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El. knyga: Multivariate Survival Analysis and Competing Risks

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(Imperial College, University of London, UK)
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Crowder (U. of London) presents an expanded and corrected edition of his 2001 Classical Competing Risks. He introduces an area of statistics that has its roots in death and disease among humans and animals, though he stretches the examples a bit to make the treatment more cheerful. Prerequisites include courses in basic probability and statistics, and in mathematics to the level of basic calculus and analysis. He has tried to encompass probability modeling, the necessary mathematical manipulations, and the practicalities of data analysis, all of which he thinks statisticians of the future will need. Annotation ©2012 Book News, Inc., Portland, OR (booknews.com)

Multivariate Survival Analysis and Competing Risks introduces univariate survival analysis and extends it to the multivariate case. It covers competing risks and counting processes and provides many real-world examples, exercises, and R code. The text discusses survival data, survival distributions, frailty models, parametric methods, multivariate data and distributions, copulas, continuous failure, parametric likelihood inference, and non- and semi-parametric methods.

There are many books covering survival analysis, but very few that cover the multivariate case in any depth. Written for a graduate-level audience in statistics/biostatistics, this book includes practical exercises and R code for the examples. The author is renowned for his clear writing style, and this book continues that trend. It is an excellent reference for graduate students and researchers looking for grounding in this burgeoning field of research.

Recenzijos

"Dr. Crowders book provides a comprehensive overview of various distributions and models that have been proposed for multivariate survival data and competing risks data. The book is divided into four parts. The material discussed in each topic is succinct, which provides enough detail to follow and includes relevant references for interested audience to dig deeper. The accompanying real-world data examples and R code are extremely handy for anyone who would like to explore those methods. Despite the advanced topics it covers, the book is a pleasant read, as the presentation style is entertaining. All together the book is very suitable for an advanced survival analysis course for its broad range of topics." Yu Cheng, University of Pittsburgh, in the LIDA-IG Newsletter, January 2018

" the book is interesting to read, [ and] the authors writing style is both entertaining and to the point " ISCB News, December 2015

" a nice addition to the previous edition is the inclusion of R code and datasets, which are available online. this book is a useful addition to the literature, which undergraduate as well as graduate students in statistics will appreciate." Australian & New Zealand Journal of Statistics, 56(4), 2014

"Crowder is known for his clear expositions and chatty style, and this book does not disappoint. It is a pleasant read. The introduction to R will be useful, as will the exercises at the end of each chapter. With its exercises and easy style, this book is very suitable as an upper-level text. It is easy to jump into later chapters without much back pedaling and this makes it a useful reference work." Roger M. Cooke, Journal of the American Statistical Association, September 2014, Vol. 109

Preface xxi
Part I Univariate Survival Analysis
1 Scenario
3(8)
1.1 Survival Data
3(2)
1.1.1 Waiting Times
3(1)
1.1.2 Discrete Time
4(1)
1.1.3 Censoring
4(1)
1.2 Some Small Data Sets
5(2)
1.2.1 Strengths of Cords
5(1)
1.2.2 Cancer Survival
6(1)
1.2.3 Catheter Infection
6(1)
1.3 Inspecting the Data with R
7(2)
1.4 Fitting Models with R
9(1)
1.5 Simulating Data with R
9(2)
2 Survival Distributions
11(18)
2.1 Continuous Lifetimes
11(1)
2.2 Some Continuous Survival Distributions
12(3)
2.2.1 The Exponential Distribution
12(1)
2.2.2 The Weibull Distribution
13(1)
2.2.3 The Pareto Distribution
13(1)
2.2.4 Other Distributions
14(1)
2.2.5 The Shape of Hazard
14(1)
2.3 Discrete Lifetimes
15(1)
2.4 Some Discrete Survival Distributions
16(1)
2.4.1 The Geometric Distribution
16(1)
2.4.2 The Negative Binomial Distribution
17(1)
2.5 Mixed Discrete-Continuous Survival Distributions
17(3)
2.5.1 From Discrete to Continuous
18(1)
2.5.2 Rieman--Stieltjes Integrals
19(1)
2.6 Reliability Topics
20(2)
2.6.1 Reliability of Systems
20(1)
2.6.2 k-out-of-n Systems
21(1)
2.6.3 Survival Aspect
21(1)
2.6.4 Degradation Processes
21(1)
2.6.5 Stress and Strength
22(1)
2.7 Exercises
22(2)
2.7.1 Survival Distributions
22(1)
2.7.2 Reliability of Systems
23(1)
2.7.3 Degradation Processes
24(1)
2.7.4 Stress and Strength
24(1)
2.8 Hints and Solutions
24(5)
2.8.1 Survival Distributions
24(2)
2.8.2 Reliability of Systems
26(1)
2.8.3 Degradation Processes
26(1)
2.8.4 Stress and Strength
26(3)
3 Continuous Time-Parametric Inference
29(28)
3.1 Parametric Inference: Frequentist and Bayesian
29(3)
3.1.1 Frequentist Approach
29(1)
3.1.2 Bayesian Approach
30(1)
3.1.3 Proceed with Caution
31(1)
3.2 Random Samples
32(7)
3.2.1 Type-I Censoring
33(1)
3.2.2 Type-II Censoring
33(1)
3.2.3 Left Truncation
34(1)
3.2.4 Probabilities of Observation versus Censoring
34(1)
3.2.5 Weibull Lifetimes
35(1)
3.2.6 Strengths of Cords
36(2)
3.2.7 Survival of Breast Cancer Patients
38(1)
3.3 Regression Models
39(5)
3.3.1 Business Start-Ups
40(1)
3.3.2 Proportional Hazards (PH)
41(1)
3.3.3 Accelerated Life (AL)
42(1)
3.3.4 Proportional Odds (PO)
42(1)
3.3.5 Mean Residual Life (MRL)
42(1)
3.3.6 Catheter Infection
43(1)
3.4 Goodness of Fit
44(3)
3.4.1 Enhanced Models
45(1)
3.4.2 Uniform Residuals
45(1)
3.4.3 Cox--Snell Residuals
46(1)
3.4.4 Right-Censored Times
46(1)
3.4.5 Other Residuals
46(1)
3.4.6 Tests on Residuals
47(1)
3.5 Frailty and Random Effects
47(4)
3.5.1 Frailty
47(1)
3.5.2 Recovering the Frailty Distribution
48(1)
3.5.3 Discrete Random Effects and Frailty
49(2)
3.5.4 Accommodating Zero Frailty
51(1)
3.6 Time-Dependent Covariates
51(2)
3.7 Exercises
53(2)
3.7.1 Regression Models
53(1)
3.7.2 Residuals
53(1)
3.7.3 Discrete Frailty
54(1)
3.8 Hints and Solutions
55(2)
3.8.1 Regression Models
55(1)
3.8.2 Residuals
55(2)
4 Continuous Time: Non- and Semi-Parametric Methods
57(20)
4.1 Random Samples
57(5)
4.1.1 The Kaplan--Meier Estimator
57(2)
4.1.2 Strengths of Cords
59(1)
4.1.3 The Integrated and Cumulative Hazard Functions
60(1)
4.1.4 Interval-Censored Data
60(2)
4.2 Explanatory Variables
62(3)
4.2.1 Cox's Proportional Hazards Model
62(1)
4.2.2 Cox's Partial Likelihood
62(1)
4.2.3 Inference
63(1)
4.2.4 Computation
64(1)
4.2.5 Catheter Infection
64(1)
4.3 Some Further Aspects
65(4)
4.3.1 Stratification
65(1)
4.3.2 Tied Lifetimes
66(1)
4.3.3 The Baseline Survivor Function
66(1)
4.3.4 The Log-Rank Test
67(1)
4.3.5 Schoenfeld Residuals
68(1)
4.3.6 Time-Dependent Covariates
68(1)
4.3.7 Interval-Censored Data
69(1)
4.4 Task Completion Times
69(3)
4.5 Accelerated Life Models
72(3)
4.6 Exercises
75(1)
4.6.1 Random Samples
75(1)
4.6.2 Partial Likelihood
75(1)
4.6.3 Applications
75(1)
4.6.4 Accelerated Life Models
76(1)
4.7 Hints and Solutions
76(1)
4.7.1 Random Samples
76(1)
4.7.2 Accelerated Life Models
76(1)
5 Discrete Time
77(24)
5.1 Random Samples: Parametric Methods
77(4)
5.1.1 Geometric Lifetimes
77(1)
5.1.2 Career Promotions
78(2)
5.1.3 Probabilities of Observation versus Censoring
80(1)
5.2 Random Samples: Non- and Semi-Parametric Estimation
81(3)
5.2.1 Career Promotions
82(1)
5.2.2 Large-Sample Theory
83(1)
5.3 Explanatory Variables
84(6)
5.3.1 Likelihood Function
84(1)
5.3.2 Geometric Waiting Times
85(1)
5.3.3 The Driving Test
85(2)
5.3.4 Proportional Hazards
87(1)
5.3.5 Proportional Odds
87(2)
5.3.6 The Driving Test
89(1)
5.3.7 The Baseline Odds
89(1)
5.4 Interval-Censored Data
90(2)
5.4.1 Cancer Survival Data
90(2)
5.5 Frailty and Random Effects
92(3)
5.5.1 Geometric Distribution
92(1)
5.5.2 Random Effects
93(1)
5.5.3 Beta-Geometric Distribution
93(1)
5.5.4 Cycles to Pregnancy
94(1)
5.5.5 The Driving Test
95(1)
5.6 Exercises
95(2)
5.6.1 Random Samples
95(1)
5.6.2 Explanatory Variables
96(1)
5.6.3 Gamma and Beta Distributions
96(1)
5.7 Hints and Solutions
97(4)
5.7.1 Random Samples
97(4)
Part II Multivariate Survival Analysis
6 Multivariate Data and Distributions
101(10)
6.1 Some Small Data Sets
101(4)
6.1.1 Repeated Response Times
101(1)
6.1.2 Paired Response Times
102(1)
6.1.3 Lengths and Strengths of Fibres
102(1)
6.1.4 Household Energy Usage
102(3)
6.2 Multivariate Survival Distributions
105(3)
6.2.1 Joint and Marginal Distributions
105(1)
6.2.2 Conditional Distributions
105(1)
6.2.3 Dependence and Association
105(1)
6.2.4 Hazard Functions and Failure Rates
106(1)
6.2.5 Gumbel's Bivariate Exponential
107(1)
6.3 Exercises
108(1)
6.3.1 Joint and Marginal Distributions
108(1)
6.3.2 Dependence and Association
108(1)
6.4 Hints and Solutions
109(2)
6.4.1 Joint and Marginal Distributions
109(1)
6.4.2 Dependence and Association
109(2)
7 Some Models and Methods
111(10)
7.1 The Multivariate Log-Normal Distribution
111(1)
7.2 Applications
112(2)
7.2.1 Household Energy Usage
112(1)
7.2.2 Repeated Response Times
113(1)
7.3 Bivariate Exponential
114(1)
7.3.1 Discrete-Time Version
114(1)
7.4 Bivariate Exponential
115(1)
7.4.1 Discrete-Time Version
116(1)
7.5 Some Other Bivariate Distributions
116(2)
7.5.1 Block and Basu (1974)
116(1)
7.5.2 Lawrance and Lewis (1983)
116(1)
7.5.3 Arnold and Brockett (1983)
117(1)
7.5.4 Cowan (1987)
117(1)
7.5.5 Yet More Distributions
117(1)
7.6 Non- and Semi-Parametric Methods
118(2)
7.7 Exercises
120(1)
8 Frailty, Random Effects, and Copulas
121(18)
8.1 Frailty: Construction
121(1)
8.2 Some Frailty-Generated Distributions
122(5)
8.2.1 Multivariate Burr
122(1)
8.2.2 Multivariate Weibull
123(1)
8.2.3 Distribution 3
124(1)
8.2.4 Multivariate Beta-Geometric
125(1)
8.2.5 Multivariate Gamma-Poisson
126(1)
8.2.6 Marshall--Olkin Families
126(1)
8.3 Applications
127(4)
8.3.1 Paired Response Times
127(2)
8.3.2 Household Energy Usage
129(1)
8.3.3 Cycles to Pregnancy
130(1)
8.4 Copulas: Structure
131(2)
8.5 Further Details
133(2)
8.6 Applications
135(1)
8.6.1 Clayton Copula
135(1)
8.7 Exercises
136(2)
8.7.1 Frailty-Generated Distributions
136(1)
8.7.2 Applications
137(1)
8.7.3 Copulas
137(1)
8.8 Hints and Solutions
138(1)
8.8.1 Copulas
138(1)
9 Repeated Measures
139(22)
9.1 Pure Frailty Models: Applications
139(4)
9.1.1 Lengths and Strengths of Fibres
140(1)
9.1.2 Visual Acuity
141(2)
9.2 Models with Serial Correlation: Application
143(2)
9.2.1 Repeated Response Times
144(1)
9.3 Matched Pairs
145(1)
9.4 Discrete Time: Applications
146(6)
9.4.1 Proportional Odds
146(1)
9.4.2 Beta-Geometric Model
147(1)
9.4.3 Bird Recapture
147(2)
9.4.4 Antenatal Knowledge
149(3)
9.5 Milestones: Applications
152(4)
9.5.1 Educational Development
152(1)
9.5.2 Pill Dissolution Rates
153(1)
9.5.3 Timber Slip
153(1)
9.5.4 Loan Default
154(2)
9.6 Exercises
156(5)
9.6.1 Bird Recapture Data
156(1)
9.6.2 Some Background for the Bivariate Beta Distribution
156(1)
9.6.3 Antenatal Data
157(1)
9.6.4 Binomial Waiting Times
158(1)
9.6.5 Milestones Data
158(3)
10 Recurrent Events
161(18)
10.1 Some Recurrence Data
161(2)
10.2 The Event Rate
163(2)
10.3 Basic Recurrence Processes
165(3)
10.3.1 Poisson Processes
165(1)
10.3.2 Renewal Processes
166(1)
10.3.3 Recurrence of Medical Condition
166(1)
10.3.4 Simulation
167(1)
10.4 More Elaborate Models
168(1)
10.4.1 Poisson Process
168(1)
10.4.2 Intensity Functions
168(1)
10.5 Other Fields of Application
169(2)
10.5.1 Repair and Warranty Data
169(1)
10.5.2 Sports Data
169(1)
10.5.3 Institution Data
170(1)
10.5.4 Alternating Periods
170(1)
10.6 Event Counts
171(3)
10.6.1 Continuous Time: Poisson Process
171(1)
10.6.2 Discrete Time
172(1)
10.6.3 Multivariate Negative Binomial
172(2)
10.7 Quasi-Life Tables
174(2)
10.7.1 Estimation
174(2)
10.8 Exercises
176(1)
10.9 Hints and Solutions
177(2)
11 Multi-State Processes
179(24)
11.1 Markov Chain Models
179(4)
11.1.1 Discrete Time
179(1)
11.1.2 Continuous Time
180(1)
11.1.3 Hidden Markov Chains
181(1)
11.1.4 Estimation
182(1)
11.2 The Wiener Process
183(1)
11.3 Wear and Tear and Lack of Care
184(1)
11.4 Cumulative Models
185(5)
11.4.1 Compound Poisson Process
186(2)
11.4.2 Compound Birth Process
188(1)
11.4.3 Gamma Process
188(2)
11.4.4 Customer Lifetime Value
190(1)
11.5 Some Other Models and Applications
190(4)
11.5.1 Empirical Equation Models
190(2)
11.5.2 Models for the Stress Process
192(1)
11.5.3 Stress-Strength Models
193(1)
11.5.4 Other Models and Applications
193(1)
11.6 Exercises
194(4)
11.6.1 Markov Chains
194(1)
11.6.2 Wiener Process
195(2)
11.6.3 Cumulative Damage Models
197(1)
11.6.4 Compound Poisson Process
197(1)
11.6.5 Compound Birth Process
197(1)
11.6.6 Gamma Process
198(1)
11.7 Hints and Solutions
198(5)
11.7.1 Markov Chains
198(1)
11.7.2 Wiener Process
198(1)
11.7.3 Cumulative Damage Models
199(1)
11.7.4 Compound Poisson Process
199(1)
11.7.5 Compound Birth Process
199(4)
Part III Competing Risks
12 Continuous Failure Times and Their Causes
203(12)
12.1 Some Small Data Sets
203(1)
12.1.1 Gubbins
203(1)
12.1.2 Catheter Infection
203(1)
12.1.3 Superalloy Testing
204(1)
12.2 Basic Probability Functions: Continuous Time
204(3)
12.2.1 Exponential Mixture
206(1)
12.3 Hazard Functions
207(2)
12.3.1 Exponential Mixture
207(1)
12.3.2 Lemma
208(1)
12.3.3 Weibull Sub-Hazards
208(1)
12.4 Proportional Hazards
209(2)
12.4.1 Weibull Sub-Hazards
209(1)
12.4.2 Theorem
210(1)
12.5 Regression Models
211(1)
12.5.1 Proportional Hazards (PH)
211(1)
12.5.2 Accelerated Life (AL)
211(1)
12.5.3 Proportional Odds (PO)
211(1)
12.5.4 Mean Residual Life (MRL)
212(1)
12.6 Examples
212(2)
12.6.1 Exponential Mixture
212(2)
12.7 Exercises
214(1)
12.8 Hints and Solutions
214(1)
13 Continuous Time: Parametric Inference
215(20)
13.1 The Likelihood for Competing Risks
215(3)
13.1.1 Forms of the Likelihood Function
215(1)
13.1.2 Uncertainty about C
216(1)
13.1.3 Uncertainty about T
216(1)
13.1.4 Maximum Likelihood Estimates
217(1)
13.1.4.1 Weibull Sub-Hazards
217(1)
13.1.4.2 Exponential Mixture
217(1)
13.2 Model Checking
218(1)
13.2.1 Goodness of Fit
218(1)
13.2.2 Uniform Residuals
218(1)
13.3 Inference
219(2)
13.4 Some Applications
221(5)
13.4.1 Gubbins
221(1)
13.4.2 Survival Times of Mice
222(2)
13.4.3 Fracture Toughness
224(1)
13.4.4 Length of Hospital Stay
225(1)
13.5 Some Examples of Hazard Modelling
226(5)
13.5.1 Exponential Mixture
226(1)
13.5.2 Gumbel's Bivariate Exponential
227(1)
13.5.3 Bivariate Makeham Distribution
227(1)
13.5.4 Kimber and Grace: The Dream Team
227(2)
13.5.5 A Clinical Trial
229(2)
13.6 Masked Systems
231(2)
13.7 Exercises
233(2)
13.7.1 Applications
233(2)
14 Latent Lifetimes
235(18)
14.1 Basic Probability Functions
235(3)
14.1.1 Tsiatis's Lemma
235(1)
14.1.2 Gumbel's Bivariate Exponential
236(2)
14.2 Some Examples
238(4)
14.2.1 Freund's Bivariate Exponential
238(1)
14.2.2 Frailty Models
238(1)
14.2.3 Multivariate Burr (MB)
239(1)
14.2.4 Multivariate Weibull (MW)
239(2)
14.2.5 A Stochastic Process Model
241(1)
14.2.6 Other Applications
241(1)
14.3 Further Aspects
242(2)
14.3.1 Latent Failure Times versus Hazard Functions
242(1)
14.3.2 Marginals versus Sub-Distributions
242(1)
14.3.3 Peterson's Bounds
243(1)
14.4 Independent Risks
244(3)
14.4.1 Gail's Theorem
245(2)
14.4.2 Other Applications
247(1)
14.5 The Makeham Assumption
247(2)
14.5.1 Proportional Hazards
248(1)
14.6 A Risk-Removal Model
249(1)
14.7 A Degradation Process
250(1)
14.7.1 Wiener Process
251(1)
14.7.2 Compound Poisson and Compound Birth Processes
251(1)
14.7.3 Gamma Process
251(1)
14.8 Exercises
251(1)
14.9 Hints and Solutions
252(1)
15 Continuous Time: Non- and Semi-Parametric Methods
253(12)
15.1 The Kaplan--Meier Estimator
253(4)
15.1.1 Survival Times of Mice
255(2)
15.2 Actuarial Approach
257(2)
15.3 Proportional Hazards and Partial Likelihood
259(2)
15.3.1 The Proportional Hazards (PH) Model
259(1)
15.3.2 The Partial Likelihood
259(1)
15.3.3 A Clinical Trial
260(1)
15.4 The Baseline Survivor Functions
261(1)
15.5 Other Methods and Applications
262(3)
16 Discrete Lifetimes
265(22)
16.1 Basic Probability Functions
265(2)
16.1.1 Geometric Mixture
266(1)
16.2 Latent Lifetimes and Sub-Odds Functions
267(4)
16.2.1 Sub-Odds Theorem
268(3)
16.3 Some Examples
271(3)
16.3.1 Discrete Version of Gumbel
271(1)
16.3.2 Discrete Version of Freund
272(1)
16.3.3 Discrete Version of Marshall--Olkin
273(1)
16.3.4 Mixture Models
273(1)
16.4 Parametric Estimation
274(3)
16.4.1 Likelihood Function
275(1)
16.4.2 Discrete Marshall--Olkin
275(1)
16.4.3 Psychiatric Wards
276(1)
16.5 Non-Parametric Estimation from Random Samples
277(6)
16.5.1 Gubbins
279(2)
16.5.2 Interval-Censored Data
281(1)
16.5.3 Superalloy Testing
281(2)
16.6 Asymptotic Distribution of Non-Parametric Estimators
283(1)
16.7 Proportional Odds and Partial Likelihood
284(2)
16.7.1 Psychiatric Wards
285(1)
16.8 Exercises
286(1)
16.9 Hints and Solutions
286(1)
17 Latent Lifetimes: Identifiability Crises
287(24)
17.1 The Cox--Tsiatis Impasse
287(3)
17.1.1 Tsiatis's Theorem
287(2)
17.1.2 Gumbel's Bivariate Exponential
289(1)
17.2 More General Identifiablility Results
290(6)
17.2.1 Miller's Theorem
290(1)
17.2.2 The LPQ Theorem
291(4)
17.2.3 The Marshall--Olkin Distribution
295(1)
17.3 Specified Marginals
296(3)
17.4 Discrete Lifetimes
299(3)
17.4.1 Discrete Freund
301(1)
17.4.2 Discrete Marshall--Olkin
301(1)
17.4.3 A Test for Independence of Risks
301(1)
17.5 Regression Case
302(2)
17.5.1 Heckman and Honore's Theorem
302(1)
17.5.2 Gumbel's Bivariate Exponential
303(1)
17.6 Censoring of Survival Data
304(2)
17.7 Parametric Identifiability
306(5)
Part IV Counting Processes in Survival Analysis
18 Some Basic Concepts
311(12)
18.1 Probability Spaces
311(1)
18.2 Conditional Expectation
312(2)
18.3 Filtrations
314(1)
18.4 Martingales in Discrete Time
315(2)
18.4.1 Likelihood Ratios
316(1)
18.5 Martingales in Continuous Time
317(2)
18.6 Counting Processes
319(1)
18.7 Product Integrals
320(3)
19 Survival Analysis
323(8)
19.1 A Single Lifetime
323(2)
19.1.1 The Intensity Process
323(1)
19.1.2 Parametric Likelihood Function
324(1)
19.2 Independent Lifetimes
325(1)
19.3 Competing Risks
326(2)
19.4 Right-Censoring
328(3)
20 Non- and Semi-Parametric Methods
331(10)
20.1 Survival Times
331(2)
20.2 Competing Risks
333(1)
20.3 Large-Sample Results
334(1)
20.3.1 Consistency
334(1)
20.3.2 Asymptotic Normality
334(1)
20.3.3 Confidence Intervals
334(1)
20.4 Hypothesis Testing
335(3)
20.4.1 Single-Sample Case
335(1)
20.4.2 Several Samples
336(2)
20.5 Regression Models
338(3)
20.5.1 Intensity Models and Time-Dependent Covariates
338(1)
20.5.2 Proportional Hazards Model
339(1)
20.5.3 Martingale Residuals
339(2)
Appendix A Terms, Notations, and Abbreviations 341(2)
Appendix B Basic Likelihood Methods 343(4)
Appendix C Some Theory for Partial Likelihood 347(4)
Appendix D Numerical Optimisation of Functions 351(4)
References 355(20)
Epilogue to First Edition 375(2)
Index 377
Martin J. Crowder
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