CHAPMAN & HALL/CRC
Interdisciplinary Statistics Series
Series editors: N. Keiding, B.J.T. Morgan, C.K. Wikle, P. van der Heijden
Published titles
AGE-PERIOD-COHORT ANALYSIS: NEW MODELS, METHODS, AND EMPIRICAL APPLICATIONS Y. Yang and K. C. Land
ANALYSIS OF CAPTURE-RECAPTURE DATA R. S. McCrea and B. J.T. Morgan
AN INVARIANT APPROACH TO STATISTICAL ANALYSIS OF SHAPES
S. Lele and J. Richtsmeier
ASTROSTATISTICS G. Babu and E. Feigelson
BAYESIAN ANALYSIS FOR POPULATION ECOLOGY R. King, B. J. T. Morgan, O. Gimenez, and S. P. Brooks
BAYESIAN DISEASE MAPPING: HIERARCHICAL MODELING IN SPATIAL EPIDEMIOLOGY, SECOND EDITION A. B. Lawson
BIOEQUIVALENCE AND STATISTICS IN CLINICAL PHARMACOLOGY
S. Patterson and B. Jones
CAPTURE-RECAPTURE METHODS FOR THE SOCIAL AND MEDICAL SCIENCES
D. Böhning, P. G. M. van der Heijden, and J. Bunge
CLINICAL TRIALS IN ONCOLOGY, THIRD EDITION S. Green, J. Benedetti, A. Smith, and J. Crowley
CLUSTER RANDOMISED TRIALS R.J. Hayes and L.H. Moulton
CORRESPONDENCE ANALYSIS IN PRACTICE, THIRD EDITION M. Greenacre
THE DATA BOOK: COLLECTION AND MANAGEMENT OF RESEARCH DATA M. Zozus
DESIGN AND ANALYSIS OF QUALITY OF LIFE STUDIES IN CLINICAL TRIALS, SECOND EDITION D.L. Fairclough
DYNAMICAL SEARCH L. Pronzato, H. Wynn, and A. Zhigljavsky
FLEXIBLE IMPUTATION OF MISSING DATA S. van Buuren
GENERALIZED LATENT VARIABLE MODELING: MULTILEVEL, LONGITUDINAL, AND STRUCTURAL EQUATION MODELS A. Skrondal and S. Rabe-Hesketh
GRAPHICAL ANALYSIS OF MULTI-RESPONSE DATA K. Basford and J. Tukey
INTRODUCTION TO COMPUTATIONAL BIOLOGY: MAPS, SEQUENCES, AND GENOMES M. Waterman
MARKOV CHAIN MONTE CARLO IN PRACTICE W. Gilks, S. Richardson, and D. Spiegelhalter
MEASUREMENT ERROR ANDMISCLASSIFICATION IN STATISTICS AND EPIDEMIOLOGY: IMPACTS AND BAYESIAN ADJUSTMENTS P. Gustafson
MEASUREMENT ERROR: MODELS, METHODS, AND APPLICATIONS
J. P. Buonaccorsi
MEASUREMENT ERROR: MODELS, METHODS, AND APPLICATIONS
J. P. Buonaccorsi
MENDELIAN RANDOMIZATION: METHODS FOR USING GENETIC VARIANTS IN CAUSAL ESTIMATION S.Burgess and S.G. Thompson
META-ANALYSIS OF BINARY DATA USINGPROFILE LIKELIHOOD D. Böhning, R. Kuhnert, and S. Rattanasiri
MISSING DATA ANALYSIS IN PRACTICE T. Raghunathan
MODERN DIRECTIONAL STATISTICS C. Ley and T. Verdebout
POWER ANALYSIS OF TRIALS WITH MULTILEVEL DATA M. Moerbeek and S. Teerenstra
SPATIAL POINT PATTERNS: METHODOLOGY AND APPLICATIONS WITH R A. Baddeley, E Rubak, and R. Turner
STATISTICAL ANALYSIS OF GENE EXPRESSION MICROARRAY DATA T. Speed
STATISTICAL ANALYSIS OF QUESTIONNAIRES: A UNIFIED APPROACH BASED ON R AND STATA F. Bartolucci, S. Bacci, and M. Gnaldi
STATISTICAL AND COMPUTATIONAL PHARMACOGENOMICS R. Wu and M. Lin
STATISTICS IN MUSICOLOGY J. Beran
STATISTICS OF MEDICAL IMAGING T. Lei
STATISTICAL CONCEPTS AND APPLICATIONS IN CLINICAL MEDICINE
J. Aitchison, J.W. Kay, and I.J. Lauder
STATISTICAL AND PROBABILISTIC METHODS IN ACTUARIAL SCIENCE
P.J. Boland
STATISTICAL DETECTION AND SURVEILLANCE OF GEOGRAPHIC CLUSTERS
P. Rogerson and I. Yamada
STATISTICAL METHODS IN PSYCHIATRY AND RELATED FIELDS: LONGITUDINAL, CLUSTERED, AND OTHER REPEATED MEASURES DATA R. Gueorguieva
STATISTICS FOR ENVIRONMENTAL BIOLOGY AND TOXICOLOGY A. Bailer and W. Piegorsch
STATISTICS FOR FISSION TRACK ANALYSIS R.F. Galbraith
SURVIVAL ANALYSIS WITH INTERVAL-CENSORED DATA: A PRACTICAL APPROACH WITH EXAMPLES IN R, SAS, AND BUGS
K. Bogaerts, A. Komárek, and E. Lesaffre
VISUALIZING DATA PATTERNS WITH MICROMAPS D.B. Carr and L.W. Pickle
Interdisciplinary Statistics Series
SURVIVAL ANALYSIS with INTERVAL-CENSORED DATA
A Practical Approach with Examples in R, SAS, and BUGS
Kris Bogaerts
Arnošt Komárek
Emmanuel Lesaffre
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1.2.2Intervalandleftcensoring................
1.2.3Somespecialcasesofintervalcensoring........
1.2.4Doublyintervalcensoring................
1.5.1Likelihoodforinterval-censoreddata..........
1.6.1Homograftstudy.....................
1.6.7SignalTandmobielstudy................
1.7.1R..............................
2.1.3SASsolution.......................
2.2Comparisonoftwosurvivaldistributions...........
2.2.1Reviewofsignificancetests...............
2.3.1.1Modeldescriptionandestimation......
2.3.1.2Modelchecking.................
2.3.1.3Rsolution...................
2.3.1.4SASsolution..................
2.3.2Acceleratedfailuretimemodel.............
2.3.2.1Modeldescriptionandestimation......
3.1Nonparametricmaximumlikelihood..............
3.1.1Estimation........................
3.1.2Asymptoticresults....................
3.1.3Rsolution.........................
3.1.4SASsolution.......................
3.2.1Estimation........................
3.2.5SASsolution.......................
3.3.1Logsplinedensityestimation..............
3.3.1.1Asmoothapproximationtothedensity...
3.3.1.2Maximumlikelihoodestimation.......
3.3.1.3Rsolution...................
3.3.2ClassicalGaussianmixturemodel...........
3.3.3PenalizedGaussianmixturemodel...........
4.1Nonparametriccomparisonofsurvivalcurves.........
4.1.1Weightedlog-ranktest:derivation...........
4.1.2Weightedlog-ranktest:linearform...........
4.1.3Weightedlog-ranktest:derivedfromthelinear transformationmodel..................
4.1.4Weightedlog-ranktest:the G ,γ family........
4.1.7SASsolution.......................
4.2Samplesizecalculation.....................
4.3Concludingremarks.......................
5Theproportionalhazardsmodel
5.1Parametricapproaches.....................
5.1.1Maximumlikelihoodestimation.............
5.1.3SASsolution.......................
5.2Towardssemiparametricapproaches..............
5.2.1Piecewiseexponentialbaselinesurvivalmodel.....
5.2.1.1Modeldescriptionandestimation......
5.2.1.2Rsolution...................
5.2.1.3SASsolution..................
5.2.2SemiNonParametricapproach..............
5.2.2.1Modeldescriptionandestimation......
5.2.2.2SASsolution..................
5.2.3Spline-basedsmoothingapproaches...........
5.2.3.1Twospline-basedsmoothingapproaches...
5.2.3.2Rsolution...................
5.2.3.3SASsolution..................
5.3Semiparametricapproaches...................
5.3.1Finkelstein’sapproach..................
5.3.2Farrington’sapproach..................
5.3.3Iterativeconvexminorantalgorithm..........
5.3.4Groupedproportionalhazardsmodel..........
5.3.5Practicalapplications..................
5.3.5.1Twoexamples.................
5.3.5.2Rsolution...................
5.3.5.3SASsolution..................
5.4Multipleimputationapproach.................
5.4.1Dataaugmentationalgorithm..............
5.4.2Multipleimputationforinterval-censoredsurvivaltimes
5.4.2.1Rsolution...................
5.4.2.2SASsolution..................
5.5Modelchecking..........................
5.5.1CheckingthePHmodel.................
5.5.2Rsolution.........................
5.5.3SASsolution.......................
5.6Samplesizecalculation.....................
6.1Parametricmodel........................
6.1.1Maximumlikelihoodestimation.............
6.1.3SASsolution.......................
6.2PenalizedGaussianmixturemodel...............
6.2.1Penalizedmaximumlikelihoodestimation.......
6.2.2Rsolution.........................
6.3SemiNonParametricapproach.................
6.5.1Computationalapproach.................
6.5.2SASsolution.......................
6.6Concludingremarks.......................
7.1Nonparametricestimationofthebivariatesurvivalfunction.
7.1.1TheNPMLEofabivariatesurvivalfunction.....
7.1.2Rsolution.........................
7.1.3SASsolution.......................
7.2Parametricmodels........................
7.2.1Modeldescriptionandestimation............
7.2.3SASsolution.......................
7.3Copulamodels..........................
7.3.1Background........................
7.3.2Estimationprocedures..................
7.3.3Rsolution.........................
7.4Flexiblesurvivalmodels.....................
7.4.1ThepenalizedGaussianmixturemodel........
7.4.2SASsolution.......................
7.5Estimationoftheassociationparameter............
7.5.1Measuresofassociation.................
7.5.2Estimatingmeasuresofassociation...........
7.5.3Rsolution.........................
7.5.4SASsolution.......................
7.6Concludingremarks.......................
8.1Doublyinterval-censoreddata.................
8.1.1Background........................
8.1.2Rsolution.........................
8.2.1Frailtymodels......................
8.2.1.1Rsolution...................
8.2.1.2SASsolution..................
8.2.2Amarginalapproachtocorrelatedsurvivaltimes...
8.2.2.1Independenceworkingmodel.........
8.2.2.2SASsolution..................
8.3Abiplotforinterval-censoreddata...............
8.3.1Classicalbiplot......................
8.3.2Extensiontointerval-censoredobservations......
9.1Bayesianinference........................
9.1.1ParametricversusnonparametricBayesianapproaches
9.1.2Bayesiandataaugmentation...............
9.1.3MarkovchainMonteCarlo...............
9.1.4Credibleregionsandcontourprobabilities.......
9.1.5Selectingandcheckingthemodel............
9.2.1Bayesiannonparametricmodellingofthehazard
9.2.2Bayesiannonparametricmodellingofthedistribution
9.3Bayesiansoftware........................
9.3.1WinBUGSandOpenBUGS...............
9.3.3
9.4Applicationsforright-censoreddata..............
9.4.1Parametricmodels....................
9.4.1.1BUGSsolution.................
9.4.1.2SASsolution..................
9.4.2NonparametricBayesianestimationofasurvivalcurve
9.4.2.1Rsolution...................
9.4.3SemiparametricBayesiansurvivalanalysis.......
10.1Bayesianparametricmodelling.................
10.2Bayesiansmoothingmethods..................
11.3SemiparametricPHmodel...................
12.1BayesianparametricAFTmodel................
12.1.1JAGSsolution......................
12.2AFTmodelwithaclassicalGaussianmixtureasanerror
12.2.1Rsolution.........................
12.3AFTmodelwithapenalizedGaussianmixtureasanerror
12.4BayesiansemiparametricAFTmodel.............
13.1.1Parametricsharedfrailtymodels............
13.1.1.1JAGSsolution.................
13.1.1.2SASsolution..................
13.1.2Flexiblesharedfrailtymodels..............
13.1.3Semiparametricsharedfrailtymodels.........
13.2.1.1JAGSsolution.................
13.2.1.2SASsolution..................
13.2.2Bivariatecopulamodels.................
13.2.3Flexiblebivariatemodels................
13.2.3.1Rsolution...................
13.2.4Semiparametricbivariatemodels............
13.2.4.1Rsolution...................
13.2.5Multivariatecase.....................
13.3Doublyintervalcensoring....................
13.3.1ParametricmodellingofunivariateDI-censoreddata.
13.3.1.1JAGSsolution.................
13.3.2FlexiblemodellingofunivariateDI-censoreddata...
13.3.2.1Rsolution...................
13.3.3SemiparametricmodellingofunivariateDI-censored data............................
13.3.3.1Rsolution...................
13.3.4ModellingofmultivariateDI-censoreddata......
13.4Concludingremarks.......................
14.1.1Competingrisksandmultistatemodels........
14.1.2Survivalmodelswithacuredsubgroup.........
14.1.3Multilevelmodels.....................
14.1.4Informativecensoring..................
14.1.5Interval-censoredcovariates...............
14.1.6Jointlongitudinalandsurvivalmodels.........
14.1.7Spatial-temporalmodels.................
C.4Inversegammaprior:
C.5Wishartprior:Wishart(
C.6InverseWishartprior:Wishart(R,k).............
C.7LinkbetweenBeta,DirichletandDirichletProcessprior.. 508
D.1icensBKLpackage........................
D.2Icenspackage...........................
D.3intervalpackage.........................
D.4survivalpackage.........................
D.5logsplinepackage........................
D.6smoothSurvpackage......................
D.7mixAKpackage.........................
D.8bayesSurvpackage........................
E.4PROCICPHREG........................
F.1IterativeConvexMinorant(ICM)algorithm.........
F.2Regionsofpossiblesupportforbivariateinterval-censoreddata 536
F.2.1AlgorithmofGentlemanandVandal(2001)...... 536
F.2.2AlgorithmofBogaertsandLesaffre(2004)....... 537
F.2.3HeightmapalgorithmofMaathuis(2005)....... 538
F.3Splines.............................. 539
F.3.1Polynomialfitting....................
F.3.2Polynomialsplines....................
F.3.3Naturalcubicsplines................... 541
F.3.4Truncatedpowerseries.................. 541
F.3.5B-splines......................... 541
F.3.6M-splinesandI-splines.................. 543
F.3.7Penalizedsplines(P-splines)............... 543
ListofFigures
1.1Right,leftandintervalcensoring................ 6
1.2Doublyintervalcensoring....................
1.3Breastcancerstudyoftheradiotherapy-onlygroup.Median timetobreastretraction..................... 9
1.4One(true)simulateddatasetfromeithersettingusedforthe illustrationofmid-pointimputation.............. 11
1.5Breastcancerstudy.Observedintervalsinmonthsfortimeto breastretractionofearlybreastcancerpatientspertreatment group............................... 21
1.6AIDSclinicaltrial.Observedintervalsinmonthsfortimeto CMVsheddingandtimetoMACcolonization........ 23
1.7Sensoryshelflifestudy.Shelflifeofyoghurtstoredat42◦Cin hours............................... 24
1.8Mobilestudy.Interval-censoredtimesofpreviousandofcurrentmobilephonepurchase................... 26
1.9Mastitisstudy.Timefromparturitiontomastitisindaysby location.............................. 28
1.10SignalTandmobielstudy.FDInumberingsystemofdeciduous andpermanentteeth.......................
2.1Homograftstudy.Kaplan-Meiercurveofhomograftfailureaccordingtotypeofgraft..................... 37
2.2Homograftstudy.Estimatedsurvivalcurvesfor14-year-old patientsbasedonthePHmodel................ 49
2.3Homograftstudy.Martingaleresidualsvs.linearpredictor.. 50
2.4ImpactofacovariateonthehazardofaPHandAFTmodel 54
3.1DeterminationofregionsofpossiblesupportfortheTurnbull
3.2Orderingoftwointerval-censoredobservationswithtiedendpoints.............................. 64
3.3Breastcancerstudyoftheradiotherapy-onlygroup.NPMLE ofthecumulativedistributionandsurvivalfunctions..... 69
3.4SignalTandmobielstudy.Log-normalmodelforemergenceof tooth44ofboys.........................
3.5SignalTandmobielstudy(boys).Probabilityplotoflognormalmodelforemergenceoftooth44............ 79
3.6Breastcancerstudy(radiotherapy-alonegroup).Distribution ofthetimetobreastretractionestimatedusingthelogspline method.............................. 87
3.7Severaldensitiesexpressedastwo-orfour-componentGaussianmixtures........................... 90
3.8SeveraldensitiesexpressedashomoscedasticGaussianmixtures............................... 94
3.9SignalTandmobielstudy(boys).Distributionofthetimeto emergenceoftooth44estimatedusingthepenalizedGaussian mixture............................. 98
3.10SignalTandmobielstudy(boys).Distributionofthestandardizedlog-timetoemergenceoftooth44estimatedusingthe penalizedGaussianmixturecomparedtoparametricdensities. 102
4.1SignalTandmobielstudy.NPMLEofthesurvivalfunctionsfor emergenceofpermanenttooth44intwogroupsaccordingto baselineDMFstatusofprimarytooth84andinthreegroups accordingtoocclusalplaquestatusofpermanenttooth46.. 114
5.1Breastcancerstudy.ValidationofPHassumptionfortreatmentviaatransformationofthesurvivalfunction...... 173
5.2Breastcancerstudy.ValidationofPHassumptionfortreatmentviaresiduals........................ 174
5.3Sensoryshelflifestudy.Baselinesurvivalfunctionfordifferent models............................... 177
6.1SignalTandmobielstudy.ParametricAFTmodelbasedsurvivalfunctionsforemergenceofpermanenttooth44ingender byDMFgroups.......................... 184
6.2SignalTandmobielstudy.ParametricAFTmodelbasedsurvivalfunctionsforemergenceofpermanenttooth44compared toNPMLE............................ 186
6.3SignalTandmobielstudy.PGMAFTmodelsbasedestimated errordensitiescomparedtoadensityofthestandardnormal distribution............................ 200
6.4SignalTandmobielstudy.PGMAFTmodelsbasedestimated survivalfunctionsforemergenceofpermanenttooth44comparedtoNPMLE......................... 203
6.5SignalTandmobielstudy.Mean-scaleAFTmodelforemergenceoftooth44withaPGMerrordistribution.Estimated densityofthestandardizederrortermcomparedtoparametricdensities............................ 208
6.6SignalTandmobielstudy.Themean-scalePGMAFTmodel basedestimatedsurvivalandhazardfunctions........ 212
7.1Graphicalrepresentationof4bivariateinterval-censoredobservations............................ 221
7.2Artificialexamplewithmoreregionsofsupportthanobservations............................... 223
7.3Anartificialdatasetwith6observedrectanglesandtheir corresponding4regionsofsupport.............. 223
7.4DensityplotsofClayton,normalandPlackettcopula.... 235
7.5SignalTandmobielstudy.DensityofpenalizedGaussianmixturemodelforemergenceofpermanentteeth14and24... 242
7.6SignalTandmobielstudy.Theestimatedcross-ratiofunction forthemaxillarfirstpremolarsforboys............ 249
8.1Threecasesofdoublyinterval-censoredsurvivaltimes.... 256
8.2Mobilestudy.Forestplotshowingtheimpactofgender,householdsizeandageonthetimetochangephone........ 258
8.3SignalTandmobielstudy.Biplot................ 277
9.1Illustrationofgammaprocess:Tenrealizationsof G(cH ∗,c) 298
9.2IllustrationoftheDirichletprocess:Tenrealizationsof DP{c Weibull(1.5,7)} 300
9.3Homograftstudy.MCMCdiagnostics............. 313
9.4Homograftstudy(aortichomograftpatients).Nonparametric Bayesianestimateofsurvivalfunction............ 317
9.5Homograftstudy.SemiparametricPHmodelBayesianestimatesofsurvivalfunctions................... 320
10.1SignalTandmobielstudy.DiagnosticplotsofBayesiananalysisusingthe SAS procedure LIFEREG 334
10.2SignalTandmobielstudy(boys,emergenceoftooth44).Posteriordensitiesofselectedparameters,distributionofthe emergencetimeestimatedusingtheclassicalGaussianmixture................................ 338
10.3Breastcancerstudyoftheradiotherapy-onlygroup.NonparametricBayesianestimateofsurvivalfunction......... 346
10.4SignalTandmobielstudy.Estimateddensity,survivaldistributionandhazardfunctionfrom DPMdencens oftheemergence timefortooth44ofboysfortwoprecisionparameters... 348
10.5SignalTandmobielstudy.Imputedemergencetimesfortooth 44ofboysfrom2ndsolutionobtainedfromthe R package DPpackage togetherwithobservedintervals......... 352
11.1Breastcancerstudy.DiagnosticplotsofBayesiananalysisusing runjags 361
11.2Breastcancerstudy.Q-Qplotstocontrastthe‘truelatent’survivaltimeswiththe‘model-basedreplicated’survival times............................... 363
11.3Breastcancerstudy.PPCscorrespondingtotherangeand maxgaptest........................... 364
11.4Breastcancer:SmoothandWeibullsurvivalfunctions.... 371
11.5SignalTandmobielstudy.Estimatedpiecewiseconstantdynamicregressioncoefficientsobtainedfrom dynsurv 374
11.6SignalTandmobielstudy.Estimatedpiecewiseconstantbaselinehazardfunctionsobtainedfrom dynsurv.......... 376
11.7SignalTandmobielstudy.Frequencyofjumppointsobtained from dynsurv........................... 379
12.1SignalTandmobielstudy.CGMAFTmodelbasedposterior predictiveerrordensitycomparedtoadensityofthestandard normaldistribution........................ 393
12.2SignalTandmobielstudy.CGMAFTmodelbasedposteriorpredictivesurvivalfunctionsforemergenceofpermanent tooth44comparedtoNPMLE................. 394
12.3SignalTandmobielstudy.CGMAFTmodelbasedposterior predictivehazardfunctionsforemergenceofpermanenttooth 44................................. 395
12.4SignalTandmobielstudy.PGMAFTmodelbasedposterior predictiveerrordensitycomparedtoadensityofthestandard normaldistribution........................ 407
12.5SignalTandmobielstudy.PGMAFTmodelbasedposteriorpredictivesurvivalfunctionsforemergenceofpermanent tooth44comparedtoCGMAFTbasedestimates...... 408
12.6SignalTandmobielstudy.SemiparametricBayesianAFT modeltraceplotsandmarginalposteriordensities...... 415
12.7SignalTandmobielstudy.Posteriorpredictiveerrordistribution................................ 417
12.8SignalTandmobielstudy.Posteriorpredictivesurvivaldistribution............................... 418
13.1Mastitisstudy.Log-normalfrailtydistributionwithWeibull baselinehazard:Estimatedrandomeffectsandtruesurvival times............................... 426
13.2SignalTandmobielstudy.Estimatederror(left)andrandom effectsdensity(right)forNorm-PGMandPGM-Normmodels toevaluatetheemergencetimesofteeth14,24,34and44.. 435
13.3SignalTandmobielstudy.Estimatedpredictiveincidence curveandhazardfunctionfromPGM-PGMmodeltocompare emergencetimesoftooth24forboyswithcariesonprimary predecessorornot........................ 436
13.4SignalTandmobielstudy.Imputedemergencetimesofteeth 14and24assumingabivariatelog-normaldistribution.... 441
13.5SignalTandmobielstudy.Contourplotsoftheestimatederror distributionsforemergencetimesofteeth14and24andteeth 24and34............................. 449
13.6SignalTandmobielstudy.Emergencedistributionsfortooth 14andtooth24forfourcovariatecombinations........ 452
13.7Mobilestudy.Histogramlog(time)purchase1stmobilephone andlog(gaptime)between2purchases............. 460
13.8Mobilestudy.Estimatederrordensityfortimeoffirstpurchaseandestimatederrordensityforgaptime........ 462
13.9Mobilestudy.Estimatedincidencefunctiontobuyanewmobilephonesplitupintoagegroups............... 463
B.1Threesurvivaldistributions...................
F.1Graphcorrespondingtothedatapresentedin Figure7.3 537
F.2Heightmapcorrespondingtotheexampledatapresentedin Figure7.3 ............................ 538
F.3Truncatedpolynomialofdegree1andB-splinesofdegree3 on[0, 10]............................. 542
F.4M-splinesoforder2and3................... 544
Notation
S(t)survivalfunction S
F (t)cumulativedistributionfunction F
f (t)densityfunction f (t)
(t)hazardfunction (t)
H(t)cumulativehazardfunction H(t)
[l,u]closedintervalwithlowerlimit l andupperlimit u
(l,u]half-openintervalwithlowerlimit l andupperlimit u
l,u open,half-openorclosedintervalwithlowerlimit l andupper limit u
δ censoringindicator
L(·)likelihood
(·)log{L(·)},log-likelihood
D collecteddata
DP Dirichletprocess
Dir p(δ1,...,δp) p-dimensionalDirichletdistributionwithparameters δ1,...,δp N (µ,σ2)normaldistributionwithmean µ andvariance σ2
ϕ densityofthestandardnormalGaussiandistribution N (0, 1)
ϕµ,σ2 densityoftheGaussiandistribution N (µ,σ2)
ΦstandardcumulativedistributionfunctionoftheGaussiandistribution N (0, 1)
Φµ,σ2 cumulativedistributionfunctionoftheGaussiandistribution N (µ,σ2)
Φρ standardbivariatecumulativedistributionfunctionofthe Gaussiandistributionwithcorrelation ρ
G(ζ,γ)gammadistributionwithashapeparameter ζ andarateparameter γ (withthemean ζ/γ)
N p(µ, Σ) p-dimensionalnormaldistributionwithmeanvector µ and covariancematrix Σ
I(·)indicatorfunctionequalto1iftheexpressionbetweenparenthesesistrue,and0otherwise
lp penalizedlog-likelihood
∆k( )k-orderforwarddifferencefunction
I identitymatrix
I Hessianmatrix
RanF rangeofthefunction F
C(u,v)copula
˘
C(u,v)survivalcopula
˘
CC
θ (u,v)Claytoncopulawithparameter θ
˘
CG
ρ (u,v)Gaussiancopulawithparameter ρ
˘ CP
θ (u,v)Plackettcopulawithparameter θ
· Euclideanlength
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