Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 20Page 21Page 22Page 231Multivariate survival analysis (chapter 13)Issues here similar to issues for multivariate orlongitudinal data analysisexamples:multiple events in same individualrecurrent events:subjects with kidney transplants, acute rejectionepisodes times until different events in same individual: time until first acute rejection episode, kidneyfailuretime until learn to read, learn to writeseparate individualstime until infection for husband/wife; siblings2let i denote clustering unit, k unit within cluster (e.g.,couple, subject; individual, event of type k inindividual)for recurrent failure-times, kth event in ith subjectdiscuss modeling/analytic optionsdiscuss problems with these optionsModels:population-averaged/marginal: subject/stratum-specific: both types of models potentially valid for dataquestions:which model is desirable?what model parameters are estimable? How?3What is wrong with ignoring the clustering unit i?(e.g., using standard fixed effects models without a termfor cluster)41. Observations within clusters are more likely to besimilar to each other with respect to hazards/risksthan observations in different clustersConsequently, outcomes are not i.i.d. (Evengiven baseline covariates X)treating them as i.i.d. leads to improper estimates ofvariancesometimes considered the problem of longitudinal studies2. causal inference setting: cluster i may be associatedwith risk, treatment/exposurethus, confounding by cluster5analytic options:1. Standard fixed effects models including indicatorvariables for cluster iIf number of clusters I is large, estimatorsinconsistent (even if model correctly specified)May be acceptable if interested in stratum-specificcomparison of other variables, strata are large; thenneed not report coefficients for strata2. Fixed effects model including stratum-specificvariables perhaps related to risk of outcome(E.g., stratum-specific probability of exposure/propensity score, risk score, etc.)danger of model misspecification-leads toinconsistent/biased estimates of subject specificparameters63. Stratified Cox models:separate baseline hazard for each stratum, estimatecommon coefficientsestimates of regression coefficients $ do not requireestimates of the baseline hazardacceptable for stratum/subject specific parameters ifhave multiple units at risk at given time (i.e., not formodels of recurrent events in individual)Analogous to conditional logistic regression (in fact,the likelihood can be the same)Analogues not fully available for parametric modelsto my knowledge (i.e, need to estimate stratum-specific parameters)74. Frailty modelsrandom effects models which assume that stratum-specific effect comes from some distributionproportional hazards models:wi are frailties/random effects; assumed to come fromsome standard distribution with mean 0, variance 1Here ui are frailties with mean 1, unknown variancecan also have frailties for accelerated failure-timemodels:8Most common model for frailties ui gamma frailty:Joint survival function for subjects in ith group:see book for derivation; involves taking the expectationof with respect to uiderive from this log-likelihood for $ and 2maximize likelihood to estimateeasily done if assume parametric form for baseline hazardmore complicated for semiparametric model; see book fordetails9advantages:broadly applicable (to all types of problem discussedhere)problems:distributional assumptions for random effectsin estimation of causal effects, will not necessarilycompletely control confounding by groupassumption is that random effects not associated withregressors Xif residual confounding by group,10marginal 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 partiallikelihood estimatoruse adjusted “Sandwich” variance estimatorderivation (loose):estimating equations: also have11now, we have, sincegroups i are assumed independenthowever, to estimate , wemust account for nonindependence of observations kwithin cluster iform given in book12advantages: easily fitwidely applicableproblem: does not account for confounding by group13transition models (longitudinal data analysis terminology)for recurrent event history account for nonindependence by conditioning on previousevent historye.g., may have different time-varying covariates for samesubjecttwo issues:for baseline covariates, effect of covariate on recurrentfailures may be explained by effect on first failure; thus,may be likely to have event # by covariate interactionmay still want to use robust variance estimator in caseresidual correlation not accounted for by modelspecification14annotated artificial examples using stataall examples: strata size 2one treatment variable a( frailtyrelative hazard for A within strata: 2first example: one subject treated (A=1) one not in eachpairsecond example: each subject assigned to treatment withprobability 0.5Third example: subject’s probability ofexposure/treatment depends on frailty; subjects withlarger frailties have lower probability A=1frailty here ((0.5)15. 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 .5093432------------------------------------------------------------------------------same point estimate, smaller standard error with robust variance estimate16. stcox a, strata(xind)