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Applied Statistics I - International Student Edition
Basic Bivariate Techniques

Third Edition

- Rebecca M. Warner - University of New Hampshire, USA

February 2020 | 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