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Statistics 262: Intermediate Biostatistics

Statistics 262: Intermediate Biostatistics. April 20, 2004: Introduction to Survival Analysis. Jonathan Taylor and Kristin Cobb. What is survival analysis?. Statistical methods for analyzing longitudinal data on the occurrence of events.

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Statistics 262: Intermediate Biostatistics

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  1. Statistics 262: Intermediate Biostatistics April 20, 2004: Introduction to Survival Analysis Jonathan Taylor and Kristin Cobb Satistics 262

  2. What is survival analysis? • Statistical methods for analyzing longitudinal data on the occurrence of events. • Events may include death, injury, onset of illness, recovery from illness (binary variables) or transition above or below the clinical threshold of a meaningful continuous variable (e.g. CD4 counts). • Accommodates data from randomized clinical trial or cohort study design. Satistics 262

  3. Intervention Control Randomized Clinical Trial (RCT) Disease Random assignment Disease-free Target population Disease-free, at-risk cohort Disease Disease-free TIME

  4. Treatment Control Randomized Clinical Trial (RCT) Cured Random assignment Not cured Target population Patient population Cured Not cured TIME

  5. Treatment Control Randomized Clinical Trial (RCT) Dead Random assignment Alive Target population Patient population Dead Alive TIME

  6. Cohort study (prospective/retrospective) Disease Exposed Disease-free Target population Disease-free cohort Disease Unexposed Disease-free TIME

  7. Objectives of survival analysis • Estimate time-to-event for a group of individuals, such as time until second heart-attack for a group of MI patients. • To compare time-to-event between two or more groups, such as treated vs. placebo MI patients in a randomized controlled trial. • To assess the relationship of co-variables to time-to-event, such as: does weight, insulin resistance, or cholesterol influence survival time of MI patients? Note: expected time-to-event = 1/incidence rate Satistics 262

  8. Examples of survival analysis in medicine Satistics 262

  9. On hormones Cumulative incidence On placebo RCT: Women’s Health Initiative (JAMA, 2001)

  10. Prospective cohort study: From April 15, 2004 NEJM: Use of Gene-Expression Profiling to Identify Prognostic Subclasses in Adult Acute Myeloid Leukemia Satistics 262

  11. Retrospective cohort study:From December 2003 BMJ: Aspirin, ibuprofen, and mortality after myocardial infarction: retrospective cohort study Satistics 262

  12. Why use survival analysis? 1. Why not compare mean time-to-event between your groups using a t-test or linear regression? -- ignores censoring 2. Why not compare proportion of events in your groups using logistic regression? --ignores time Satistics 262

  13. Cox regression vs.logistic regression Distinction between rate and proportion: • Incidence (hazard) rate: number of new cases of disease per population at-risk per unit time (or mortality rate, if outcome is death) • Cumulative incidence: proportion of new cases that develop in a given time period Satistics 262

  14. Cox regression vs.logistic regression Distinction between hazard/rate ratio and odds ratio/risk ratio: • Hazard/rate ratio: ratio of incidence rates • Odds/risk ratio: ratio of proportions By taking into account time, you are taking into account more information than just binary yes/no. Gain power/precision. Logistic regression aims to estimate the odds ratio; Cox regression aims to estimate the hazard ratio Satistics 262

  15. Rates vs. risks Relationship between risk and rates: Satistics 262

  16. Rates vs. risks For example, if rate is 5 cases/1000 person-years, then the chance of developing disease over 10 years is: Compare to .005(10) = 5% The loss of persons at risk because they have developed disease within the period of observation is small relative to the size of the total group. Satistics 262

  17. Rates vs. risks If rate is 50 cases/1000 person-years, then the chance of developing disease over 10 years is: Compare to .05(10) = 50% Satistics 262

  18. Exponential density function for waiting time until the event (constant hazard rate) Rates vs. risks Relationship between risk and rates (derivation): Preview: Waiting time distribution will change if the hazard rate changes as a function of time: h(t) Satistics 262

  19. Survival Analysis: Terms • Time-to-event: The time from entry into a study until a subject has a particular outcome • Censoring: Subjects are said to be censored if they are lost to follow up or drop out of the study, or if the study ends before ends before they die or have an outcome of interest. They are counted as alive or disease-free for the time they were enrolled in the study. • If dropout is related to both outcome and treatment, dropouts may bias the results

  20. Right Censoring (T>t) Common examples • Termination of the study • Death due to a cause that is not the event of interest • Loss to follow-up We know that subject survived at least to time t. Satistics 262

  21. Left censoring (T<t) • The origin time, not the event time, is known only to be less than some value. • For example, if you are studying menarche and you begin following girls at age 12, you may find that some of them have already begun menstruating. Unless you can obtain information about the start date for those girls, the age of menarche is left-censored at age 12. *from:Allison, Paul. Survival Analysis. SAS Institute. 1995. Satistics 262

  22. Interval censoring (a<T<b) • When we know the event has occurred between two time points, but don’t know the exact dates. • For example, if you’re screening subjects for HIV infection yearly, you may not be able to determine the exact date of infection.* *from:Allison, Paul. Survival Analysis. SAS Institute. 1995. Satistics 262

  23. Data Structure: survival analysis • Time variable: ti = time at last disease-free observation or time at event • Censoring variable: ci =1 if had the event; ci =0 no event by time ti Satistics 262

  24. Choice of origin Satistics 262

  25. Satistics 262

  26. Describing survival distributions • Ti the event time for an individual, is a random variable having a probability distribution. • Different models for survival data are distinguished by different choice of distribution for Ti. Satistics 262

  27. Survivor function (cumulative distribution function) Cumulative failure function Survival analysis typically uses complement, or the survivor function: Example: If t=100 years, S(t=100) = probability of surviving beyond 100 years. Satistics 262

  28. Corresponding density function The probability of the failure time occurring at exactly time t (out of the whole range of possible t’s). Also written: Satistics 262

  29. Hazard function In words: the probability that if you survive to t, you will succumb to the event in the next instant. Derivation: Satistics 262

  30. Relating these functions: Satistics 262

  31. Introduction to Kaplan-Meier • Non-parametric estimate of survivor function. • Commonly used to describe survivorship of study population/s. • Commonly used to compare two study populations. • Intuitive graphical presentation. Satistics 262

  32. Subject A Subject B Subject C Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data (right-censored)

  33. 100% Probability of surviving to just before 4 months is 100% = 5/5 Fraction surviving this death = 4/5 Subject E dies at 4 months  Time in months  Corresponding Kaplan-Meier Curve

  34. Subject A 2. subject A drops out after 6 months Subject B Subject C 3. subject C dies at 7 months X Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data

  35. 100% Fraction surviving this death = 2/3 subject C dies at 7 months  Time in months  Corresponding Kaplan-Meier Curve

  36. Subject A 2. subject A drops out after 6 months Subject B 4. Subjects B and D survive for the whole year-long study period Subject C 3. subject C dies at 7 months X Subject D Subject E 1. subject E dies at 4 months X Beginning of study End of study  Time in months  Survival Data

  37. 100%  Time in months  Corresponding Kaplan-Meier Curve Product limit estimate of survival = P(surviving/at-risk through failure 1) * P(surviving/at-risk through failure 2) = 4/5 * 2/3= .5333

  38. The product limit estimate • The probability of surviving in the entire year, taking into account censoring • = (4/5) (2/3) = 53% • NOTE:  40% (2/5) because the one drop-out survived at least a portion of the year. • AND <60% (3/5) because we don’t know if the one drop-out would have survived until the end of the year.

  39. KM estimator, formally Satistics 262

  40. Comparing 2 groups

  41. Caveats • Survival estimates can be unreliable toward the end of a study when there are small numbers of subjects at risk of having an event.

  42. Small numbers left WHI and breast cancer

  43. Overview of SAS PROCS • LIFETEST - Produces life tables and Kaplan-Meier survival curves. Is primarily for univariate analysis of the timing of events. • LIFEREG – Estimates regression models with censored, continuous-time data under several alternative distributional assumptions. Does not allow for time-dependent covariates. • PHREG– Uses Cox’s partial likelihood method to estimate regression models with censored data. Handles both continuous-time and discrete-time data and allows for time-dependent covariables Satistics 262

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