Preface

Acknowledgements

1 Introduction

1.1 Historical notes

1.2 Defining competing risks

1.3 Use of the Kaplan–Meier method in the presence of competing risks

1.4 Testing in the competing risk framework

1.5 Sample size calculation

1.6 Examples

1.6.1 Tamoxifen trial

1.6.2 Hypoxia study

1.6.3 Follicular cell lymphoma study

1.6.4 Bone marrow transplant study

1.6.5 Hodgkin’s disease study

2 Survival—basic concepts

2.1 Introduction

2.2 Definitions and background formulae

2.2.1 Introduction

2.2.2 Basic mathematical formulae

2.2.3 Common parametric distributions

2.2.4 Censoring and assumptions

2.3 Estimation and hypothesis testing

2.3.1 Estimating the hazard and survivor functions

2.3.2 Nonparametric testing: log-rank and Wilcoxon tests

2.3.3 Proportional hazards model

2.4 Software for survival analysis

2.5 Closing remarks

3 Competing risks—definitions

3.1 Recognizing competing risks

3.1.1 Practical approaches

3.1.2 Common endpoints in medical research

3.2 Two mathematical definitions

3.2.1 Competing risks as bivariate random variable

3.2.2 Competing risks as latent failure times

3.3 Fundamental concepts

3.3.1 Competing risks as bivariate random variable

3.3.2 Competing risks as latent failure times

3.3.3 Discussion of the two approaches

3.4 Closing remarks

4 Descriptive methods for competing risks data

4.1 Product-limit estimator and competing risks

4.2 Cumulative incidence function

4.2.1 Heuristic estimation of the CIF

4.2.2 Nonparametric maximum likelihood estimation of the CIF

4.2.3 Calculating the CIF estimator

4.2.4 Variance and confidence interval for the CIF estimator

4.3 Software and examples

4.3.1 Using R

4.3.2 Using SAS

4.4 Closing remarks

5 Testing a covariate

5.1 Introduction

5.2 Testing a covariate

5.2.1 Gray’s method

5.2.2 Pepe and Mori’s method

5.3 Software and examples

5.3.1 Using R

5.3.2 Using SAS

5.4 Closing remarks

6 Modelling in the presence of competing risks

6.1 Introduction

6.2 Modelling the hazard of the cumulative incidence function

6.2.1 Theoretical details

6.2.2 Model-based estimation of the CIF

6.2.3 Using R

6.3 Cox model and competing risks

6.4 Checking the model assumptions

6.4.1 Proportionality of the cause-specific hazards

6.4.2 Proportionality of the hazards of the CIF

6.4.3 Linearity assumption

6.5 Closing remarks

7 Calculating the power in the presence of competing risks

7.1 Introduction

7.2 Sample size calculation when competing risks are not present

7.3 Calculating power in the presence of competing risks

7.3.1 General formulae

7.3.2 Comparing cause-specific hazards

7.3.3 Comparing hazards of the subdistributions

7.3.4 Probability of event when the exponential distribution is not a valid assumption

7.4 Examples

7.4.1 Introduction

7.4.2 Comparing the cause-specific hazard

7.4.3 Comparing the hazard of the subdistribution

7.5 Closing remarks

8 Other issues in competing risks

8.1 Conditional probability function

8.1.1 Introduction

8.1.2 Nonparametric estimation of the CP function

8.1.3 Variance of the CP function estimator

8.1.4 Testing a covariate

8.1.5 Using R

8.1.6 Using SAS

8.2 Comparing two types of risk in the same population

8.2.1 Theoretical background

8.2.2 Using R

8.2.3 Discussion

8.3 Identifiability and testing independence

8.4 Parametric modelling

8.4.1 Introduction

8.4.2 Modelling the marginal distribution

8.4.3 Modelling the Weibull distribution

9 Food for thought

Problem 1: Estimation of the probability of the event of interest

Problem 2: Testing a covariate

Problem 3: Comparing the event of interest between two groups when the competing risks are different for each group

Problem 4: Information needed for sample size calculations

Problem 5: The effect of the size of the incidence of competing risks on the coefficient obtained in the model

Problem 6: The KLY test and the non-proportionality of hazards

Problem 7: The KLY and Wilcoxon tests

A: Theoretical background

A.1 Nonparametric maximum likelihood estimation for the survivor function in the discrete case

A.2 Confidence interval for survivor function

A.3 The Variance for Gray's test

A.4 Derivation of the parameters for the exponential latent failure time model

A.5 Likelihood of a mixture of exponentials in the bivariate approach

B: Analysing competing risks data using R and SAS

B.1 The R software and

**cmprsk** package

B.1.1 Downloading and installation

B.1.2 Getting help

B.1.3 Operators in R

B.1.4 Objects in R

B.1.5 The **cmprsk** package and datasets

B.2 Importing datasets in SAS

B.3 Other programs written for R

B.3.1 CIF variance based on the delta method

B.3.2 Pepe–Mori test for the difference between two CIFs

B.3.3 Conditional probability and its variance

B.3.4 Plotting the conditional probability

B.3.5 Testing the conditional probability

B.3.6 Calculating the conditional probability of observing the event of interest in a time period knowing that the patient was free of any event at the beginning of the period

B.3.7 Comparing two types of risk in the same population

B.3.8 Calculating the power

B.4 SAS macros for competing risk analysis

B.4.1 Cumulative incidence and conditional probability

B.4.2 Pepe–Mori test for the comparison of two CIFs

B.4.2 Pepe–Mori test for the comparison of two CPs

References

Index