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Statistical Thinking for the 21st Century
A Systematic Way of Thinking About Data
Russell A. Poldrack
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Front Matter
Colophon
Preface
1
Introduction
1.1
What is statistical thinking?
1.2
Dealing with statistics anxiety
1.3
What can statistics do for us?
1.4
The big ideas of statistics
1.4.1
Learning from data
1.4.2
Aggregation
1.4.3
Uncertainty
1.4.4
Sampling from a population
1.5
Causality and statistics
1.6
Learning objectives
1.7
Suggested readings
2
Working with Data
2.1
What are data?
2.1.1
Qualitative data
2.1.2
Quantitative data
2.1.2.1
Types of numbers
2.2
Discrete versus continuous measurements
2.3
What makes a good measurement?
2.3.1
Reliability
2.3.2
Validity
2.4
Learning Objectives
2.5
Suggested readings
2.6
Appendix
2.6.1
Scales of measurement
3
Summarizing Data
3.1
Why summarize data?
3.2
Summarizing data using tables
3.2.1
Frequency distributions
3.2.2
Cumulative distributions
3.2.3
Plotting histograms
3.2.4
Histogram bins
3.3
Idealized representations of distributions
3.3.1
Skewness
3.3.2
Long-tailed distributions
3.4
Learning objectives
3.5
Suggested readings
4
Data Visualization
4.1
Anatomy of a plot
4.2
Principles of good visualization
4.2.1
Show the data and make them stand out
4.2.2
Maximize the data/ink ratio
4.2.3
Avoid chartjunk
4.2.4
Avoid distorting the data
4.3
Accommodating human limitations
4.3.1
Perceptual limitations
4.4
Correcting for other factors
4.5
Learning objectives
4.6
Suggested readings and videos
5
Fitting Models to Data
5.1
What is a model?
5.2
Statistical modeling: An example
5.2.1
Improving our model
5.3
What makes a model "good"?
5.4
Can a model be too good?
5.5
Summarizing data using the mean
5.5.1
Summarizing data robustly using the median
5.6
The mode
5.7
Variability: How well does the mean fit the data?
5.8
Using simulations to understand statistics
5.9
Z-scores
5.9.1
Interpreting Z-scores
5.9.2
Standardized scores
5.9.3
Using Z-scores to compare distributions
5.10
Learning objectives
5.11
Appendix
5.11.1
Proof that the sum of errors from the Mean is zero
6
Probability
6.1
What is probability?
6.2
How do we determine probabilities?
6.2.1
Personal belief
6.2.2
Empirical frequency
6.2.3
Classical probability
6.2.4
Solving de Méré’s problem
6.3
Probability distributions
6.3.1
Cumulative probability distributions
6.4
Conditional probability
6.5
Computing conditional probabilities from data
6.6
Independence
6.7
Reversing a conditional probability: Bayes’ rule
6.8
Learning from data
6.9
Odds and odds ratios
6.10
What do probabilities mean?
6.11
Learning objectives
6.12
Suggested readings
6.13
Appendix
6.13.1
Derivation of Bayes’ rule
7
Sampling
7.1
How do we sample?
7.2
Sampling error
7.3
Standard error of the mean
7.4
The Central Limit Theorem
7.5
Learning objectives
7.6
Suggested readings
8
Resampling and Simulation
8.1
Monte Carlo simulation
8.2
Randomness in statistics
8.3
Generating random numbers
8.4
Using Monte Carlo simulation
8.5
Using simulation for statistics: The bootstrap
8.5.1
Computing the bootstrap
8.6
Learning objectives
8.7
Suggested readings
9
Hypothesis Testing
9.1
Null Hypothesis Statistical Testing (NHST)
9.2
Null hypothesis statistical testing: An example
9.3
The process of null hypothesis testing
9.3.1
Step 1: Formulate a hypothesis of interest
9.3.2
Step 2: Specify the null and alternative hypotheses
9.3.3
Step 3: Collect some data
9.3.4
Step 4: Fit a model to the data and compute a test statistic
9.3.5
Step 5: Determine the probability of the observed result under the null hypothesis
9.3.5.1
P-values: A very simple example
9.3.5.2
Computing p-values using the
t
distribution
9.3.5.3
Computing p-values using randomization
9.3.5.4
Randomization: a simple example
9.3.5.5
Randomization: BMI/activity example
9.3.6
Step 6: Assess the “statistical significance” of the result
9.3.6.1
Hypothesis testing as decision-making: The Neyman-Pearson approach
9.4
What does a significant result mean?
9.4.1
Does it mean that the probability of the null hypothesis being true is .01?
9.4.2
Does it mean that the probability that you are making the wrong decision is .01?
9.4.3
Does it mean that if you ran the study again, you would obtain the same result 99% of the time?
9.4.4
Does it mean that you have found a practically important effect?
9.5
NHST in a modern context: Multiple testing
9.6
Learning objectives
9.7
Suggested readings
10
Quantifying Effects and Designing Studies
10.1
Confidence intervals
10.1.1
Confidence intervals using the normal distribution
10.1.2
Confidence intervals using the t distribution
10.1.3
Confidence intervals and sample size
10.1.4
Computing confidence intervals using the bootstrap
10.1.5
Relation of confidence intervals to hypothesis tests
10.2
Effect sizes
10.2.1
Cohen’s D
10.2.2
Pearson’s r
10.2.3
Odds ratio
10.3
Statistical power
10.3.1
Power analysis
10.4
Learning objectives
10.5
Suggested readings
11
Bayesian Statistics
11.1
Generative models
11.2
Bayes’ theorem and inverse inference
11.3
Doing Bayesian estimation
11.3.1
Specifying the prior
11.3.2
Collect some data
11.3.3
Computing the likelihood
11.3.4
Computing the marginal likelihood
11.3.5
Computing the posterior
11.4
Estimating posterior distributions
11.4.1
Specifying the prior
11.4.2
Collect some data
11.4.3
Computing the likelihood
11.4.4
Computing the marginal likelihood
11.4.5
Computing the posterior
11.4.6
Maximum a posteriori (MAP) estimation
11.4.7
Credible intervals
11.4.8
Effects of different priors
11.5
Choosing a prior
11.6
Bayesian hypothesis testing
11.6.1
Bayes factors
11.6.2
Bayes factors for statistical hypotheses
11.6.2.1
One-sided tests
11.6.2.2
Interpreting Bayes Factors
11.6.3
Assessing evidence for the null hypothesis
11.7
Learning objectives
11.8
Suggested readings
11.9
Appendix
11.9.1
Rejection sampling
12
Modeling Categorical Relationships
12.1
Example: Candy colors
12.2
Pearson’s chi-squared test
12.3
Contingency tables and the two-way test
12.4
Standardized residuals
12.5
Odds ratios
12.6
Bayes factor
12.7
Categorical analysis beyond the 2 X 2 table
12.8
Beware of Simpson’s paradox
12.9
Learning objectives
12.10
Additional readings
13
Modeling Continuous Relationships
13.1
An example: Hate crimes and income inequality
13.2
Is income inequality related to hate crimes?
13.3
Covariance and correlation
13.3.1
Hypothesis testing for correlations
13.3.2
Robust correlations
13.4
Correlation and causation
13.4.1
Causal graphs
13.5
Learning objectives
13.6
Suggested readings
13.7
Appendix
13.7.1
Quantifying inequality: The Gini index
13.7.2
Bayesian correlation analysis
14
The General Linear Model
14.1
Linear regression
14.1.1
Regression to the mean
14.1.2
The relation between correlation and regression
14.1.3
Standard errors for regression models
14.1.4
Statistical tests for regression parameters
14.1.5
Quantifying goodness of fit of the model
14.2
Fitting more complex models
14.3
Interactions between variables
14.4
Beyond linear predictors and outcomes
14.5
Criticizing our model and checking assumptions
14.6
What does “predict” really mean?
14.6.1
Cross-validation
14.7
Learning objectives
14.8
Suggested readings
14.9
Appendix
14.9.1
Estimating linear regression parameters
15
Comparing Means
15.1
Testing the Value of a Single Mean
15.2
Comparing Two Means
15.3
The t-test as a Linear Model
15.3.1
Effect Sizes for Comparing Two Means
15.4
Bayes Factor for Mean Differences
15.5
Comparing Paired Observations
15.5.1
Sign Test
15.5.2
Paired t-test
15.6
Comparing More Than Two Means
15.6.1
Analysis of Variance
15.7
Learning Objectives
15.8
Appendix
15.8.1
The Paired t-test as a Linear Model
16
Multivariate statistics
16.1
Multivariate data: An example
16.2
Visualizing multivariate data
16.2.1
Scatterplot of matrices
16.2.2
Heatmap
16.3
Clustering
16.3.1
K-means clustering
16.3.2
Hierarchical clustering
16.4
Dimensionality reduction
16.4.1
Principal component analysis
16.4.2
Factor analysis
16.4.3
Determining the number of factors
16.5
Learning objectives
16.6
Suggested readings
17
Practical Statistical Modeling
17.1
The process of statistical modeling
17.1.1
1: Specify your question of interest
17.1.2
2: Identify or collect the appropriate data
17.1.3
3: Prepare the data for analysis
17.1.4
4. Determine the appropriate model
17.1.5
5. Fit the model to the data
17.1.6
6. Criticize the model to make sure it fits properly
17.1.7
7. Test hypothesis and quantify effect size
17.2
What about possible confounds?
17.2.1
Determine the appropriate model
17.3
Getting help
18
Doing Reproducible Research
18.1
How we think science should work
18.2
How science (sometimes) actually works
18.3
The reproducibility crisis in science
18.3.1
Positive predictive value and statistical significance
18.3.2
The winner’s curse
18.4
Questionable research practices
18.4.1
ESP or QRP?
18.5
Doing reproducible research
18.5.1
Pre-registration
18.5.2
Reproducible practices
18.5.3
Replication
18.6
Doing reproducible data analysis
18.7
Conclusion: Doing better science
18.8
Learning objectives
18.9
Suggested Readings
Backmatter
References
Index
Colophon
Colophon
©2019–2026 Russell A. Poldrack
This work is licensed under the Creative Commons Attribution-NonCommercial 4.0 International License. To view a copy of this license, visit
CreativeCommons.org
🔗
