Parametric survival models

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1 Parametric survival models ST3242: Introduction to Survival Analysis Alex Cook October 2008 ST3242 : Parametric survival models 1/17

2 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

3 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

4 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

5 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

6 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

7 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

8 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

9 Last time in survival analysis Reintroduced parametric models Distinguished parametric semi-parametric non-parametric Recapped MLE of parameters & CIs Two examples: exponential model easy, Weibull model hard Numerical method of finding MLE Today: continue reintroduction & AFT models Comparing parametric models Censored data Visual assessment of suitability of parametric model Regression parametric models: AFT ST3242 : Parametric survival models 2/17

10 Comparing models As expected! We have multiple possible parametric models (exponential vs log-logistic vs log-normal) Which one is best for our data? If models are nested: use likelihood ratio test If non-nested: use AIC ST3242 : Parametric survival models 3/17

11 Comparing models As expected! We have multiple possible parametric models (exponential vs log-logistic vs log-normal) Which one is best for our data? If models are nested: use likelihood ratio test If non-nested: use AIC ST3242 : Parametric survival models 3/17

12 Comparing models As expected! We have multiple possible parametric models (exponential vs log-logistic vs log-normal) Which one is best for our data? If models are nested: use likelihood ratio test If non-nested: use AIC ST3242 : Parametric survival models 3/17

13 Comparing models As expected! We have multiple possible parametric models (exponential vs log-logistic vs log-normal) Which one is best for our data? If models are nested: use likelihood ratio test If non-nested: use AIC ST3242 : Parametric survival models 3/17

14 Example ST3242 : Parametric survival models 4/17

15 Warning! You might want to use AIC to compare parametric with Cox Can t! Data for parametric model: event times Data for Cox models: rank event times Cox model trying to explain simpler data set: unfair comparison ST3242 : Parametric survival models 5/17

16 Warning! You might want to use AIC to compare parametric with Cox Can t! Data for parametric model: event times Data for Cox models: rank event times Cox model trying to explain simpler data set: unfair comparison ST3242 : Parametric survival models 5/17

17 Warning! You might want to use AIC to compare parametric with Cox Can t! Data for parametric model: event times Data for Cox models: rank event times Cox model trying to explain simpler data set: unfair comparison ST3242 : Parametric survival models 5/17

18 Warning! You might want to use AIC to compare parametric with Cox Can t! Data for parametric model: event times Data for Cox models: rank event times Cox model trying to explain simpler data set: unfair comparison ST3242 : Parametric survival models 5/17

19 Warning! You might want to use AIC to compare parametric with Cox Can t! Data for parametric model: event times Data for Cox models: rank event times Cox model trying to explain simpler data set: unfair comparison ST3242 : Parametric survival models 5/17

20 Censored data It is trivial to deal with censored data Consider right-censored data only True failure time is T Observe (t, δ) If δ = 1, we know T = t, i.e. the individual failed at the time we have observed. If δ = 0, we know T > t, i.e. the individual was right-censored at the time we have observed. ST3242 : Parametric survival models 6/17

21 Censored data It is trivial to deal with censored data Consider right-censored data only True failure time is T Observe (t, δ) If δ = 1, we know T = t, i.e. the individual failed at the time we have observed. If δ = 0, we know T > t, i.e. the individual was right-censored at the time we have observed. ST3242 : Parametric survival models 6/17

22 Censored data It is trivial to deal with censored data Consider right-censored data only True failure time is T Observe (t, δ) If δ = 1, we know T = t, i.e. the individual failed at the time we have observed. If δ = 0, we know T > t, i.e. the individual was right-censored at the time we have observed. ST3242 : Parametric survival models 6/17

23 Censored data It is trivial to deal with censored data Consider right-censored data only True failure time is T Observe (t, δ) If δ = 1, we know T = t, i.e. the individual failed at the time we have observed. If δ = 0, we know T > t, i.e. the individual was right-censored at the time we have observed. ST3242 : Parametric survival models 6/17

24 Censored data If we had observed the set of T i for i = 1,..., m, the likelihood would be... ST3242 : Parametric survival models 7/17

25 Validity of parametric models AIC or LRT give relative model GOF not absolute model GOF log-logistic vs exponential Just because the log-logistic fits better than the exponential does not mean the log-logistic adequately describes the data ST3242 : Parametric survival models 8/17

26 Validity of parametric models AIC or LRT give relative model GOF not absolute model GOF log-logistic vs exponential Just because the log-logistic fits better than the exponential does not mean the log-logistic adequately describes the data ST3242 : Parametric survival models 8/17

27 Validity of parametric models AIC or LRT give relative model GOF not absolute model GOF log-logistic vs exponential Just because the log-logistic fits better than the exponential does not mean the log-logistic adequately describes the data ST3242 : Parametric survival models 8/17

28 Validity of parametric models Graphical method to assess abs GOF Plots that should be linear if model is true f 1 {S(t)} = f 2 (θ) + f 3 (θ)f 4 (t). Then plot f 1 {Ŝ(t)} using Kaplan Meier versus f 4(t) ST3242 : Parametric survival models 9/17

29 Validity of parametric models Graphical method to assess abs GOF Plots that should be linear if model is true f 1 {S(t)} = f 2 (θ) + f 3 (θ)f 4 (t). Then plot f 1 {Ŝ(t)} using Kaplan Meier versus f 4(t) ST3242 : Parametric survival models 9/17

30 Validity of parametric models Graphical method to assess abs GOF Plots that should be linear if model is true f 1 {S(t)} = f 2 (θ) + f 3 (θ)f 4 (t). Then plot f 1 {Ŝ(t)} using Kaplan Meier versus f 4(t) ST3242 : Parametric survival models 9/17

31 Example ST3242 : Parametric survival models 10/17

32 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

33 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

34 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

35 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

36 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

37 Regression parametric models Perfectly possible to do regression within parametric models PHM accelerated failure times Exponential model covariate x No covariates: h(t) = λ PHM: h(t) = exp(β 0 + β 1 x) = e β 0 e β 1x = h 0 e β 1x AFT: h(t) = 1/ exp(α 0 + α 1 x) = h 0 e α 1x PHM and AFT or PHM not AFT or AFT not PHM or neither ST3242 : Parametric survival models 11/17

38 Accelerated Failure Time PICTURE ST3242 : Parametric survival models 12/17

39 Accelerated Failure Time Assume: t i to S(t i ) = σ for one individual i is a constant times the time t j to S(t j ) = σ for j Same constant regardless of which quantile S(t i ) = S(ψ(x i, x j )t j ) ψ(x i, x j ) is the acceleration factor for i relative to j. ST3242 : Parametric survival models 13/17

40 Accelerated Failure Time Assume: t i to S(t i ) = σ for one individual i is a constant times the time t j to S(t j ) = σ for j Same constant regardless of which quantile S(t i ) = S(ψ(x i, x j )t j ) ψ(x i, x j ) is the acceleration factor for i relative to j. ST3242 : Parametric survival models 13/17

41 Accelerated Failure Time Assume: t i to S(t i ) = σ for one individual i is a constant times the time t j to S(t j ) = σ for j Same constant regardless of which quantile S(t i ) = S(ψ(x i, x j )t j ) ψ(x i, x j ) is the acceleration factor for i relative to j. ST3242 : Parametric survival models 13/17

42 Accelerated Failure Time Assume: t i to S(t i ) = σ for one individual i is a constant times the time t j to S(t j ) = σ for j Same constant regardless of which quantile S(t i ) = S(ψ(x i, x j )t j ) ψ(x i, x j ) is the acceleration factor for i relative to j. ST3242 : Parametric survival models 13/17

43 Accelerated Failure Time Assume: t i to S(t i ) = σ for one individual i is a constant times the time t j to S(t j ) = σ for j Same constant regardless of which quantile S(t i ) = S(ψ(x i, x j )t j ) ψ(x i, x j ) is the acceleration factor for i relative to j. ST3242 : Parametric survival models 13/17

44 Example ST3242 : Parametric survival models 14/17

45 Accelerated Failure Time Models: Covariates affect scale not shape Exponential: S(t, x 1, x 2 ) = exp{ t exp( α 0 α 1 x 1 α 2 x 2 )} Weibull: S(t, x 1, x 2 ) = exp{ t κ exp( α 0 α 1 x 1 α 2 x 2 )} Log-logistic: S(t, x 1, x 2 ) = {1 + t κ exp( α 0 α 1 x 1 α 2 x 2 )} 1 ST3242 : Parametric survival models 15/17

46 Accelerated Failure Time Models: Covariates affect scale not shape Exponential: S(t, x 1, x 2 ) = exp{ t exp( α 0 α 1 x 1 α 2 x 2 )} Weibull: S(t, x 1, x 2 ) = exp{ t κ exp( α 0 α 1 x 1 α 2 x 2 )} Log-logistic: S(t, x 1, x 2 ) = {1 + t κ exp( α 0 α 1 x 1 α 2 x 2 )} 1 ST3242 : Parametric survival models 15/17

47 Accelerated Failure Time Models: Covariates affect scale not shape Exponential: S(t, x 1, x 2 ) = exp{ t exp( α 0 α 1 x 1 α 2 x 2 )} Weibull: S(t, x 1, x 2 ) = exp{ t κ exp( α 0 α 1 x 1 α 2 x 2 )} Log-logistic: S(t, x 1, x 2 ) = {1 + t κ exp( α 0 α 1 x 1 α 2 x 2 )} 1 ST3242 : Parametric survival models 15/17

48 Accelerated Failure Time Models: Covariates affect scale not shape Exponential: S(t, x 1, x 2 ) = exp{ t exp( α 0 α 1 x 1 α 2 x 2 )} Weibull: S(t, x 1, x 2 ) = exp{ t κ exp( α 0 α 1 x 1 α 2 x 2 )} Log-logistic: S(t, x 1, x 2 ) = {1 + t κ exp( α 0 α 1 x 1 α 2 x 2 )} 1 ST3242 : Parametric survival models 15/17

49 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

50 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

51 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

52 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

53 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

54 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

55 Accelerated Failure Time Can: Estimate parameters as before (eg MLE) Obtain CIs using asymptotic normality Compare models using LRT or AIC Assess validity of parametric forms graphically Model building approaches Don t extend model to deal with violation of PHA Stratify over covariates by giving each different shape parameter ST3242 : Parametric survival models 16/17

56 Coming up Next time: How to do parametric analyses on the computer Last time: Frailty models ST3242 : Parametric survival models 17/17

57 Coming up Next time: How to do parametric analyses on the computer Last time: Frailty models ST3242 : Parametric survival models 17/17

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