From Bivariate Through Multivariate Techniques

Transcription

1 A p p l i e d S T A T I S T I C S From Bivariate Through Multivariate Techniques R e b e c c a M. W a r n e r University of New Hampshire DAI HOC THAI NGUYEN TRUNG TAM HOC LIEU *)SAGE Publications '55' Los Angeles London New Delhi Singapore

2 Copyright 2008 by Sage Publications, Inc. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. For information: Sage Publications, Inc Teller Road Thousand Oaks, California Sage Publications Ltd. 1 Oliver's Yard 55 City Road London EC1Y ISP United Kingdom Sage Publications India Pvt. Ltd. B 1/11 Mohan Cooperative Industrial Area Mathura Road, New Delhi India Sage Publications Asia-Pacific Pte. Ltd. 33 Pekin Street #02-01 Far East Square Singapore Printed in the United States of America Library of Congress Cataloging-in-Publication Data Warner, Rebecca M. Applied statistics: from bivariate through multivariate techniques/rebecca M. Warner. p. cm. Includes bibliographical references and index. ISBN-13: (cloth) 1. Social sciences Statistical methods. 2. Psychology Statistical methods. 3. Multivariate analysis. I. Title. HA31.35.W '35 dc This book is printed on acid-free paper Acquisitions Editor: Associate Editor: Editorial Assistant: Production Editor: Copy Editors: Typesetter: Indexer: Cover Designer: Marketing Manager: Vicki Knight Sean Connelly Lauren Habib Laureen A. Shea Linda Gray and QuADS C&M Digitals (P) Ltd. Will Ragsdale Candice Harman Stephanie Adams

3 C o n t e n t s Preface xxj Acknowledgments xxv Chapter 1. Review of Basic Concepts Introduction A Simple Example of a Research Problem Discrepancies Between Real and Ideal Research Situations Samples and Populations Descriptive Versus Inferential Uses of Statistics Levels of Measurement and Types of Variables The Normal Distribution Research Design Experimental Design Quasi-Experimental Design Nonexperimental Research Design Between-Subjects Versus Within-Subjects or Repeated Measures Parametric Versus Nonparametric Statistics Additional Implicit Assumptions Selection of an Appropriate Bivariate Analysis Summary 29 Comprehension Questions 37 Chapter 2. Introduction to SPSS: Basic Statistics, Sampling Error, and Confidence Intervals Introduction Research Example: Description of a Sample of HR Scores Sample Mean (M) Sum of Squared Deviations and Sample Variance (s 2 ) Degrees of Freedom (df) for a Sample Variance Why Is There Variance? Sample Standard Deviation (s) Assessment of Location of a Single X Score Relative to a Distribution of Scores 59

4 2.9 A Shift in Level of Analysis: The Distribution of Values of M Across Many Samples From the Same Population An Index of Amount of Sampling Error: The Standard Error of the Mean (a M ) Effect of Sample Size (AO on the Magnitude of the Standard Error (a u ) Sample Estimate of the Standard Error of the Mean (SE M ) The Family of f Distributions Confidence Intervals The General Form of a CI Setting Up a CI for M When a Is Known Setting Up a CI for M When the Value of a Is Not Known Reporting CIs Summary 75 Appendix on SPSS 76 Comprehension Questions 77 Chapter 3. Statistical Significance Testing The Logic of Null Hypothesis Significance Testing (NHST) Type I Versus Type II Error Formal NHST Procedures: The z Test for a Null Hypothesis About One Population Mean Obtaining a Random Sample From the Population of Interest Formulating a Null Hypothesis (H 0 ) for the One-Sample z Test Formulating an Alternative Hypothesis (#,) Choosing a Nominal Alpha Level Determining the Range of z Scores Used to Reject H Determining the Range of Values of M Used to Reject H Reporting an "Exact"/) Value Common Research Practices Inconsistent With Assumptions and Rules for NHST Use of Convenience Samples Modification of Decision Rules After the Initial Decision Conducting Large Numbers of Significance Tests Impact of Violations of Assumptions on Risk of Type I Error Strategies to Limit Risk of Type I Error Use of Random and Representative Samples Adherence to the Rules for NHST Limit the Number of Significance Tests Bonferroni-Corrected Per-Comparison Alpha Levels Replication of Outcome in New Samples Cross-Validation Interpretation of Results Interpretation of Null Results Interpretation of Statistically Significant Results When Is a f Test Used Instead of a z Test? Effect Size Evaluation of "Practical" (vs. Statistical) Significance Formal Effect Size Index: Cohen's Little d 104

5 3.9 Statistical Power Analysis Numerical Results for a One-Sample t Test Obtained From SPSS Guidelines for Reporting Results Summary Logical Problems With NHST Other Applications of the t Ratio What Does It Mean to Say > <.05"? 122 Comprehension Questions 123 Chapter 4. Preliminary Data Screening Introduction: Problems in Real Data Quality Control During Data Collection Example of an SPSS Data Worksheet Identification of Errors and Inconsistencies Missing Values Empirical Example of Data Screening for Individual Variables Frequency Distribution Tables Removal of Impossible or Extreme Scores Bar Chart for a Categorical Variable Histogram for a Quantitative Variable Identification and Handling of Outliers Screening Data for Bivariate Analyses Bivariate Data Screening for Two Categorical Variables Bivariate Data Screening for One Categorical and One Quantitative Variable Bivariate Data Screening for Two Quantitative Variables Nonlinear Relations Data Transformations Verifying That Remedies Had the Desired Effects Multivariate Data Screening Reporting Preliminary Data Screening Summary and Checklist for Data Screening 176 Comprehension Questions 179 Chapter 5. Comparing Group Means Using the Independent Samples t Test Research Situations Where the Independent Samples f Test Is Used A Hypothetical Research Example Assumptions About the Distribution of Scores on the Quantitative Dependent Variable Quantitative, Approximately Normally Distributed Equal Variances of Scores Across Groups (the Homogeneity of Variance Assumption) Independent Observations Both Between and Within Groups Robustness to Violations of Assumptions Preliminary Data Screening Issues in Designing a Study Formulas for the Independent Samples t Test 191

6 5.6.1 The Pooled Variances f Test Computation of the Separate Variances t Test and Its Adjusted df Evaluation of Statistical Significance of a t Ratio Confidence Interval Around M, - M Conceptual Basis: Factors That Affect the Size of the t Ratio Design Decisions That Affect the Difference Between Group Means,M t -M Design Decisions That Affect Pooled Within-Group Variance, Design Decisions About Sample Sizes, n, and n Summary: Factors That Influence the Size of t Effect Size Indexes for r Eta Squared (if) Cohen's d Point Biserial r (rj Statistical Power and Decisions About Sample Size for the Independent Samples t Test Describing the Nature of the Outcome SPSS Output and Model Results Section Summary 209 Comprehension Questions 211 Chapter 6. One-Way Between-Subjects Analysis of Variance Research Situations Where One-Way Between-Subjects Analysis of Variance (ANOVA) Is Used Hypothetical Research Example Assumptions About Scores on the Dependent Variable for One-Way Between-S ANOVA Issues in Planning a Study Data Screening Partition of Scores Into Components Computations for the One-Way Between-S ANOVA Comparison Between the Independent Samples t Test and One-Way Between-S ANOVA Summarizing Information About Distances Between Group Means: Computing MS bamm Summarizing Information About Variability of Scores Within Groups: Computing MS^^ The F Ratio: Comparing MS^^ With MS^ Patterns of Scores Related to the Magnitudes of MS^^ and AfS wilhin Expected Value of F When H 0 Is True Confidence Intervals (CIs) for Group Means Effect-Size Index for One-Way Between-S ANOVA Statistical Power Analysis for One-Way Between-S ANOVA Nature of Differences Among Group Means Planned Contrasts Post Hoc or "Protected" Tests SPSS Output and Model Results 241

7 6.12 Summary 248 Comprehension Questions 251 Chapter 7. Bivariate Pearson Correlation Research Situations Where Pearson r Is Used Hypothetical Research Example Assumptions for Pearson r Preliminary Data Screening Design Issues in Planning Correlation Research Computation of Pearson r Statistical Significance Tests for Pearson r Testing the Hypothesis That p XY = Testing Other Hypotheses About p XY Assessing Differences Between Correlations Reporting Many Correlations: Need to Control Inflated Risk of Type I Error Limiting the Number of Correlations Cross-Validation of Correlations Bonferroni Procedure: A More Conservative Alpha Level for Tests of Individual Correlations Setting Up CIs for Correlations Factors That Influence the Magnitude and Sign of Pearson r Pattern of Data Points in the X, Y Scatter Plot Biased Sample Selection: Restricted Range or Extreme Groups Correlations for Samples That Combine Groups Control of Extraneous Variables Disproportionate Influence by Bivariate Outliers Shapes of Distributions of X and Y Curvilinear Relations Transformations of Data Attenuation of Correlation Due to Unreliability of Measurement Part-Whole Correlations Aggregated Data Pearson r and r 2 as Effect Size Indexes Statistical Power and Sample Size for Correlation Studies Interpretation of Outcomes for Pearson r "Correlation Does Not Necessarily Imply Causation" (So What Does It Imply?) Interpretation of Significant Pearson r Values Interpretation of a Nonsignificant Pearson r Value SPSS Output and Model Results Write-Up Summary 304 Comprehension Questions 305 Chapter 8. Alternative Correlation Coefficients Correlations for Different Types of Variables Two Research Examples 312

8 8.3 Correlations for Rank or Ordinal Scores Correlations for True Dichotomies Point Biserial r(r pb ) Phi Coefficient (0) Correlations for Artificially Dichotomized Variables Biserial r(r b ) Tetrachoric r(r tet ) Assumptions and Data Screening for Dichotomous Variables Analysis of Data: Dog Ownership and Survival After a Heart Attack Chi-Square Test of Association (Computational Methods for Tables of Any Size) Other Measures of Association for Contingency Tables SPSS Output and Model Results Write-Up Summary 334 Comprehension Questions 335 Chapter 9. Bivariate Regression Research Situations Where Bivariate Regression Is Used A Research Example: Prediction of Salary From Years of Job Experience Assumptions and Data Screening Issues in Planning a Bivariate Regression Study Formulas for Bivariate Regression Statistical Significance Tests for Bivariate Regression Setting Up Confidence Intervals Around Regression Coefficients Factors That Influence the Magnitude and Sign of b Factors That Affect the Size of the b Coefficient Comparison of Coefficients for Different Predictors or for Different Groups Effect Size/Partition of Variance in Bivariate Regression Statistical Power Raw Score Versus Standard Score Versions of the Regression Equation Removing the Influence of X From the Y Variable by Looking at Residuals From Bivariate Regression Empirical Example Using SPSS Information to Report From a Bivariate Regression Summary 369 Comprehension Questions 374 Chapter 10. Adding a Third Variable: Preliminary Exploratory Analyses Three-Variable Research Situations First Research Example Exploratory Statistical Analyses for Three-Variable Research Situations Separate Analysis of X v Y Relationship for Each Level of the Control Variable X Partial Correlation Between X, and Y, Controlling for X Understanding Partial Correlation as the Use of Bivariate Regression to Remove Variance Predictable by X 2 From Both X, and Y 389

9 10.7 Computation of Partial r From Bivariate Pearson Correlations Intuitive Approach to Understanding Partial r Significance Tests, Confidence Intervals, and Statistical Power for Partial Correlations Statistical Significance of Partial r Confidence Intervals for Partial r Effect Size, Statistical Power, and Sample Size Guidelines for Partial r Interpretation of Various Outcomes for r Y] 2 and r Y Two-Variable Causal Models Three-Variable Models: Some Possible Patterns of Association Among X,, Y, and X X, and YAre Not Related Whether You Control for X 2 or Not X 2 Is Irrelevant to the X,, Y Relationship When You Control for X 2, the X,, Y Correlation Drops to 0 or Close to Completely Spurious Correlation Completely Mediated Association Between X, and Y When You Control for X 2, the Correlation Between X, and Y Becomes Smaller (but Does Not Drop to 0 and Does Not Change Sign) X 2 Partly Accounts for the X,, Y Association, or X, and X 2 Are Correlated Predictors of Y X 2 Partly Mediates thex,, YRelationship When You Control for X 2, the X,, Y Correlation Becomes Larger Than r 1Y or Becomes Opposite in Sign Relative to r 1Y Suppression of Error Variance in a Predictor Variable A Second Type of Suppression "None of the Above" Mediation Versus Moderation Preliminary Analysis to Identify Possible Moderation Preliminary Analysis to Detect Possible Mediation Experimental Tests for Mediation Models Model Results Summary 419 Comprehension Questions 421 Chapter 11. Multiple Regression With Two Predictor Variables Research Situations Involving Regression With Two Predictor Variables Hypothetical Research Example Graphic Representation of Regression Plane Semipartial (or "Part") Correlation Graphic Representation of Partition of Variance in Regression With Two Predictors 428

10 11.6 Assumptions for Regression With Two Predictors Formulas for Regression Coefficients, Significance Tests, and Confidence Intervals Formulas for Standard Score Beta Coefficients Formulas for Raw Score (b) Coefficients Formula for Multiple R and Multiple R Test of Significance for Overall Regression: Overall F Test for H 0 : R = Test of Significance for Each Individual Predictor: f Test for H 0 : b- = Confidence Interval for Each b Slope Coefficient SPSS Regression Results Conceptual Basis: Factors That Affect the Magnitude and Sign of P and b Coefficients in Multiple Regression With Two Predictors Tracing Rules for Causal Model Path Diagrams Comparison of Equations for p\ b, pr, and sr Nature of Predictive Relationships Effect Size Information in Regression With Two Predictors Effect Size for Overall Model Effect Size for Individual Predictor Variables Statistical Power Issues in Planning a Study Sample Size Selection of Predictor Variables Multicollinearity Among Predictors Range of Scores Use of Regression With Two Predictors to Test Mediated Causal Models Results Summary 458 Comprehension Questions 462 Chapter 12. Dummy Predictor Variables and Interaction Terms in Multiple Regression Research Situations Where Dummy Predictor Variables Can Be Used Empirical Example Screening for Violations of Assumptions Issues in Planning a Study Parameter Estimates and Significance Tests for Regressions With Dummy Variables Group Mean Comparisons Using One-Way Between-S ANOVA Gender Differences in Mean Salary College Differences in Mean Salary Three Methods of Coding for Dummy Variables Regression With Dummy-Coded Dummy Predictor Variables Two-Group Example With a Dummy-Coded Dummy Variable Multiple-Group Example With Dummy-Coded Dummy Variables 481