Statistics for the Behavioural Sciences An Introduction to Frequentist and Bayesian Approaches

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 updatedeResources 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.

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 S 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 (s2) and the standard deviation (s) 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 s is known 7.4. Testing hypotheses about means when s 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 onlycontent) 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 nonindependentPearson correlation coefficients r 10.10. Partial correlation 10.11. Factors affecting the Pearson correlation coefficient r 10.12. The point biserial correlation rpb 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 ß 11.6. Further on the relationship between linear regression and Pearson’s r: r2 as a measure of effect size 11.7. F

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.