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Applied Statistics II
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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

 
Summary

 
 
2. Advanced Data Screening: Outliers and Missing Values
Introduction

 
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

 
Summary

 
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

 
Summary

 
 
4. Regression Analysis and Statistical Control
Introduction

 
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

 
Results

 
Summary

 
 
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

 
Summary

 
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

 
Results

 
Summary

 
 
7. Moderation: Interaction in Multiple Regression
Terminology

 
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

 
Summary

 
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

 
Summary

 
 
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

 
Results

 
One-Way ANOVA on Scores on Discriminant Functions

 
Summary

 
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

 
Summary

 
 
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

 
Summary

 
Appendix 13 A The CES-D Scale

 
Appendix 13 B Web Resources About Psychological Measurement

 
 
14. More About Repeated Measures
Introduction

 
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

 
Summary

 
 
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

 
Summary

 
 
17. Additional Statistical Techniques
Introduction

 
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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