Continuous Time Survival Analysis

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Information about Continuous Time Survival Analysis
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Published on November 16, 2007

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Continuous time survival analysis ALDA, Chapters 13, 14, and 15:  Judith D. Singer & John B. Willett Harvard Graduate School of Education Continuous time survival analysis ALDA, Chapters 13, 14, and 15 “Time is nature’s way of keeping everything from happening at once” Woody Allen What we will cover:  What we will cover Moving from discrete to continuous time (§13.1-13.3)—new definitions and estimation strategies (and hazard gets harder to estimate) The cumulative hazard function (§13.4)—a relative of hazard that’s easy to estimate Specifying and fitting the continuous time hazard model to data (§14.2)—Cox regression—a.k.a. the proportional hazards model—is straightforward to implement and interpret Evaluating the results of model fitting (§14.3)—interpreting parameter estimates, testing hypotheses, & evaluating goodness of fit Including time-varying predictors (§15.1)—It’s not always as easy as in discrete-time, but it can be done Evaluating and relaxing the proportionality assumption (§15.2). You can test and, if necessary, relax the proportional hazards assumption—effectively fitting non-proportional proportional hazards models! © Singer & Willett, page 2 What happens when we record event occurrence in continuous time?:  What happens when we record event occurrence in continuous time? We know the precise instant when events occur: e.g., Jane took her first drink at 6:19pm the day after release from an alcohol treatment program There exist an infinite number of these instants because any division of continuous time can always be made finer—e.g., femtoseconds (10-15), attoseconds (10-18), yoctoseconds (10-24) The probability of observing any particular event time is infinitesimally small and approaches 0 as time’s divisions get finer—we must therefore redefine hazard. The probability of “ties” is also (theoretically) infinitesimally small (assumed to be 0), but in reality ties are inevitable (because the metric for measuring time is never truly continuous) (ALDA, Section 13.1, pp 469-475) © Singer & Willett, page 3 Defining continuous time survivor and hazard functions:  Defining continuous time survivor and hazard functions Survivor function definition is essentially unchanged: S(tij)=Pr[Ti>tj]: The survival probability for individual i at time tj is the probability that his/her event time will exceed tj Hazard function definition must change: Hazard now assesses the conditional risk—at that particular moment—that an individual who has not yet done so will experience the event Can’t be defined as a conditional probability because that probability  0 Instead, divide time into an infinite number of vanishingly small intervals [tj, tj+t) Compute the probability that Ti falls in this interval as t 0 Tips for interpreting continuous time hazard It’s a rate per unit time, not a probability You must be explicit about the unit of time (60 mph, 60k/year) Unlike probabilities, rates can exceed 1 (has implications for modeling; instead of modeling logit hazard we model log hazard) Intuition using repeatable events—estimate the number of events in a finite period—e.g., monthly hazard = .10  annual hazard = 1.2 © Singer & Willett, page 4 Notation: T is a continuous random variable; Ti is individual i’s event time; tj clocks the infinite number of instants when events can occur Kaplan-Meier estimates of the survivor function:  Kaplan-Meier estimates of the survivor function (ALDA, Section 13.3, p 483-491) © Singer & Willett, page 5 What about hazard? Introducing a relative: The Cumulative Hazard Function:  What about hazard? Introducing a relative: The Cumulative Hazard Function (ALDA, Section 13.4, pp 488-491) To develop your intuition, let’s first move from h(t) to H(t). Because h(t) is constant, H(t) increases linearly as the same fixed amount of risk—the constant value of hazard—is added to the prior cumulative level at each successive instant (making H(t) linear). © Singer & Willett, page 6 From cumulative hazard to hazard: Developing your intuition:  From cumulative hazard to hazard: Developing your intuition (ALDA, Section 13.4.1, pp 488-491) © Singer & Willett, page 7 The cumulative hazard function in practice:  The cumulative hazard function in practice (ALDA, Section 13.4.2, p 491-494) Estimation methods Nelson-Aalen method: uses the conditional probabilities of event occurrence from the KM method -ln S(t) method —through calculus, it can be shown that H(tj)= -ln S(tj). Conclusion: Hazard is initially low, increases until around the 5th second, and then decreases again © Singer & Willett, page 8 Developing your data analytic intuition for continuous time event data:  Developing your data analytic intuition for continuous time event data (ALDA, Section 13.6, pp 497-502) Relapse common: ML=22 weeks, final S(t)=.2149 Departure inevitable—either retirement or death, ML=16 yrs Onset very rare: no ML; S(t)=.92 at age 54 During 3 yrs, many MDs stay; no ML; S(t)=.57 at 129 wks © Singer & Willett, page 9 Example for fitting Cox regression models:  Example for fitting Cox regression models Sample: 194 inmates released from a minimum security prison Research design Each was followed for up to 3 years Event: Whether and, if so, when they were re-arrested Arrest recorded to the nearest day N=106 (54.6%) were re-arrested Data source: Kristin Henning and colleagues, Criminal Justice and Behavior (1996) © Singer & Willett, page 10 Towards a continuous time hazard model: Inspecting within-group sample survivor and cumulative hazard functions:  Towards a continuous time hazard model: Inspecting within-group sample survivor and cumulative hazard functions (ALDA, Section 14.1.1, pp. 504-507) Survivor functions: Recidivism is high in both groups, although those with a history of person-related crimes are at greater risk (ML of 13.1 vs. 17.3) What should the model look like? Intuitively, it should resemble a DT model, in which a transformation of hazard is expressed as the sum: (1) a baseline function; and (2) a weighted linear combination of predictors. Pragmatically, because we lack a complete picture of hazard, we’ll develop the model using H(t). After doing so, we’ll use transformation to re-specify the model in terms of hazard itself. Realistically, to do so, we still have to deal with the fact that H(t) is bounded (by 0 from below). Transform by taking logarithms Usually regularizes distances between functions—stretches distances between small values and compresses distances between large values Not bounded at all (although you need to get used to negative #’s) © Singer & Willett, page 11 What population model might have generated these sample data? Sample log cumulative hazard functions  parameterizing the CT hazard model:  What population model might have generated these sample data? Sample log cumulative hazard functions  parameterizing the CT hazard model (ALDA, Section 14.1.1, pp. 504-127) © Singer & Willett, page 12 PERSONAL=1 PERSONAL=0 Taking antilogs to specify the Cox regression model in terms of cumulative hazard:  Taking antilogs to specify the Cox regression model in terms of cumulative hazard (ALDA, Section 14.1.2, pp. 507-512) © Singer & Willett, page 13 Hazard function representation of the Cox regression model:  Hazard function representation of the Cox regression model (ALDA, Section 14.1.3, pp. 512-516) © Singer & Willett, page 14 Fitting the Cox regression model to data:  Fitting the Cox regression model to data (ALDA, Section 14.2, pp 516-523) Estimation: In addition to specifying a particular model for hazard, Cox developed an ingenious method for model fitting data: partial maximum likelihood estimation (available in all major stat packages (See § 14.2). © Singer & Willett, page 15 Interpreting parameter estimates in a fitted Cox regression model:  Interpreting parameter estimates in a fitted Cox regression model Strategy for interpreting parameter estimates: Each assesses the effect of a 1-unit difference in the associated predictor on log hazard (controlling for all other predictors in the model) (ALDA, Section 14.3.1, p. 524-528) © Singer & Willett, page 16 Interpreting hazard ratios in a fitted Cox regression model:  Interpreting hazard ratios in a fitted Cox regression model (ALDA, Section 14.3.1, p. 524-528) © Singer & Willett, page 17 Including time-varying predictors in a Cox regression model:  Including time-varying predictors in a Cox regression model (ALDA, Section 15.1, pp 544-545) Model specification is easy: Just add the subscript j to the time-varying predictors Data demands can be high (sometimes insurmountable) You need to know the value of the time-varying predictor—for everyone still at risk— at every moment when someone experiences the event Requirement holds whether there are 10, 100 or 1,000 unique event times Same requirement as in discrete-time, but it was unproblematic there because: Number of unique event times was relatively small Event occurrence and predictors are typically assessed on the same schedule In continuous time, you typically can’t set the data collection schedule to coincide with event occurrence for everyone still at risk © Singer & Willett, page 18 Including TV non-reversible dichotomies in a Cox regression model: Data example:  Including TV non-reversible dichotomies in a Cox regression model: Data example (ALDA, Section 15.1.1, pp 545-551) Sample: 1,658 men interviewed twice (in 1974 and 1985) -- 382 (23.0%) started using cocaine between ages 17 and 41 Data source: Burton and colleagues (1996), Journal of Health and Social Behavior © Singer & Willett, page 19 Interpreting the results of fitting Cox regression models with time-varying predictors:  Interpreting the results of fitting Cox regression models with time-varying predictors (ALDA, Section 15.1.1, pp 545-551) © Singer & Willett, page 20 Might the effect of a predictor vary over TIME?:  Might the effect of a predictor vary over TIME? (ALDA, Section 15.3.2, pp 564-570) Sample: 174 teens admitted to a psychiatric hospital Research Design: True experiment Half (n=88) had traditional psychiatric treatment and services (TREAT=0) The other half (n=86) were randomly selected to participate in an innovative program that provided coordinated mental health services regardless of setting (in- or out-patient) Everyone tracked for up to 3 months to determine whether and, if so, when they were released RQ: Does provision of comprehensive mental health services reduce the length of hospital stay? Data source: Michael Foster and colleagues (1996), Evaluation and Program Planning © Singer & Willett, page 21 How might we detect a violation of the proportional hazards assumption?:  How might we detect a violation of the proportional hazards assumption? (ALDA, Section 15.3.2, pp 564-570) In discrete-time, we could plot the sample hazard functions to see if there was a violation, but in continuous time, we can’t plot sample hazard functions Solution: Plot sample cumulative hazard functions because the model equivalence means that these plots can tell what we need to know about potential violations of the proportionality assumption If the proportionality assumption is violated for a predictor, then there is an interaction between the predictor and TIME. © Singer & Willett, page 22 Fitting non-proportional Cox regression models:  Fitting non-proportional Cox regression models (ALDA, Section 15.3.2, pp 564-570) © Singer & Willett, page 23

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