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Applied Statistics I
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Applied Statistics I
Basic Bivariate Techniques

Third Edition


January 2020 | 672 pages | SAGE Publications, Inc
Applied Statistics I: Basic Bivariate Techniques has been created from the first half of Rebecca M. Warner's popular Applied Statistics: From Bivariate Through Multivariate Techniques. The author's contemporary approach differs from some of the well-worn texts in the market, and reflects current thinking in the field. It spends less time on statistical significance testing, and moves in the direction of the "new statistics" by focusing more on confidence intervals and effect size. Instructors of upper undergraduate or beginning graduate level courses will find that the greater focus on basic concepts such as partition of variance and effect size is more useful to students, particularly as preparation for more advanced courses. Spending less time on statistical significance testing allows for more time to be devoted to more interesting and useful statistics that students will see in journal articles (such as correlation and regression). This introductory statistics text includes examples in SPSS, together with datasets on an accompanying website. A companion study guide reproducing the exercises and examples in R will also be available.
 
1. Evaluating Numeric Information
Introduction

 
Guidelines for Numeracy

 
Source Credibility

 
Message Content

 
Evaluating Generalizability

 
Making Causal Claims

 
Quality Control Mechanisms in Science

 
Biases of Information Consumers

 
Ethical Issues in Data Collection and Analysis

 
Lying with Graphs and Statistics

 
Degrees of Belief

 
Summary

 
 
2. Basic Research Concepts
Introduction

 
Types of Variables

 
Independent and Dependent Variables

 
Typical Research Questions

 
Conditions for Causal Inference

 
Experimental Research Design

 
Non-experimental Research Design

 
Quasi- Experimental Designs

 
Other Issues in Design and Analysis

 
Choice of Statistical Analysis (Preview)

 
Populations and Samples: Ideal Versus Actual Situations

 
Common Problems in Interpretation of Results

 
Appendix 2 A: More About Levels of Measurement

 
Appendix 2 B: Justification for Use of Likert and Other Rating Scales as Quantitative Variables (In Some Situations)

 
 
3. Frequency Distribution Tables
Introduction

 
Use of Frequency Tables for Data Screening

 
Frequency Tables for Categorical Variables

 
Elements of Frequency Tables

 
Using SPSS to Obtain a Frequency Table

 
Mode, Impossible Score Values, and Missing Values

 
Reporting Data Screening for Categorical Variables

 
Frequency Tables for Quantitative Variables

 
Frequency Tables for Categorical Versus Quantitative Variables

 
Reporting Data Screening for Quantitative Variables

 
What We Hope to See in Frequency Tables for Categorical Variables

 
What We Hope to See in Frequency Tables for Quantitative Variables

 
Summary

 
Appendix 3 A: Getting Started in IBM SPSS ® version 25

 
Appendix 3 B: Missing Values in Frequency Tables

 
Appendix 3 C: Dividing Scores into Groups or Bins

 
 
4. Descriptive Statistics
Introduction

 
Questions about Quantitative Variables

 
Notation

 
Sample Median

 
Sample Mean (M)

 
An Important Characteristic of M: Sum of Deviations from M = 0

 
Disadvantage of M: It is Not Robust Against Influence of Extreme Scores

 
Behavior of Mean, Median and Mode in Common Real-World Situations

 
Choosing Among Mean, Median, and Mode

 
Using SPSS to Obtain Descriptive Statistics for a Quantitative Variable

 
Minimum, Maximum, and Range: Variation among Scores

 
The Sample Variance s2

 
Sample Standard Deviation (s or SD)

 
How a Standard Deviation Describes Variation Among Scores in a Frequency Table

 
Why Is There Variance?

 
Reports of Descriptive Statistics in Journal Articles

 
Additional Issues in Reporting Descriptive Statistics

 
Summary

 
Appendix 4 A Order of Arithmetic Operations

 
Appendix 4 B Rounding

 
 
5. Graphs: Bar Charts, Histograms, and Box Plots
Introduction

 
Pie Charts for Categorical Variables

 
Bar Charts for Frequencies of Categorical Variables

 
Good Practice for Construction of Bar Charts

 
Deceptive Bar Graphs

 
Histograms for Quantitative Variables

 
Obtaining a Histogram Using SPSS

 
Describing and Sketching Bell-Shaped Distributions

 
Good Practices in Setting up Histograms

 
Box Plot (Box and Whiskers Plot)

 
Telling Stories About Distributions

 
Uses of Graphs in Actual Research

 
Data Screening: Separate Bar Charts or Histograms for Groups

 
Use of Bar Charts to Represent Group Means

 
Other Examples

 
Summary

 
 
6. The Normal Distribution and z Scores
Introduction

 
Locations of Individual Scores in Normal Distributions

 
Standardized or “z” Scores

 
Converting z Scores Back into Original Units of X

 
Understanding Values of z

 
Qualitative Description of Normal Distribution Shape

 
More Precise Description of Normal Distribution Shape

 
Reading Tables of Areas for the Standard Normal Distribution

 
Dividing the Normal Distribution Into Three Regions: Lower Tail, Middle, Upper Tail

 
Outliers Relative to a Normal Distribution

 
Summary of First Part of Chapter

 
Why We Assess Distribution Shape

 
Departure from Normality: Skewness

 
Another Departure from Normality: Kurtosis

 
Overall Normality

 
Practical Recommendations

 
Reporting Information About Distribution Shape, Missing Values, Outliers, and Descriptive Statistics for Quantitative Variables

 
Summary

 
Appendix 6 A: The Mathematics of the Normal Distribution

 
Appendix 6 B: How to Select and Remove Outliers in SPSS

 
Appendix 6 C: Quantitative Assessments of Departure from Normality

 
Appendix 6 D: Why Are Some Real-World Variables Approximately Normally Distributed?

 
 
7. Sampling Error and Confidence Intervals
Descriptive Versus Inferential Uses of Statistics

 
Notations for Samples Versus Populations

 
Sampling Error and the Sampling Distribution for Values of M

 
Prediction Error

 
Sample Versus Population (Revisited)

 
The Central Limit Theorem: Characteristics of the Sampling Distribution of M

 
Factors that Influence Population Standard Error

 
Effect of N on Value of the Population Standard Error

 
Describing the Location of a Single Outcome for M Relative to a Population Sampling Distribution (Setting Up a z Ratio)

 
What We Do When ?? Is Unknown

 
The Family of t Distributions

 
Tables for t Distributions

 
Using Sampling Error to Set Up a Confidence Interval

 
How to Interpret a Confidence Interval

 
Empirical Example: Confidence Interval for Body Temperature

 
Other Applications for CIs

 
Error Bars in Graphs of Group Means

 
Summary

 
 
8. The One-Sample t test: Introduction to Statistical Significance Tests
Introduction

 
Significance Tests as Yes/No Questions About Proposed Values of Population Means

 
Stating a Null Hypothesis

 
Selecting an Alternative Hypothesis

 
The One-Sample t Test

 
Choosing an Alpha (?) Level

 
Specifying Reject Regions Based on ?, Halt and df

 
Questions for the One-Sample t Test

 
Assumptions for the Use of the One-Sample t Test

 
Rules for the Use of NHST

 
First Example: Mean Driving Speed (Nondirectional Test)

 
SPSS Analysis: One Sample t Test for Mean Driving Speed

 
“Exact” p Values

 
Reporting Results for a Two-tailed One-Sample t Test

 
The Driving Speed Data Reconsidered Using a One-Tailed Test

 
Reporting Results for a One-tailed One-Sample t Test:

 
Advantages/ Disadvantages of One Tailed Tests

 
Traditional NHST Versus New Statistics Recommendations

 
Things You Should Not Say About p Values

 
Summary

 
 
9. Issues in Significance Tests: Effect Size, Statistical Power, and Decision Errors
Beyond p Values

 
Cohen’s d: An Effect Size Index

 
Factors that Affect the Size of t Ratios

 
Statistical Significance Versus Practical Importance

 
Statistical Power

 
Type I and Type II Decision Errors

 
Meanings of “Error”

 
Use of NHST in Exploratory Versus Confirmatory Research

 
Inflated Risk of Type I Error From Multiple Tests Interpretation of Null Outcomes

 
Interpretation of Null Outcomes

 
Interpretation of Statistically Significant Outcomes

 
Understanding Past Research

 
Planning Future Research

 
Guidelines for Reporting Results

 
What You Cannot Say

 
Summary

 
Appendix 9 A Further Explanation of Statistical Power

 
 
10. Bivariate Pearson Correlation
Research Situations Where Pearson r Is Used

 
Correlation and Causal Inference

 
How Sign and Magnitude of r Describe an X, Y Relationship

 
Setting Up Scatter Plots With Examples of Perfect Linearity

 
Most Associations Are Not Perfect

 
Different Situations In Which r = 0

 
Assumptions for Use of Pearson r

 
Preliminary Data Screening for Pearson r

 
Effect of Extreme Bivariate Outliers

 
Research Example

 
Data Screening for Research Example

 
Computation of Pearson r

 
How Computation for Correlation Is Related to Pattern of Data Points in the Scatter Plot

 
Testing the Hypothesis That ?0 = 0

 
Reporting Many Correlations and Inflated Risk of Type I Error

 
Obtaining CIs for Correlations

 
Pearson’s r and r2 as Effect-Size Indexes and Partition of Variance

 
Statistical Power and Sample Size for Correlation Studies

 
Interpretation of Outcomes for Pearson’s r

 
SPSS Example

 
Results Sections for One and Several Pearson r Values

 
Reasons to Be Skeptical of Correlations

 
Summary

 
Appendix 10 A: Nonparametric Alternatives to Pearson r

 
Appendix 10 B: Setting Up a 95% CI for Pearson r

 
Appendix 10 C: Testing Significance of Differences Between Correlations

 
Appendix 10 D: Factors That Artifactually Influence the Magnitude of Pearson’s r

 
Appendix 10 E: Analysis of Non Linear Relationships

 
 
11. Bivariate Regression
Research Situations Where Bivariate Regression is Used

 
New Information Provided by Regression

 
Regression Equations and Lines

 
Two Versions of Regression Equations

 
Steps in Regression Analysis

 
Preliminary Data Screening

 
Formulas for Bivariate Regression Coefficients

 
Statistical Significance Tests for Bivariate Regression

 
Confidence Intervals for Regression Coefficients

 
Effect Size and Statistical Power

 
Empirical Example Using SPSS: Salary Data

 
SPSS Output: Salary Data

 
Plotting the Regression Line: Salary Data

 
Results Section: Salary Data

 
Using Regression Equation to Predict Score for Individual: Joe’s Hr Data

 
Partition of SS in Bivariate Regression: Joe’s Hr Data

 
Issues in Planning a Bivariate Regression Study

 
Plotting Residuals

 
Standard Error of the Estimate, sy.x

 
Summary

 
Appendix 11 A OLS Derivation of Equation for Regression Coefficients

 
Appendix 11 B Fully Worked Example for SS values: Joe’s HR Data

 
 
12. The Independent Samples t Test
Research Situations Where the Independent Samples t Test is Used

 
Hypothetical Research Example

 
Assumptions for Use of the Independent Samples t Test

 
Preliminary Data Screening: Evaluating Violations of Assumptions and Getting to Know Your Data

 
Computation of Independent Samples t Test

 
Statistical Significance of Independent Samples t Test

 
Confidence Interval Around (M1 – M2)

 
SPSS Commands for Independent Samples t Test

 
SPSS Output for Independent Samples t Test

 
Effect-Size Indexes for t

 
Factors that Influence the Size of t

 
Results Section

 
Graphing Results: Means and CIs

 
Decisions About Sample Size for the Independent Samples t Test

 
Issues in Designing a Study

 
Summary

 
Appendix 12 A: A Nonparametric Alternative to the Independent Samples t Test

 
 
13. One-Way Between-S Analysis of Variance
Research Situations Where Between-S One-Way ANOVA is Used

 
Questions in One-Way Between S ANOVA

 
Hypothetical Research Example

 
Assumptions and Data Screening for One-Way ANOVA

 
Computations for One-Way Between-S ANOVA

 
Patterns of Scores and Magnitudes of SSbetween and SSwithin

 
Confidence Intervals (CIs) For Group Means

 
Effect Sizes for One-Way Between-S ANOVA

 
Statistical Power Analysis for One-Way Between-S ANOVA

 
Planned Contrasts

 
Post Hoc or “Protected” Tests

 
One Way Between S ANOVA Procedure in SPSS

 
Output from SPSS for One Way Between S ANOVA

 
Reporting Results from One Way Between S ANOVA

 
Issues in Planning a Study

 
Summary

 
Appendix A ANOVA Model and Division of Scores Into Components

 
Appendix B Expected Value of F When H0 is True

 
Appendix C Comparison of ANOVA to t Test

 
Appendix D Nonparametric Alternative to One Way Between S ANOVA

 
 
14. Paired Samples t-Test
Independent Versus Paired Samples Designs

 
Between-S and Within-S or Paired Groups Designs

 
Types of Paired Samples

 
Hypothetical Study: Effects of Stress on Heart Rate

 
Review: Data Organization for Independent Samples

 
New: Data Organization for Paired Samples

 
A First Look at Repeated Measures Data

 
Calculation of Difference (d) Scores

 
Null Hypothesis for Paired Samples t Test

 
Assumptions for Paired Samples t Test

 
Formulas for Paired Samples t Test

 
SPSS Paired Samples t Test Procedure

 
Comparison of Results For Independent Samples t and Paired Samples t Tests

 
Effect Size and Power

 
Some Design Problems in Repeated Measures Designs

 
Results for Paired Samples t-Test: Stress and HR

 
Further Evaluation of Assumptions for Larger Dataset

 
Summary

 
Appendix A Nonparametric Alternative to Paired Samples t: Wilcoxon Signed Rank Test

 
 
15. One Way Repeated Measures ANOVA
Introduction

 
Null Hypothesis for Repeated Measures ANOVA

 
Preliminary Assessment of Repeated Measures Data

 
Computations for One-Way Repeated Measures ANOVA

 
Use of SPSS Reliability Procedure for One Way Repeated Measures ANOVA

 
Partition of SS in Between-S Versus Within-S ANOVA

 
Assumptions for Repeated Measures ANOVA

 
Choices of Contrasts in GLM Repeated Measures

 
SPSS GLM Procedure for Repeated Measures ANOVA

 
Output for GLM Repeated Measures ANOVA

 
Paired Samples t Tests as Follow Up

 
Results

 
Effect Size

 
Statistical Power

 
Counterbalancing in Repeated Measures Studies

 
More Complex Designs

 
Summary

 
Appendix 15 A Test for Person by Treatment Interaction

 
 
16. Factorial Analysis of Variance (Between – S)
Research Situations Where Factorial Design Is Used

 
Questions in Factorial ANOVA

 
Null Hypotheses in Factorial ANOVA

 
Screening for Violations of Assumptions

 
Hypothetical Research Situation

 
Computations for Between-S Factorial ANOVA

 
Computation of SS, df, and MS in Two Way Factorial

 
Effect Size Estimates for Factorial ANOVA

 
Statistical Power

 
Follow-Up Tests

 
Factorial ANOVA Using the SPSS GLM Procedure

 
SPSS Output

 
Results

 
Design Decisions and Magnitudes of SS Terms

 
Summary

 
Appendix 16 A: Unequal Cell ns in Factorial ANOVA

 
Appendix 16 B: Weighted Versus Unweighted Means

 
Appendix 16 C: Model for Factorial ANOVA

 
Appendix 16 D: Fixed Versus Random Factors

 
 
17. Chi Square Analysis of Contingency Tables
Evaluating Association Between Two Categorical Variables

 
First Example: Contingency Tables for Titanic Data

 
What is Contingency?

 
Conditional and Unconditional Probabilities

 
Null Hypothesis for Contingency Table Analysis

 
Second Empirical Example: Dog Ownership Data

 
Preliminary Examination of Dog Ownership Data

 
Expected Cell Frequencies If H0 True

 
Computation of Chi Squared Significance Test

 
Evaluation of Statistical Significance of ?2.

 
Effect Sizes for Chi Squared

 
Chi Squared Example Using SPSS

 
Output from Crosstabs Procedure

 
Reporting Results

 
Assumptions and Data Screening For Contingency Tables

 
Other Measures of Association for Contingency Tables

 
Summary

 
Appendix 17 A: Margin of Error For Percentages in Surveys

 
Appendix 17 B: Contingency Tables With Repeated Measures: McNemar Test

 
Appendix 17 C: Fisher Exact Test

 
Appendix 17 D: How Marginal Distributions for X and Y Constrain Maximum Value of ??

 
Appendix 17 E: Other Uses of ?2

 
 
18. Selection of Bivariate Analyses and Review of Key Concepts
Selecting Appropriate Bivariate Analyses

 
Types of Independent and Dependent Variables (Categorical Versus Quantitative)

 
Parametric Versus Nonparametric Analyses

 
Comparisons of Means or Medians Across Groups (Categorical IV and Quantitative DV)

 
Problems with Selective Reporting of Evidence and Analyses

 
Limitations of Statistical Significance Tests and p Values

 
Statistical Versus Practical Significance

 
Generalizability Issues

 
Causal Inference

 
Results Sections

 
Beyond Bivariate Analyses: Adding Variables

 
Some Multivariable or Multivariate Analyses

 
Degrees of Belief

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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