partial Correlations

Comparing Outcomes for ry1.2 and ry1

Introduction to Path Models

Possible Paths Among X1, Y, and X2

One Possible Model: X1 and Y are Not Related Whether You Control for X2 or Not

Possible Model: Correlation Between X1 and Y is the Same Whether X2 is Statistically Controlled or Not (X2 is Irrelevant to the X1, Y Relationship)

When You Control for X2, Correlation Between X1 and Y Drops to 0

When You Control for X2, the Correlation Between X1 and Y Becomes Smaller (But Does not Drop to 0 or Change Sign)

Some Forms of Suppression: When You Control for X2, r1y.2 Becomes Larger Than r1y or Opposite in Sign to r1y.

“None of the Above”

Results Section


4. Regression Analysis and Statistical Control

Hypothetical Research Example

Graphic Representation of Regression Plane

Semipartial (or “Part”) Correlation

Partition of Variance In Y in Regression with Two Predictors

Assumptions for Regression With Two Predictors

Formulas for Regression With Two Predictors

SPSS Regression

Conceptual Basis: Factors that Affect the Magnitude and Sign of ? and b in Regression With Two Predictors

Tracing Rules for Path Models

Comparison of Equations for ?, b, pr, and sr

Nature of Predictive Relationships

Effect Size Information in Regression with Two Predictors

Statistical Power

Issues in Planning a Study



5. Multiple Regression with Multiple Predictors
Research Questions

Empirical Example

Screening for Violations of Assumptions

Issues in Planning a Study

Computation of Regression Coefficients with k Predictor Variables

Methods of Entry for Predictor Variables

Variance Partitioning in Standard Regression Versus Hierarchical and Statistical Regression

Significance Test for an Overall Regression Model

Significance Tests for Individual Predictors in Multiple Regression

Effect Size

Changes in F and R as Additional Predictors Are Added to a Model in Sequential or Statistical Regression

Statistical Power

Nature of the Relationship Between Each X Predictor and Y (Controlling for Other Predictors)

Assessment of Multivariate Outliers in Regression

SPSS Examples and Results


Appendix 5 A Use of Matrix Algebra to Estimate Regression Coefficients for Multiple Predictors

Appendix 5 B Tables for Wilkinson and Dallal (1981) Test of Significance of Multiple R2 in Forward Statistical Regression

6. Dummy Predictor Variables in Multiple Regression
What Dummy Variables Are and When They Are Used

Empirical Example

Screening for Violations of Assumptions

Issues in Planning a Study

Parameter Estimates and Significance Tests for Regressions with Dummy Predictor Variables

Group Mean Comparisons Using One-Way Between-S ANOVA

Three Methods of Coding for Dummy Variables

Regression Models That Include Both Dummy and Quantitative Predictor Variables

Effect Size and Statistical Power

Nature of the Relationship and/or Follow-Up Tests



7. Moderation: Interaction in Multiple Regression

Interaction Between Two Categorical Predictors: Factorial ANOVA

Interaction Between One Categorical and One Quantitative Predictor

Preliminary Data Screening: One Categorical and One Quantitative Predictor

Scatterplot for Preliminary Assessment of Possible Interaction Between Categorical and Quantitative Predictor

Regression to Assess Statistical Significance of Interaction Between One Categorical and One Quantitative Predictor

Interaction Analysis With More Than Three Categories

Example With Different Data: Significant Sex by Years Interaction

Follow-Up: Analysis of Simple Main Effects

Interaction Between Two Quantitative Predictors

SPSS Example of Interaction Between Two Quantitative Predictors

Results for Interaction of Age and Habits as Predictors of Symptoms

Graphing Interaction for Two Quantitative Predictors

Results Section for Interaction of Two Quantitative Predictors

Additional Issues and Summary

Appendix 7 A Graphing Interactions Between Quantitative Variables “By Hand”

8. Analysis of Covariance
Research Situations for ANCOVA

Empirical Example

Screening for Violations of Assumptions

Variance Partitioning in ANCOVA

Issues in Planning a Study

Formulas for ANCOVA

Computation of Adjusted Effects and Adjusted Y* Means

Conceptual Basis: Factors that Affect the Magnitude of SSAadj and SSresidual and the Pattern of Adjusted Group Means

Effect Size

Statistical Power

Nature of the Relationship and Follow-Up Tests: Information to Include in the Results Section

SPSS Analysis and Results

Additional Discussion of ANCOVA Results


Appendix 8 A Alternative Methods for the Analysis of Pretest/Posttest Data

9. Mediation
Definition of Mediation

Hypothetical Research Example

Limits of “Causal” Models

Questions in a Mediation Analysis

Issues in Designing a Mediation Analysis Study

Assumptions in Mediation Analysis and Preliminary Data Screening

Path Coefficient Estimation

Conceptual Issues: Assessment of Direct Versus Indirect Paths

Evaluating Statistical Significance

Effect Size Information

Sample Size and Statistical Power

Additional Examples of Mediation

Note About Use of Structural Equation Modeling Programs to Test Mediation Hypotheses

Results Section


10. Discriminant Analysis
Research Situations and Research Questions

Introduction to Empirical Example

Screening for Violations of Assumptions

Issues in Planning a Study

Equations for Discriminant Analysis

Conceptual Basis: Factors That Affect the Magnitude of Wilks’s L

Effect Size

Statistical Power and Sample Size Recommendations

Follow-Up Tests to Assess What Pattern of Scores Best Differentiates Groups


One-Way ANOVA on Scores on Discriminant Functions


Appendix 10 A The Eigenvalue/ Eigenvector Problem

Appendix 10 B Additional Equations for Discriminant Analysis

11. Multivariate Analysis of Variance (MANOVA)
Research Situations and Research Questions

First Research Example: One-Way MANOVA

Why Include Multiple Outcome Measures?

Equivalence of MANOVA and DA

The General Linear Model

Assumptions and Data Screening

Issues in Planning a Study

Conceptual Basis of MANOVA

Multivariate Test Statistics

Factors that Influence the Magnitude of Wilks’s Lambda

Effect Size for MANOVA

Statistical Power and Sample Size Decisions

One Way MANOVA: Career Group Data

2 x 3 Factorial MANOVA: Career Group Data

Significant Interaction in a 3 x 6 MANOVA

Comparison of Univariate Versus Multivariate Follow Up Analyses


12. Exploratory Factor Analysis
Research Situations

Path Model for Factor Analysis

Factor Analysis as a Method of Data Reduction

Introduction of Empirical Example

Screening for Violations of Assumptions

Issues in Planning a Factor-Analytic Study

Computation of Factor Loadings

Steps in the Computation of Principal Components and Factor Analysis

Analysis One: Principal Components Analysis of Three Items Retaining All Three Components

Analysis Two: Principal Component Analysis of Three Items Retaining Only the First Component

Principal Components Versus Principal Axis Factoring

Analysis 3: PAF of Nine Items, Two Factors Retained, No Rotation

Geometric Representation of Factor Rotation

Factor Analysis as Two Sets of Multiple Regressions

Final Analysis/ Analysis 4: PAF With Varimax Rotation

Questions to Address in the Interpretation of Factor Analysis

Results Section for Analysis 4: PAF With Varimax Rotation

Factor Scores Versus Unit-Weighted Composites

Summary of Issues in Factor Analysis

Appendix 12 A The Matrix Algebra of Factor Analysis

Appendix 12 B A Brief Introduction to Latent Variables in Structural Equation Modeling

13. Reliability, Validity, and Multiple-Item Scales
Assessment of Measurement Quality

Cost and Invasiveness of Measures

Empirical Examples of Reliability Assessment

Concepts from Classical Measurement Theory

Use of Multiple-Item Measures to Improve Measurement Reliability

Computation of Summated Scales

Assessment of Internal Homogeneity for Multiple-Item Measures: Cronbach’s Alpha Reliabilit Coefficient

Validity Assessment

Typical Scale Development Process

A Brief Note About Modern Measurement Theories

Reporting Reliability


Appendix 13 A The CES-D Scale

Appendix 13 B Web Resources About Psychological Measurement

14. More About Repeated Measures

Review of Assumptions for Repeated Measures ANOVA

First Example: Heart Rate/ Social Stress Study

Test for Participant by Time or Participant by Treatment Interaction

One-Way Repeated Measures Results for HR/ Social Stress Data

Testing the Sphericity Assumption

MANOVA for Repeated Measures

Results for HR and Social Stress Analysis Using MANOVA

Doubly Multivariate Repeated Measures

Mixed Model ANOVA: Between-S and Within-S Factors

Order and Sequence Effects

First Example: Order Effect as a Nuisance

Second Example: Order Effect is of Interest

Summary and Other Complex Designs

15. Structural Equation Modeling with AMOS: A Brief Introduction
What is Structural Equation Modeling?

Review of Path Models

More Complex Path Models

First Example: Mediation Structural Model

Introduction to AMOS®

Screening and Preparing Data for SEM

Specifying the SEM Model (Variable Names and Paths)

Specifying the Analysis Properties

Running the Analysis and Examining Results

Locating Bootstrapped CI Information

Sample Results for the Mediation Analysis

Selected SEM Model Terminology

SEM Goodness of Fit Indexes

Second Example: Confirmatory Factor Analysis

Third Example: Model with Both Measurement and Structural Components

Reporting SEM


16. Binary Logistic Regression
Research Situations

First Example: Dog Ownership and Odds of Death

Conceptual Basis for Binary Logistic Regression Analysis

Definition and Interpretation of Odds

A New Type of Dependent Variable: The Logit

Terms Involved in Binary Logistic Regression Analysis

Logistic Regression for First Example: Prediction of Death from Dog Ownership.

Issues in Planning and Conducting a Study

More Complex Models

Binary Logistic Regression for Second Example: Drug Dose and Sex as Predictors of Odds of Death

Comparison of Discriminant Analysis to Binary Logistic Regression


17. Additional Statistical Techniques

A Brief History of Developments in Statistics

Survival Analysis

Cluster Analyses

Time Series Analyses

Poisson and Binomial Regression for Zero-Inflated Count Data

Bayes’ Theorem

Multilevel Modeling

Some Final Words


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