Mathematics of Medical Decisions

MICHAEL C. HIGGINS is Adjunct Professor in the Division of Computational Medicine at Stanford School of Medicine. His research focuses on quantitative methods for clinical decision analysis and the development of mathematical models to improve medical outcomes.

• FOREWORD XVIIPREFACE XVIIIABOUT THE COMPANION WEBSITE XX1 Introduction: Medical Decisions 11.1 Introduction 11.2 Example of a Medical Decision 21.3 Uncertainty in Medical Decisions 91.4 Quantifying What the Possible Outcomes Mean to the Patient 131.5 Temporal Aspects of Decision Outcomes 221.6 Who Is the Decision-Maker? 251.7 Epilogue 302 Probabilistic Analysis: Quantifying Uncertainty 362.1 Introduction 362.2 Basic Principles – Probability as a Measure of Certainty 382.3 Bayes' Formula – An Example of How Probability Is Analyzed 442.4 Generalizing Probabilistic Analysis 642.5 Expected Value – Summarizing Random Variables 722.6 Epilogue 743 Decision Trees: Structuring Decision Problems 823.1 Introduction 823.2 Decision Tree Basics 823.3 Decision Tree Analysis 953.4 Leukemia Treatment Decision – A Complete, Detailed Medical Example 1013.5 Epilogue 1124 Sensitivity Analysis: Analyzing the Analysis 1154.1 Introduction 1154.2 Medical Example – Maintenance Therapy for Acute Myeloid Leukemia 1154.3 Traditional Sensitivity Analysis 1164.4 Stochastic Sensitivity Analysis 1234.5 Epilogue 1375 Determining Probabilities: Empirical and Subjective Methods 1475.1 Introduction 1475.2 Empirical Probabilities Based on Predictive Models 1475.3 Subjective Methods 1685.4 Epilogue 1866 Survival Models: Representing the Uncertainty About the Length of Life 1916.1 Introduction 1916.2 Survival Model Basics 1926.3 Constant Hazard Rate Survival Models 2006.4 Actuarial Survival Models 2046.5 Fitting Parametric Models to Observations 2146.6 Epilogue 2227 Utility Fundamentals: Risk Attitudes and Quantifying Preferences 2257.1 Introduction 2257.2 What Are Risk Attitudes? 2267.3 Demonstration of Risk Attitudes in a Medical Context 2277.4 Mathematical Foundations 2377.5 General Observations About Outcome Utilities 2437.6 More Mathematics 2487.7 Determining Outcome Utilities – Underlying Concepts 2547.8 Epilogue 2608 Parametric Utility Models: Simplifying the Representation of Risk Attitudes 2668.1 Introduction 2668.2 First Example of a Parametric Model for Outcome Utilities 2678.3 Other Parametric Utility Models 2928.4 Incorporating Risk Attitudes into Clinical Policies 3068.5 Epilogue 3159 Multidimensional Utility Models: Adjusting for the Quality of Life 3219.1 Introduction 3219.2 Example – Why Does the Quality-of-Life Matters? 3229.3 Quality-Lifetime Tradeoff Models 3269.4 Quality-Survival Tradeoff Models 3369.5 Utility Independence 3439.6 What Does It All Mean? – Utility Assessment and an Extended Example 3559.7 Epilogue 36210 Markov Models: Representing the Dynamics of Uncertainty 36610.1 Introduction 36610.2 Markov Model Basics 36710.3 Analysis of Markov Models – Direct Approach 37710.4 Nonstationary Markov Models 39310.5 Epilogue 40211 Utility for Time-Varying Outcomes: Preferences for When Things Happen 41111.1 Introduction 41111.2 Motivating Example and Notation 41111.3 Willingness-To-Pay Models 41611.4 The Markovian Utility Model 41911.5 Example 43311.6 Generalizations 44011.7 Epilogue 44712 Decision Thresholds: Deciding When to Act 45312.1 Introduction 45312.2 Threshold Probability for Treatment 45412.3 Test Threshold Probabilities for Binary Tests 46712.4 Test Threshold Probabilities for More Complex Tests 48012.5 Epilogue 48313 Receiver Operating Characteristic Curves: Graphical Analysis of Diagnostic Performance 48913.1 Introduction 48913.2 Receiver Operating Characteristic Curve Analysis 49113.3 Parameterized Cutoff Analysis 50513.4 Decision Curve Analysis 51013.5 Accounting for Nondiagnostic Effects 51213.6 Epilogue 51914 Influence Diagrams: Scalable Alternative to Decision Trees 52614.1 Introduction 52614.2 Influence Diagram Structure 52714.3 Encoding Uncertainty in an Influence Diagram 53014.4 Analyzing an Influence Diagram 53214.5 Extending Influence Diagrams 53714.6 Epilogue 54315 Analyzing Screening Protocols: Predicting How Periodic Testing Affects Patient Outcomes 54515.1 Introduction 54515.2 Cervical Cancer 54615.3 Representing the Natural History of Cervical Cancer 55115.4 Screening Protocol Analysis 56115.5 Epilogue 58416 Monetary Costs: The Elephant in the Exam Room 58916.1 Introduction 58916.2 Cost-Effectiveness and Cost–Benefit Analysis 59016.3 Shared Resource Analysis 60616.4 Epilogue 631Exercises 63417 Prospect Theory: A Possible Paradigm Shift for Decision Analysis 63617.1 Introduction 63617.2 Examples of How Decisions Are Made 63617.3 Prospect Theory 64617.4 Framing Decision Problems 65417.5 Cumulative Prospect Theory 65717.6 Assessment of Weighting Functions in a Medical Context 66617.7 Epilogue 669INDEX 671