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Statistics for the Behavioural Sciences

An Introduction to Frequentist and Bayesian Approaches

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## Book Description

This accessible textbook is for those without a mathematical background (just some notions of basic algebra are sufficient) and provides a comprehensive introduction to all topics covered in introductory behavioural science statistics courses. It includes plenty of real examples to demonstrate approaches in depth based on real psychology experiments utilizing the statistical techniques described.

New content in this thoroughly updated second edition includes an introduction to Bayesian statistics which complements the coverage of Classical/Frequentist statistics present in the first edition. It also offers practical details on how to perform analyses using JASP – a globally employed, freely downloadable statistical package. The updated eResources also feature a range of new material including additional exercises so readers can test themselves on what they have learned in the book.

This timely and highly readable text will be invaluable to undergraduate students of psychology and research methods courses in related disciplines, as well as anyone with an interest in understanding and applying the basic concepts and inferential techniques associated with statistics in the behavioural sciences.

## Table of Contents

** **

Preface edition 1

Acknowledgements edition I

Preface edition II

Acknowledgements edition II

0. Mathematics and Algebra: A Rapid-Mini Review

0.1. Operators and symbols

0.2. Orders of operations

0.3. Dealing with fractions

0.4. Variables, constants and equations

0.5. Graphs and equations

0.6. How to solve an equation with one unknown

1. Introduction and basic concepts

1.1. Why is statistics useful in the behavioural sciences?

1.2. Simple example of statistical teseting

1.3. Descriptive and inferential statistics

1.4. Descriptive and inferential statistics

1.5 What is an experiment?

1.6 Correlational studies

1.7 Irrelevant variables

2. Descriptive statistics

2.1. Organising raw data

2.2. Frequency distributions and histograms

2.3. Grouped data

2.4. Stem-and-leaf diagrams

2.5. Summarising data

2.6. Measures of central tendency: Mode, median, and mean

2.7. Advantages and disadvantages of mode, median, and mean

2.8. A useful digression on the Σ notation

2.9. Measures of dispersion (or variability)

2.10. Further on the mean, variance, and standard deviation of frequency distributions

2.11. How to calculate the combined mean and the combined variance of several samples (Web only content)

2.12. Properties of estimators

2.13. Mean and variance of linearly transformed data

2.14 Using JASP for data analysis: Descriptive statistics

3. Introduction to probability

3.1. Why are some notions of probability useful?

3.2. Some preliminary definitions and the concept of probability

3.3. Venn diagrams and probability

3.4. The addition rule and the multiplication rule of probability

3.5. Probability trees

3.6. Conditional probability

3.7. Independence and conditional probability

3.8. Bayes’s Theorem

4. Introduction to inferential statistics

4.1. Inferential statistics and probability

4.2. The Classical/Frequentist approach to inferential statistics

4.3. How the inferential statistic process operates in frequentist statistics

4.4. Reducing the risk of false positives

4.5. The risk of making false negative errors

4.6. Estimating the magnitude of the size of the parameter associated to the theory

4.7. Confidence intervals and inferential statistics.

4.8. The Bayesian approach to inferential statistics

4.9. Odds, probabilities and how to update probabilities

4.10. Chickenpox or Smallpox? This is the dilemma. Bayesian inference in practice.

4.11. The Bayes Factor: The Bayesian equivalent of significance testing

4.12. The Bayes Factor in practice

4.13. Computing the BF and interpreting its function in statistical inference

4.14. Estimating the magnitude of the size of the parameter associated to the theory: Credible intervals

4.15. Frequentist and Bayesian approaches to statistical inference: A rough comparison

5. Probability distributions and the binomial distribution

5.1. Introduction

5.2. Probability distributions

5.3. Calculating the mean (*μ*) of a probability distribution

5.4. Calculating the variance (*σ*^{2}) and the standard deviation (*σ*) of a probability distribution

5.5. Orderings (or permutations)

5.6. Combinations

5.7. The binomial distribution

5.8. Mean and variance of the binomial distribution

5.9. How to use the binomial distribution in testing hypotheses: The Frequentist approach

5.10. The sign test

5.11. Further on the binomial distribution and its use in hypothesis testing (Web only content)

5.12. Using JASP to conduct the binomial test (Frequentist approach)

5.13. The Bayesian binomial test

5.14. Using JASP to conduct the binomial test (Bayesian approach)

5.15. The selection of the prior

6. Continuous random variables and the normal distribution

6.1. Introduction

6.2. Continuous random variables and their distribution

6.3. The normal distribution

6.4. The standard normal distribution

6.5. Hypothesis testing and the normal distribution: The Frequentist approach

6.6. Type I and Type II errors

6.7. One-tailed and two-tailed statistical tests

6.8. Hypothesis testing and the normal distribution: The Bayesian approach

6.9. Using the normal distribution as an approximation of the binomial distribution (Web only content)

**7. Sampling distribution of the mean, its use in hypothesis testing and the one-sample t-test (Frequentist approach)**

7.1. Introduction

7.2. The sampling distribution of the mean and the Central Limit Theorem

7.3. Testing hypotheses about means when *σ* is known

7.4. Testing hypotheses about means when *σ* is unknown: The Student’s *t*-distribution and the one-sample *t*-test

7.5. Two-sided confidence intervals for a population mean: Estimating the size of the population mean**.**

7.6. A fundamental conceptual equation in frequentist data analysis: Magnitude of a significance test = Size of the effect × Size of the study

7.7. Statistical power analysis: A brief introduction and its application to the one-sample* t*-test

7.8. Power calculations for the one-sample *t*-test

7.9. Using JASP to conduct the one sample t-test (Frequentist approach)

8. Comparing a pair of means: the matched- and the independent-samples *t*-test (Frequentist approach)

8.1. Introduction

8.2. The matched-samples *t*-test

8.3. Confidence intervals for a population mean

8.4. Counterbalancing

8.5. The sampling distribution of the difference between pairs of means and the independent-samples *t*-test

8.6 The independent-samples *t*-test

8.7. An application of the independent-samples *t*-test

8.8. Confidence intervals for the difference between two population means

8.9. The robustness of the independent-samples* t*-test

8.10. An example of the violation of the assumption of homogeneity of variances (Web only content)** **

8.11. Ceiling and floor effects

8.12. Matched-samples or independent-samples *t*-test: Which of these two tests should

be used?

8.13. A fundamental conceptual equation in data analysis: Magnitude of a significance test = Size of the effect × Size of the study

8.14. Power analysis for the independent-samples and the paired-samples *t*-test

8.15. Using JASP to conduct the paired and the independent sample *t*-test (Frequentist approach)

9. The Bayesian approach to the *t-*test

9.1. Introduction

9.2. An illustration of how to calculate the Bayes Factor for the one-sample *t*-test case

9.3. Credible intervals (i.e. the Bayesian version of Frequentist confidence intervals)

9.4. Using JASP to perform the one-sample *t*-test and the selection of the distribution to model your prior

9.5. JASP in practice: The Bayesian one-sample *t*-test

9.6. JASP in practice: The Bayesian paired-samples *t*-test

9.7. JASP in practice: The Bayesian independent-samples *t*-test

9.8. Bayesian *t*-test using Dienes’ calculator

10. Correlation

10.1. Introduction

10.2. Linear relationships between two continuous variables

10.3. More on linear relationships between two variables

10.4. The covariance between two variables

10.5. The Pearson product-moment correlation coefficient *r*

10.6. Hypothesis testing on the Pearson correlation coefficient *r *

10.7. Confidence intervals for the Pearson correlation coefficient

10.8. Testing the significance of the difference between two independent Pearson

correlation coefficients *r*

10.9. Testing the significance of the difference between two nonindependent

Pearson correlation coefficients *r*

10.10. Partial correlation

10.11. Factors affecting the Pearson correlation coefficient *r *

10.12. The point biserial correlation *r*_{pb}

10.13. The Spearman Rank correlation coefficient

10.14. Kendall’s coefficient of concordance *W*

10.15. Power calculation for correlation coefficients

10.16. Power calculation for the difference between two independent Pearson correlation coefficients *r*** **

10.17. Using JASP to perform correlation analyses (Frequentist approach)

10.19. Using JASP to perform correlation analyses (Bayesian approach)

11. Regression

11.1. Introduction

11.2. The regression line

11.3. Linear regression and correlation

11.4. Hypothesis testing on the slope *b*

11.5. Confidence intervals for the population regression slope *β*

11.6. Further on the relationship between linear regression and Pearson’s *r*: *r*^{2} as a measure of effect size

11.7. Further on the error of prediction

11.8. Why the term regression?

11.9. Using JASP to conduct a Linear regression analysis (Frequentist approach)

11.10. Using JASP to conduct a Linear regression analysis (Bayesian approach)

**12. The chi-square distribution and the analysis of categorical data **

12.1. Introduction

12.2. The chi-square (*χ*^{2}) distribution

12.3. The Pearson’s chi-square test

12.4. The Pearson’s *χ*^{2} goodness of fit test

12.5. Pearson’s *χ*^{2 }test used in assessing how well the distribution of a set of data fits a prescribed distribution (Web only content)

12.6. Further on the goodness of fit test (Web only content)

12.7. Assumptions underlying the use of Pearson's *χ*^{2 }test

12.8. Compacting a set of data for the goodness of fit test

12.9. Pearson’s *χ*^{2} test and the analysis of 2 × 2 contingency tables

12.10. Further on the degrees of freedom and the calculation of the expected frequencies

for any contingency table

12.11. The analysis of *R* × *C* contingency tables

12.12. One- and two-tailed tests

12.13. How to measure the strength of the association between variables in a contingency

table

12.14. A fundamental conceptual equation in data analysis: Magnitude of a significance

test = Size of the effect × Size of the study

12.15. The odds ratio and the analysis of 2 × 2 contingency tables

12.16. An important note on the inclusion of non-occurrences in contingency tables

12.17. The analysis of contingency tables using JASP (Frequentist approach)

12.17. The analysis of contingency tables using JASP (Bayesian approach)

13. Statistical tests on proportions (Web only content)

13.1. Introduction

13.2. Statistical tests on the proportion of successes in a sample

13.3. Confidence intervals for population proportions

13.4. Statistical tests on the difference between the proportions of successes from

two independent samples

13.5. Confidence intervals for the difference between two independent population

proportions

13.6. Power calculation for a single proportion

13.7. Power calculation for the difference between two independent proportions

13.8. Statistical tests on the difference between nonindependent proportions of

successes (McNemar test)

14. Nonparametric statistical tests (Web only content)

14.1. Introduction

14.2. The Wilcoxon matched-pairs signed-ranks test

14.3. The Wilcoxon rank-sum test

Appendix

References

Index

## Author(s)

### Biography

**Riccardo Russo** is a Professor of Psychology at the University of Essex, UK, and the University of Pavia, Italy. His research interests vary in applied and theoretical areas of cognitive psychology and cognitive neuroscience.

## Reviews

Praise from the Previous Edition:

"This is an outstanding introductory text that will appeal to instructors for its attention to detail, and to students for its clarity."-Thom Baguley, Loughborough University"This book explains how students can understand the basic concepts of statistical inference. I enjoyed reading it and would definitely recommend it to my students."-Naz Derakshan, University of Leeds"I like this book a lot and would recommend it to my undergraduate students."-David Clark-Carter, Staffordshire University