Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 23Page 24Page 25Page 26Page 27Page 281Multivariate survival analysis (chapter 13)Issues here similar to issues for multivariate or longitudinal data analysisclassification:multiple events in same individualrecurrent events (of same type)different types of eventsseparate individualsgive examples:2multiple events in same individualrecurrent events:subjects with kidney transplants, acute rejection episodes times until different events in same individual: time until first acute rejection episode, kidney failuretime until learn to read, learn to writeseparate individualstime until infection for husband/wife; siblings3let i denote clustering unit, k unit within cluster (e.g., couple, subject; individual,event of type k in individual)for recurrent failure-times, kth event in ith subjectdiscuss modeling/analytic optionsdiscuss problems with these options4Models:population-averaged/marginal: subject/stratum-specific: both types of models potentially valid for dataquestions:which model is desirable?what model parameters are estimable? How?5What is wrong with ignoring the clustering unit i?(e.g., using standard fixed effects models without a term for cluster)61. Observations within clusters are more likely to be similar to each other withrespect to hazards/risks than observations in different clustersConsequently, outcomes are not i.i.d. (Even given baseline covariatesX)treating them as i.i.d. leads to improper estimates of variancesometimes considered the problem of longitudinal studies2. causal inference setting: cluster i may be associated with risk,treatment/exposurethus, confounding by clusteranalytic options:7analytic options:1. Standard fixed effects models including indicator variables for cluster iIf number of clusters I is large, estimators inconsistent (even if modelcorrectly specified)May be acceptable if interested in stratum-specific comparison of othervariables, strata are large; then need not report coefficients for strata82. Fixed effects model including stratum-specific variables perhaps related torisk of outcome(E.g., stratum-specific probability of exposure/ propensity score, risk score,etc.)danger of model misspecification-leads to inconsistent/biased estimates ofsubject specific parameters93. Stratified (PH or AFT) models:separate baseline hazard for each stratum, estimate common coefficientsestimates of regression coefficients $ do not require estimates of thebaseline hazardacceptable for stratum/subject specific parameters if have multiple units atrisk at given time (i.e., not for models of recurrent events in individual)Analogous to conditional logistic regression (in fact, the likelihood can bethe same)Analogues not fully available for parametric models to my knowledge (i.e,need to estimate stratum-specific parameters)104. Frailty modelsrandom effects models which assume that stratum-specific effect comesfrom some distributionproportional hazards models:iw are frailties/random effects; assumed to come from some standarddistribution with mean 0, variance 1iHere u are frailties with mean 1, unknown variancecan also have frailties for accelerated failure-time models:11iMost common model for frailties u gamma frailty: Joint survival function for subjects in ith group:see book for derivation; involves taking the expectation of iwith respect to uderive from this log-likelihood for $ and 2maximize likelihood to estimateeasily done if assume parametric form for baseline hazardmore complicated for semiparametric model; see book for details12advantages:broadly applicable (to all types of problem discussed here)problems:distributional assumptions for random effectsin estimation of causal effects, will not necessarily completely controlconfounding by groupassumption is that random effects not associated with regressors Xif residual confounding by group,13marginal model, robust estimationstructural model does not include group iaccount for nonindependence in variancefor various models, use usual point estimatorse.g., for proportional hazards model, maximum partial likelihood estimatoruse adjusted “Sandwich” variance estimator14derivation (loose):estimating equations: also have15now, we have , since groups i areassumed independenthowever, to estimate , we must account fornonindependence of observations k within cluster iform given in book16advantages: easily fitwidely applicableproblem: does not account for confounding by group17transition models (longitudinal data analysis terminology) for recurrent eventhistory account for nonindependence by conditioning on previous event historye.g., may have different time-varying covariates for same subjecttwo issues:for baseline covariates, effect of covariate on recurrent failures may beexplained by effect on first failure; thus, may be likely to have event # bycovariate interactionmay still want to use robust variance estimator in case residual correlation notaccounted for by model specification18annotated artificial examples using stata all examples: strata size 2one treatment variable a( frailtyrelative hazard for A within strata: 2first example: one subject treated (A=1) one not in each pairsecond example: each subject assigned to treatment with probability 0.5Third example: subject’s probability of exposure/treatment depends on frailty;subjects with larger frailties have lower probability A=1frailty here ((0.5)19. infile xind a t delta tstart t3 u using d:\gausfile\nprop5.txt(10000 observations read). stcox a, nohr------------------------------------------------------------------------------ _t | _d | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- a | .4589076 .033992 13.50 0.000 .3922844 .5255308------------------------------------------------------------------------------. stcox a, cluster(xind) nohr (standard errors adjusted for clustering on xind)------------------------------------------------------------------------------ _t | Robust _d | Coef. Std. Err. z P>|z| [95% Conf. Interval]-------------+---------------------------------------------------------------- a | .4589076 .0257329 17.83 0.000 .408472