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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 semiparametric nonparametric 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 loglogistic vs lognormal) Which one is best for our data? If models are nested: use likelihood ratio test If nonnested: use AIC ST3242 : Parametric survival models 3/17
11 Comparing models As expected! We have multiple possible parametric models (exponential vs loglogistic vs lognormal) Which one is best for our data? If models are nested: use likelihood ratio test If nonnested: use AIC ST3242 : Parametric survival models 3/17
12 Comparing models As expected! We have multiple possible parametric models (exponential vs loglogistic vs lognormal) Which one is best for our data? If models are nested: use likelihood ratio test If nonnested: use AIC ST3242 : Parametric survival models 3/17
13 Comparing models As expected! We have multiple possible parametric models (exponential vs loglogistic vs lognormal) Which one is best for our data? If models are nested: use likelihood ratio test If nonnested: 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 rightcensored 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 rightcensored at the time we have observed. ST3242 : Parametric survival models 6/17
21 Censored data It is trivial to deal with censored data Consider rightcensored 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 rightcensored at the time we have observed. ST3242 : Parametric survival models 6/17
22 Censored data It is trivial to deal with censored data Consider rightcensored 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 rightcensored at the time we have observed. ST3242 : Parametric survival models 6/17
23 Censored data It is trivial to deal with censored data Consider rightcensored 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 rightcensored 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 loglogistic vs exponential Just because the loglogistic fits better than the exponential does not mean the loglogistic 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 loglogistic vs exponential Just because the loglogistic fits better than the exponential does not mean the loglogistic 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 loglogistic vs exponential Just because the loglogistic fits better than the exponential does not mean the loglogistic 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 )} Loglogistic: 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 )} Loglogistic: 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 )} Loglogistic: 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 )} Loglogistic: 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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