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Survival Analysis and the ACT study Laura Gibbons, PhD Thanks to An Introduction to Survival Analysis Using Stata
Acknowledgement • Funded in part by Grant R13 AG030995 from the National Institute on Aging • The 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 to event data. • 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: A population-based study Kristen Dams-O’Connor, Laura E Gibbons, James D Bowen, Susan M McCurry, Eric B Larson, Paul K Crane. J NeurolNeurosurg Psychiatry 2013 Feb;84(2):177-82.
TBI-LOC = Traumatic Brain Injury with Loss of Consciousness Outcomes (different ways of defining failure) • TBI-LOC during follow-up • Dementia • Death
Survival function • The number who survive out of the total number at risk • In 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 with LOC (with 95% confidence bands)
Think carefully about onset of time at risk • Study entry • Time-dependent covariates • What 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 risk • Exposure: report of first TBI-LOC at baseline None (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 is known Discrete – time interval for failure is known
ACT onsetdate for 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 aging study. • Often modeled as baseline age + time. Ties • Multiple events 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 unknown • No longer at risk • Missing data – random or informative? • Hope it’s noninformative [Distribution of censoring times is independent of event times, conditional on covariates. ~ MAR.]
Right censoring Event is unobserved due to • Drop out • Study end • Competing event (more on this later) Interval censoring • Know it occurs between visits, but not when • Assume failure time is uniformly distributed in that interval • An issue in ACT (hence onsetdate)
Left censoring The event 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 truncation Onset of 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) or • Visit date (visitdt) • Withdrawal date (withdrawdt) (The FH data does not include anyone who withdrew.) • Death date (deathdt)
Modeling Non parametric – Kaplan-Meier
Log-rank test for equality of survivor functions | Events Events p | observed expected ---------------------------+------------------------- No TBI-LOC before baseline | 66 82.70 TBI-LOC before baseline | 30 13.30 ---------------------------+------------------------- Total | 96 96.00 chi2(1) = 24.44 Pr>chi2 = 0.0000
Semi-parametric (Cox) Assumes the hazards are proportional • Looks like a reasonable assumption here, but we looked at a variety of graphs and statistics 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 TBI Late life TBI with LOCwith LOC cases/person years HR (95% CI) None prior to baseline 66/21,945 1 (Reference) < 25 15/2147 2.54 (1.42, 4.52) 25-54 6/678 3.24 (1.40, 7.52) 55-baseline 9/798 3.79 (1.89, 7.62)
Model checking • Proportional hazards assumption • Covariate form • Baseline, lag or current visit covariate • Et cetera
Parametric Can be proportional hazard models • Exponential. Constant baseline hazard. • Weibull. Hazard is monotone increasing or decreasing, depending on the values for a and b. • Gompertz. Hazard rates increase or decrease exponentially over time. • See Flexible Parametric Survival Analysis Using Stata for 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, Weibull can also be formulated as AFT models. In the Weibull model, 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-LOC and the risk of dementia • Proportional hazards assumption not tenable. • The log-logistic AFT model was the winner, reflecting an increased risk over time. • You can compare AICs to pick best model, or pick one based on your hypothesis.
Our AFT model for any dementia • Controlling for baseline age, gender and any APOE-4 alleles • Remember that TR > 1 => longer survival
TBI-LOC is NS. Older baseline age and APOE associated with shorter survival. Female and education associated with longer survival. ------------------------------------------------ _t | TR [95% Conf. Interval] -------------+---------------------------------- base4 | <25 | 1.02 0.87 1.20 25-54 | 1.04 0.78 1.38 55+ | 1.06 0.81 1.39 | 10 years age | 0.53 0.49 0.57 female | 1.15 1.05 1.26 education | 1.02 1.00 1.04 apoe4 | 0.69 0.63 0.76 ------------------------------------------------
Competing risks • Individuals are at-risk for AD, vascular dementia, other dementias • No 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? • Which is it? • One process determines dementia and another says which type • Two 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, aka unobserved heterogeneity 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. • Population hazard can fall while the individual hazards rise. • The frailer individuals have failed already, so the overall hazard rate drops. Yet time 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.
Other issues in survival analysis not covered today include • Events that can occur more than once (heart attacks, for example) • Parallel processes • Unshared frailty