Statistics at Square Two

Description:

STATISTICS AT SQUARE TWO

An easy-to-follow exploration of intermediate statistical techniques used in medical research

In the newly revised third edition of Statistics at Square Two: Understanding Modern Statistical Applications in Medicine, a team of distinguished statisticians delivers an accessible and intuitive discussion of advanced statistical methods for readers and users of scientific medical literature. This will allow readers to engage critically with modern research as the authors explain the correct interpretation of results in the medical literature.

The book includes two brand new chapters covering meta-analysis and time-series analysis as well as new references to the many checklists that have appeared in recent years to enable better reporting of contemporary research. Most examples have been updated as well, and each chapter contains practice exercises and answers. Readers will also find sample code (in R) for many of the analyses, in addition to:

• A thorough introduction to models and data, including the different types of data, statistical models, and computer-intensive methods
• Comprehensive explorations of multiple linear regression, including the interpretation of computer output, diagnostic statistics such as influential points, and many uses of multiple regression
• Practical discussions of multiple logistic regression, survival analysis, Poisson regression and random effects models including their uses, examples in the medical literature, and strategies for interpreting computer output

Perfect for anyone hoping to better understand the statistics presented in contemporary medical research, Statistics at Square Two: Understanding Modern Statistical Applications in Medicine will also benefit postgraduate students studying statistics and medicine.

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In the 16 years since the second edition of Statistics at Square Two was published, there have been many developments in statistical methodology and in methods of presenting statistics. MJC is pleased that his colleague Richard Jacques, who has considerable experience in more advanced statistical methods and teaching medical statistics to non- statisticians, has joined him as a co-author. Most of the examples have been updated and two new chapters have been added on meta-analysis and on time series analysis. In addition, reference is made to the many checklists which have appeared since the last edition to enable better reporting of research.

This book is intended to build on the latest edition of Statistics at Square One.1 It is hoped to be a vade mecum for investigators who have undergone a basic statistics course, but need more advanced methods. It is also intended for readers and users of the medical literature, but is intended to be rather more than a simple “bluffer’s guide”. It is hoped that it will encourage the user to seek professional help when necessary. Important sections in each chapter are tips on reading and reporting about a particular technique; the book emphasises correct interpretation of results in the literature. Much advanced statistical methodology is used rather uncritically in medical research, and the data and code to check whether the methods are valid are often not provided when the investigators write up their results. This text will help readers of statistics in medical research engage in constructive critical review of the literature.

Since most researchers do not want to become statisticians, detailed explanations of the methodology will be avoided. However, equations of the models are given, since they show concisely what each model is assuming. We hope the book will prove useful to students on postgraduate courses and for this reason there are a number of exercises with answers. For students on a more elementary course for health professionals we recommend Walters et al.2

The choice of topics reflects what we feel are commonly encountered in the medical literature, based on many years of statistical refereeing. The linking theme is regression models and we cover multiple regression, logistic regression, Cox regression, random effects (mixed models), ordinal regression, Poisson regression, time series regression and meta-analysis. The predominant philosophy is frequentist, since this reflects the literature and what is available in most packages. However, a discussion on the uses of Bayesian methods is given in an Appendix 4. The huge amount of work on causal modelling is briefly referenced, but is generally beyond the scope of this book.

Most of the concepts in statistical inference have been covered in Statistics at Square One.1 In order to keep this book short, reference will be made to the earlier book for basic concepts. All the analyses described in the book have been conducted in the free software R and the code is given to make the methods accessible to reserachers without commercial statistical packages.

We are grateful to Tommy Nyberg of the Biostatistics Unit, Cambridge for feedback on his survival paper and to our colleague, Jeremy Dawson, who read and commented on the final draft. Any remaining errors are our own.

Michael J. Campbell
Richard M. Jacques
Sheffield, June 2022

Table of contents :

Statistics at Square Two
Contents
Preface
1 Models, Tests and Data
1.1 Types of Data
1.2 Confounding, Mediation and Effect Modification
1.3 Causal Inference
1.4 Statistical Models
1.5 Results of Fitting Models
1.6 Significance Tests
1.7 Confidence Intervals
1.8 Statistical Tests Using Models
1.9 Many Variables
1.10 Model Fitting and Analysis: Exploratory and Confirmatory Analyses
1.11 Computer-intensive Methods
1.12 Missing Values
1.13 Bayesian Methods
1.14 Causal Modelling
1.15 Reporting Statistical Results in the Medical Literature
1.16 Reading Statistics in the Medical Literature
2 Multiple Linear Regression
2.1 The Model
2.2 Uses of Multiple Regression
2.3 Two Independent Variables
2.3.1 One Continuous and One Binary Independent Variable
2.3.2 Two Continuous Independent Variables
2.3.3 Categorical Independent Variables
2.4 Interpreting a Computer Output
2.4.1 One Continuous Variable
2.4.2 One Continuous Variable and One Binary Independent Variable
2.4.3 One Continuous Variable and One Binary Independent Variable with Their Interaction
2.4.4 Two Independent Variables: Both Continuous
2.4.5 Categorical Independent Variables
2.5 Examples in the Medical Literature
2.5.1 Analysis of Covariance: One Binary and One Continuous Independent Variable
2.5.2 Two Continuous Independent Variables
2.6 Assumptions Underlying the Models
2.7 Model Sensitivity
2.7.1 Residuals, Leverage and Influence
2.7.2 Computer Analysis: Model Checking and Sensitivity
2.8 Stepwise Regression
2.9 Reporting the Results of a Multiple Regression
2.10 Reading about the Results of a Multiple Regression
2.11 Frequently Asked Questions
2.12 Exercises: Reading the Literature
3 Multiple Logistic Regression
3.1 Quick Revision
3.2 The Model
3.2.1 Categorical Covariates
3.3 Model Checking
3.3.1 Lack of Fit
3.3.2 “Extra-binomial” Variation or “Over Dispersion”
3.3.3 The Logistic Transform is Inappropriate
3.4 Uses of Logistic Regression
3.5 Interpreting a Computer Output
3.5.1 One Binary Independent Variable
3.5.2 Two Binary Independent Variables
3.5.3 Two Continuous Independent Variables
3.6 Examples in the Medical Literature
3.6.1 Comment
3.7 Case-control Studies
3.8 Interpreting Computer Output: Unmatched Case-control Study
3.9 Matched Case-control Studies
3.10 Interpreting Computer Output: Matched Case-control Study
3.11 Example of Conditional Logistic Regression in the Medical Literature
3.11.1 Comment
3.12 Alternatives to Logistic Regression
3.13 Reporting the Results of Logistic Regression
3.14 Reading about the Results of Logistic Regression
3.15 Frequently Asked Questions
3.16 Exercise
4 Survival Analysis
4.1 Introduction
4.2 The Model
4.3 Uses of Cox Regression
4.4 Interpreting a Computer Output
4.5 Interpretation of the Model
4.6 Generalisations of the Model
4.6.1 Stratified Models
4.6.2 Time Dependent Covariates
4.6.3 Parametric Survival Models
4.6.4 Competing Risks
4.7 Model Checking
4.8 Reporting the Results of a Survival Analysis
4.9 Reading about the Results of a Survival Analysis
4.10 Example in the Medical Literature
4.10.1 Comment
4.11 Frequently Asked Questions
4.12 Exercises
5 Random Effects Models
5.1 Introduction
5.2 Models for Random Effects
5.3 Random vs Fixed Effects
5.4 Use of Random Effects Models
5.4.1 Cluster Randomised Trials
5.4.2 Repeated Measures
5.4.3 Sample Surveys
5.4.4 Multi-centre Trials
5.5 Ordinary Least Squares at the Group Level
5.6 Interpreting a Computer Output
5.6.1 Different Methods of Analysis
5.6.2 Likelihood and gee
5.6.3 Interpreting Computer Output
5.7 Model Checking
5.8 Reporting the Results of Random Effects Analysis
5.9 Reading about the Results of Random Effects Analysis
5.10 Examples of Random Effects Models in the Medical Literature
5.10.1 Cluster Trials
5.10.2 Repeated Measures
5.10.3 Comment
5.10.4 Clustering in a Cohort Study
5.10.5 Comment
5.11 Frequently Asked Questions
5.12 Exercises
6 Poisson and Ordinal Regression
6.1 Poisson Regression
6.2 The Poisson Model
6.3 Interpreting a Computer Output: Poisson Regression
6.4 Model Checking for Poisson Regression
6.5 Extensions to Poisson Regression
6.6 Poisson Regression Used to Estimate Relative Risks from a 2 × 2 Table
6.7 Poisson Regression in the Medical Literature
6.8 Ordinal Regression
6.9 Interpreting a Computer Output: Ordinal Regression
6.10 Model Checking for Ordinal Regression
6.11 Ordinal Regression in the Medical Literature
6.12 Reporting the Results of Poisson or Ordinal Regression
6.13 Reading about the Results of Poisson or Ordinal Regression
6.14 Frequently Asked Question
6.15 Exercises
7 Meta-analysis
7.1 Introduction
7.2 Models for Meta-analysis
7.3 Missing Values
7.4 Displaying the Results of a Meta-analysis
7.5 Interpreting a Computer Output
7.6 Examples from the Medical Literature
7.6.1 Example of a Meta-analysis of Clinical Trials
7.6.2 Example of a Meta-analysis of Case-control Studies
7.7 Reporting the Results of a Meta-analysis
7.8 Reading about the Results of a Meta-analysis
7.9 Frequently Asked Questions
7.10 Exercise
8 Time Series Regression
8.1 Introduction
8.2 The Model
8.3 Estimation Using Correlated Residuals
8.4 Interpreting a Computer Output: Time Series Regression
8.5 Example of Time Series Regression in the Medical Literature
8.6 Reporting the Results of Time Series Regression
8.7 Reading about the Results of Time Series Regression
8.8 Frequently Asked Questions
8.9 Exercise
Appendix 1 Exponentials and Logarithms
Appendix 2 Maximum Likelihood and Significance Tests
A2.1 Binomial Models and Likelihood
A2.2 The Poisson Model
A2.3 The Normal Model
A2.4 Hypothesis Testing: the Likelihood Ratio Test
A2.5 The Wald Test
A2.6 The Score Test
A2.7 Which Method to Choose?
A2.8 Confidence Intervals
A2.9 Deviance Residuals for Binary Data
A2.10 Example: Derivation of the Deviances and Deviance Residuals Given in Table 3.3
A2.10.1 Grouped Data
A2.10.2 Ungrouped Data
Appendix 3 Bootstrapping and Variance Robust Standard Errors
A3.1 The Bootstrap
A3.2 Example of the Bootstrap
A3.3 Interpreting a Computer Output: The Bootstrap
A3.3.1 Two-sample T-test with Unequal Variances
A3.4 The Bootstrap in the Medical Literature
A3.5 Robust or Sandwich Estimate SEs
A3.6 Interpreting a Computer Output: Robust SEs for Unequal Variances
A3.7 Other Uses of Robust Regression
A3.8 Reporting the Bootstrap and Robust SEs in the Literature
A3.9 Frequently Asked Question
Appendix 4 Bayesian Methods
A4.1 Bayes’ Theorem
A4.2 Uses of Bayesian Methods
A4.3 Computing in Bayes
A4.4 Reading and Reporting Bayesian Methods in the Literature
A4.5 Reading about the Results of Bayesian Methods in the Medical Literature
Appendix 5 R codes
A5.1 R Code for Chapter 2
A5.3 R Code for Chapter 3
A5.4 R Code for Chapter 4
A5.5 R Code for Chapter 5
A5.6 R Code for Chapter 6
A5.7 R Code for Chapter 7
A5.8 R Code for Chapter 8
A5.9 R Code for Appendix 1
A5.10 R Code for Appendix 2
A5.11 R Code for Appendix 3
Answers to Exercises
Glossary
Index
EULA