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Survival Analysisandthe ACT study
Laura Gibbons,PhDThanks toAn Introduction to Survival Analysis UsingStata
Funded in part by Grant R13 AG030995 from the National Institute on AgingThe views expressed in written conference materials or publications and by speakers and moderators do not necessarily reflect the official policies of the Department of Health and Human Services; nor does mention by trade names, commercial practices, or organizations imply endorsement by the U.S. Government.
What is survival analysis?
Time toeventdata.It’s not just a question of who gets demented, but when.Event, survive, and fall are generic terms.
ACT example
Risk for Late-life Re-injury, Dementia, and Death Among Individuals with Traumatic Brain Injury:Apopulation-based studyKristenDams-O’Connor, Laura E Gibbons,JamesD Bowen, Susan M McCurry, Eric B Larson,PaulK Crane.JNeurolNeurosurgPsychiatry2013 Feb;84(2):177-82.
TBI-LOC =Traumatic Brain Injury withLossofConsciousnessOutcomes (different ways of defining failure)TBI-LOC during follow-upDementiaDeath
Survival function
The number who survive out of the total number at riskIn this example, “failure” is a TBI-LOC after baseline.4225 participants, with 96 TBI-LOC after baseline.
Hazard function
The probability of failing given survival until this time (currently at risk)The hazard function reflects the hazard at each time point.It’s usually easier to look at the cumulative hazard graph.
Cumulative Hazard for TBI withLOC
(with 95% confidence bands)
Think carefully about onset oftimeat risk
Study entryTime-dependent covariatesWhat to do about exposures which occur before study entry (left truncation)
ACT :TBI-LOC during follow-up
Used study entry as onset of time at riskExposure: report of first TBI-LOC at baselineNone (n = 3619)At age<25 (n=371)At age 25-54 (n=104)At age 55 to baseline (n=131)No time-dependent covariates for this example
Time axis
Continuous – exact failure time isknownDiscrete – time interval for failure is known
ACTonsetdatefor dementia outcomes
The midpoint between the two study visits (biennial and/or annual) that precede the date of the consensus of dementia. The date of the consensus of dementia is defined as the earliest consensus that resulted in a positive diagnosis of dementia (DSMIV) and that was not later reversed as a false positive.
Age as the time axis
Makes sense in an agingstudy.Often modeled as baseline age +time.TiesMultipleevents occurring at the same time.Make sure your software is handling this the way you want.
Think carefully about censoring
Censoring: The event time is unknownNo longer at riskMissing data – random or informative?Hope it’snoninformative[Distribution of censoring times is independent of event times, conditional on covariates. ~ MAR.]
Right censoringEventis unobserved due toDrop outStudy endCompeting event (more on this later)Interval censoringKnowit occurs between visits, but not whenAssume failure time is uniformly distributed in that intervalAn issue in ACT (henceonsetdate)
LeftcensoringTheevent occurred before the study began.What about those whose TBI-LOC resulted in death or dementia before age 65? They are not in our study.Worry about this one.Left truncationOnsetof risk was before study entry.We used our 4-category exposure, but risk really must be defined as “TBI-LOC before age 25 and not left-censored”, etc.
ACT censoring variables
Competing event:onsetdate(dementia) orVisit date (visitdt)Withdrawal date (withdrawdt)(The FH data does not include anyone who withdrew.)Death date (deathdt)
Non parametric – Kaplan-Meier
Log-ranktest for equality of survivor functions| EventsEventsp | observed expected---------------------------+-------------------------No TBI-LOC before baseline | 66 82.70TBI-LOC before baseline | 30 13.30---------------------------+-------------------------Total | 96 96.00chi2(1) = 24.44Pr>chi2 = 0.0000
Semi-parametric (Cox)
Assumesthe hazards are proportional
Looks like a reasonable assumption here, but we lookedata variety of graphsandstatistics to make sure.
Hazard Ratios
Baseline report of age at first TBI with LOC as a predictor of TBI with LOC after study enrolment, controlling for age, sex, and years of education.Age at first TBILatelife TBI withLOCwithLOC cases/personyearsHR (95% CI)None prior tobaseline 66/21,9451(Reference)< 2515/21472.54(1.42, 4.52)25-546/6783.24(1.40, 7.52)55-baseline9/7983.79 (1.89, 7.62)
Model checking
Proportional hazards assumptionCovariate formBaseline, lag or current visit covariateEt cetera
Can be proportional hazard modelsExponential. Constant baseline hazard.Weibull. Hazard is monotone increasing or decreasing, depending on the values foraandb.Gompertz. Hazard rates increase or decrease exponentially over time.SeeFlexible Parametric Survival Analysis UsingStatafor many more.
Accelerated failure time
Risk is not constant over time.Time ratios. Ratios > 1 indicate LONGER survival.
Types of accelerated failure time(AFT) models
Gamma. 3 parameter. Most flexible. Fit a gamma model and see which parameters are relevant.Exponential,Weibullcan also be formulated as AFT models. In theWeibullmodel, the risk increases over time when β > 1.Log-normal. The hazard increases and then decreases.Log-logistic. Very similar to log-normal.
Baseline report of TBI-LOCandthe risk of dementia
Proportional hazards assumption not tenable.The log-logistic AFT model was the winner, reflecting an increased risk overtime.You can compare AICs to pick best model, or pick one based on your hypothesis.
Our AFTmodelforanydementia
Controlling for baseline age, gender and any APOE-4 allelesRemember that TR > 1 => longersurvival
------------------------------------------------_t |TR[95% Conf. Interval]-------------+----------------------------------base4 |<25 | 1.02 0.87 1.2025-54 | 1.04 0.78 1.3855+ | 1.06 0.81 1.39|10 years age | 0.53 0.49 0.57female |1.15 1.05 1.26education |1.02 1.001.04apoe4 | 0.69 0.63 0.76------------------------------------------------
TBI-LOCis NS. Older baseline age andAPOEassociated with shorter survival. Female and education associated with longer survival.
Competing risks
Individuals are at-risk for AD, vascular dementia, other dementiasNo longer at risk for one type once diagnosed with another (assuming we’re dealing with first diagnosis)Use cause-specific hazard functions and cumulative incidence functions.
What is going on in competing risks?
Whichis it?One process determines dementia and another says which typeTwo separate processes, and one event censors the other.In our analysis of TBI-LOC predicting AD, we censored at other dementia diagnoses, but we could have modeled multiple dementia outcomes (assuming adequate numbers).The competing risk model may be more accurate, because the time to AD and the time to other dementia are probably correlated.
Shared frailty, akaunobservedheterogeneity or random-effects
There may be variability in individuals’ underlying (baseline) risk for an event that is not directly measurable.One way of dealing with patients from different cities, for example.The assumption is that the effect is random and multiplicative on the hazard function.
Need to distinguish between hazard for individuals and the population average.Populationhazard can fall while the individual hazards rise.Thefrailer individuals have failed already, so the overall hazard rate drops.Yettime is passing, so each person’s risk is still rising.
In a shared frailty model, HR estimates are for time 0.Covariate effects decrease as the frail fail.Gamma frailty models. Covariate effects completely disappear over time.Inverse-Gaussian models. Covariate effects decrease but do not disappear over time.
Otherissues in survival analysis not covered today include
Eventsthat can occur more than once(heart attacks, for example)Parallel processesUnshared frailty
Questions, discussion





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Survival Analysis and the ACT study - Welcome to nginx!