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Time-to-Event Outcomes

Time-to-Event Outcomes. Chapter 7. Time to Event Outcomes. An outcome with two pieces: Time Did the event happen? (Dichotomous) Examples: Time to relapse Time to conception Time to death (AKA Survival Analysis). Hazard Functions and Survival Curves. Numerical and Graphical Summaries.

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Time-to-Event Outcomes

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  1. Time-to-Event Outcomes Chapter 7

  2. Time to Event Outcomes • An outcome with two pieces: Time Did the event happen? (Dichotomous) • Examples: Time to relapse Time to conception Time to death (AKA Survival Analysis)

  3. Hazard Functions and Survival Curves Numerical and Graphical Summaries

  4. Hazard Rates • Event rates that vary substantially over time are called hazard rates. • The hazard rate is the probability that an individual will experience an event at time t (assuming the event has not happened). The denominator for the hazard rate is the number of subjects that still have not had the event. Calculate the event rate for each time period of interest.

  5. Hazard Rate Interpretation • If the hazard rate is constant over time and it is equal to 0.03, we expect 0.03 events to occur in one time interval. • If a subject had a hazard rate of 0.03 at time t and another subject had a hazard rate of 0.015 at time t, then the second subject's risk of an event is half at time t.

  6. Hazard Function • A plot of hazard rates over time

  7. Survival Probabilities • Survival probabilities represent the probability of not having the event. The probability of having the event at a later time • The survival function (or curve) is a plot of these probabilities over time.

  8. Survival Curve

  9. Notation • Note that the notation involves t. Estimates occur at each time point. There is no single survival probability estimate or hazard rate. • Do not be mislead by the names. The event does not have to be a “bad” thing.

  10. Censoring

  11. Why is time-to-event analysis special? • Two-level outcome. • Waiting times are typically skewed. • Sometimes the event does not happen during the observation period (censored).

  12. Censoring • An observation is censored when the event does not occur when the subject is being observed. Right censoring Left censoring Interval censoring • Therefore, the outcome is really the time until the event or censoring.

  13. Kaplan-Meier Estimates • A (common) method for calculating survival probability estimates • Can handle censoring

  14. Kaplan-Meier Estimates • AKA product-limit estimates • Provide an estimate of the survival probability up to time ti= S(ti) • So, for example, the S(t4)

  15. Group Comparisons

  16. Kaplan-Meier Survival Curves

  17. Inference for Group Comparisons • Log-rank test

  18. Log-Rank TestSummary • Powerful method for analyzing data when the time to event is important (not just that the event occurred) • Works with censored data • Works with varying periods of follow-up • Allows you to test relationship between survival and a single “exposure” variable • But what if you are interested in more variables? What if there are lots of categories? What if you have continuous variables?

  19. Proportional Hazards Regression

  20. Time to Event Regression • Statistical models: Left-hand side involves the hazard Describes the relationship between regressors and event times as well as controls for confounding Parametric or not

  21. Parametric vs Not • Parametric models: Need the distribution of event times Hazard function is specified You might have heard of Weibull models or exponential models • Not: Do not know the distribution of event times Hazard function is not specified Proportional hazards (Cox) regression

  22. Proportional Hazards Model

  23. Proportional Hazards Assumption • The hazard ratios do not depend on time.

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