Survival Data Analysis Parametric Models

Transcription

1 1 Survival Data Analysis Parametric Models January 21, 2015 Sandra Gardner, PhD Dalla Lana School of Public Health University of Toronto

2 2 January 21, 2015 Agenda Basic Parametric Models Review: hazard & cumulative hazard functions; likelihood function Proportional hazards versus accelerated failure Exponential model Weibull model Log-Normal model Log-Logistic model Checking assumptions Gamma model Goodness of fit and residuals Other Models Changepoint model (piecewise exponential model ) Reference: Matthews & Farewell 1982 Gamel-Boag (cure fraction) model Reference: Frankel & Longmate 2002 Bayesian analysis

3 3 January 21, 2015 Probability density function Random survivaltime T 0 f ( t) h( t) S ( t)

4 January 21, Hazard function Specifies the instantaneous rate of failure at T=t t t T t t T t P t h t ) ( ) ( lim 0 ) ( ) ( ) ( t S t f t h See K&M Section 2.3

5 January 21, Cumulative hazard function ) ( log ) (. ) ( ) (, ] [ ) ( 0 ) ( t S t H Note du u h t H where e t P T t S t u t H

6 Likelihood Full likelihood for parametric models Assuming censoring is independent of failure and noninformative 1 1 ) ( ) ( ), Pr( ), min( ), Pr( ) ( ( t S t f t and C X wheret t L C S x f L r n i i i R i r n D i i January 21, K&M 3.5.1

7 Likelihood ] [ ] [ ] [ ] [ ) ( exp )] [ ) exp )] [ ) exp[ ) )exp[ ) ( ( L i n i i t n i i t n i t i i n i i t H t h ds s h t h ds s h ds s h t h t S t f i i i January 21, K&M 3.5.3

8 8 January 21, 2015 Parametric Survival models Fully specified model with hazard rate a function of covariates (including intercept) Proportional Hazards (PH) constant hazard ratios across time Exponential, Weibull Accelerated Failure Models (AFT) constant time ratios across survival percentiles Exponential, Weibull, Log Normal, Log Logistic

9 PH versus AFT January 21, e x t x t TR e t x h t x h HR binary is X g e ) 0, ( ) 1, ( ) 0, ( ) 1, ( PH AFT Exponential Model

10 PH versus AFT January 21, ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ' ' ' ' ' t e S X t S e t e h X t h t S X t S e t h X t h X X X e X X PH AFT Be careful of parameterization of models in texts and software.

11 11 January 21, 2015 Sample Weibull hazard plots - HR=1.5 h(t) years x 0 1

12 12 January 21, 2015 Sample Weibull survival plots - TR=.67 (or AF=1.5) S(t) years x 0 1

13 13 January 21, 2015 Error distributions f ( ) exp( exp( )) f ( ) exp( ) Y logt X f ( ) e 1 e 2 Be careful of parameterization of models in texts and software.

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15 15 January 21, 2015 Exponential Model constant hazard functions both PH and AFT model underlying error function has an extreme value function with σ=1 S( t) e t h( t) ln(.5) Median 1 Mean.69

16 n i i n i i i n i i i n i t t l t mle ˆ ] [ ) ] log[ ( ) ( exp ] [ ) L( Exponential Model January 21,

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20 20 January 21, 2015 Weibull monotone increasing or decreasing hazard functions both PH and AFT model Exponential model is special case (γ=1) S( t) e t h( t) t Median 1 ln(.5) 1

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27 Log Normal hazard functions rise to a maximum then slowly decline, AFT model only January 21, ) ln( 2 1 ) ( ) ( ) ( 2 1 ) ( ) ln( 1 ) ( e Mean e e Median t S t f t h e t t f t t S t

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34 Log Logistic hazard functions rise to a maximum then slowly decline or are monotone decreasing, AFT model only January 21, ) ( 2 1) ( 1 1 ) ( ) ( ) ( ) (1 ) ( 1 1 ) ( Median t t t S t f t h t t t f t t S

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41 41 January 21, 2015 Example Data Set Patients diagnosed with brain cancer are randomized to a treatment group versus placebo. N=222, with only 15 censored cases Mean age around 48 years and 64% male. Other covariates are available in data set.

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43 43 January 21, 2015 Overall Survival Estimated median=27.4 and mean=44.5

44 44 January 21, logs(t) Plot Plot versus t If a straight line then exponential model (H(t)=λt)

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46 46 January 21, 2015 log-logs(t) Plot Plot versus log(t) If a straight line then Weibull model H(t)=λt γ logh(t)=log(λ)+γlog(t)

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48 48 January 21, 2015 Probit Plot Plot Ф -1 (1-S(t)) versus log(t) If a straight line then Log Normal model S(t)=1-Ф((log(t)-u)/σ)

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50 50 January 21, 2015 Logit Plot Plot log((1-s(t)) /S(t)) versus log(t) Plot of odds of having the event by time t If a straight line then Log Logistic model S(t)=1/(1+αt γ )

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52 52 January 21, 2015 Other options Non-parametric smoothing of hazard function Probability plots Likelihood ratio tests of nested models (Gamma) Check distribution of t or log(t) for the noncensored cases

53 53 January 21, 2015 Smoothed Hazard Function

54 54 January 21, 2015 Smoothed Hazard Function (reduced range)

55 55 January 21, 2015 SAS Code (SAS 9.3 using ODS graphics) proc lifereg data=sda.brain; model weeks*event(0)=/d=exponential; probplot; inset; title 'LifeReg: Overall Survival - Probability Plot (Exponential)'; run; proc lifetest data=sda.brain plot=(survival(atrisk outside) hazard logsurv loglogs) notable; time weeks*event(0); title 'LifeTest: Overall Survival: hazard'; run;

56 56 January 21, 2015 Exponential Probability Plot

57 57 January 21, 2015 Weibull Probability Plot

58 58 January 21, 2015 Log Normal Probability Plot

59 59 January 21, 2015 Log Logistic Probability Plot

60 60 January 21, 2015 Gamma Model SAS fits the generalized 3-parameter model it can fit a Weibull (exponential) and log-normal model (test using likelihood ratio test) it can also fit a model with a U-shaped hazard function Survivor and hazard functions involve incomplete gamma functions

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64 64 January 21, 2015 Unadjusted model

65 65 January 21, 2015 Basic data summary Variable Sum event 207 weeks 9426 Estimated rate: 207/9426= ln( )= overall median % CI (23.14, 31.43) mean SE= 3.285

66 66 January 21, 2015 Exponential (Intercept only) -2 Log Likelihood= , AIC= Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 Scale Weibull Scale Weibull Shape Lagrange Multiplier Statistics Parameter Chi-Square Pr > ChiSq Scale λ = exp( ) = S(t) = exp(-λ*t) h(t) = λ Median = -ln(.5)/λ = 31.4 Mean = 1/λ = 45.5

67 67 January 21, 2015 Weibull(Intercept only) -2 Log Likelihood= AIC= Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 Scale * 1/shape Weibull Scale Weibull Shape * gamma λ = exp( * ) = γ = S(t) = exp(-λ*t**γ) h(t) = γ*λ*(t**(γ-1)) Median = (-ln(.5)/λ)**(1/γ) = 32.3

68 68 January 21, 2015 Log Normal(Intercept only) -2 Log Likelihood= , AIC= Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 Scale u = σ = S(t) = 1-Φ((ln(t)-u)/σ) f(t) = 1/(sqrt(2*π)*t*σ)*exp(-1/2*((ln(t)-u)/σ)**2) h(t) = f(t)/s(t) Median = exp(u) = 28.9 Mean = exp(u+0.5σ**2) = 45.7

69 69 January 21, 2015 Log Logistic (Intercept only) -2 Log Likelihood= , AIC= Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 Scale α = exp( / ) = γ = 1/ = S(t) = 1/(1+α*t**γ) f(t) = (α*γ*t**(γ-1))/(1+α*t**γ)**2 h(t) = f(t)/s(t) Median = (1/α)**(1/γ) = 27.8

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71 71 January 21, 2015 Gamma model (Intercept only) -2 Log Likelihood= , AIC= Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 Scale Shape If shape parameter is 0 then log-normal model If shape parameter is 1 then Weibull model If shape =1 and scale=1 then exponential model If shape and scale are equal, then standard gamma distribution Likelihood ratio test: Gamma vs log normal chi-square = = 7.667, p=0.006

72 72 January 21, 2015 Gamma hazard (Allison LIFEHAZ macro) Gamma-unadjusted

73 73 January 21, 2015 Comparison of Gamma model and Kaplan-Meier curve S(t) weeks PLOT Gamma KM

74 74 January 21, 2015 Model Comparison Model -2logL AIC AICC BIC Exponential Weibull LogNormal LogLogistic Gamma

75 75 January 21, 2015 Akaike Information Criteria AIC=-2log(Likelihood)+2(p+k) K&M k=1 (exponential) k=2 for Weibull, log logistic and log normal k=3 for generalized gamma In our example, AIC for gamma (606.3) is close to AIC for log-logistic (608.3). AICC AIC 2 p( n p 1) p 1 BIC 2log L p log( n)

76 76 January 21, 2015 Adjusted model Use preferred model building strategy to add covariates into the model (to be discussed further next month) Choosing two binary covariates for illustration Treated (treat=1); not treated (treat=0) Age <50 (age50=0) and age 50 (age50=1)

77 77 January 21, 2015 Covariates: median survival median treat=no : (20.57, 28.00) median treat=yes : (26.29, 37.00) median age<50 : (27.14, 39.71) median age>=50 : (19.00, 27.29)

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81 81 January 21, 2015 Exponential (treatment) Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 treat Scale Weibull Shape HR=exp(-beta)=exp( )=0.83 TR=exp(beta)=exp(0.1853)=1.20

82 82 January 21, 2015 Exponential (Hazard Ratio) Note closed form solution for hazard ratio can be calculated from the summary data below (unadjusted for other covariates): No treatment: 104/4300= (note log(0.0242)=-3.722) and Yes, treated: 103/5126= HR: / = Log(HR): log(0.0201/0.0242)= TR: exp(0.1856)=1.204 (output from proc means) treat Obs Variable Sum No 112 event 104 weeks 4300 Yes 110 event 103 weeks 5126

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86 86 January 21, 2015 Exponential/Weibull (age grouped) Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 age λ = exp(-( *age50)) Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 age Scale Weibull Shape λ = exp( *( *age50)) HR = exp(-beta*1.0688) = 1.69 TR = exp(beta) = 0.61 AF = 1.64

87 87 January 21, 2015 Goodness of fit Sample plots How well does model match Kaplan-Meier curves? Cox-Snell residuals Log-log(SDF) or cumulative hazard of residuals is a straight line? ˆ Other residuals: e.g. normal deviate residuals, see Nardi & Schemper r j and H ( T r Z where Hˆ is estimated j j ) distributed j exp(1) from data logti SAS output : log( S( ' xib ))

88 88 January 21, 2015 Goodness of fit Martingale residuals Klein & Moeschberger: estimate of the excess number of deaths seen in the data, but not predicted by model δ j -H(T j Z j ) i.e. δ j -r j Deviance residuals Klein & Moeschberger: more symmetric about 0 Transformed martingale residuals

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92 92 January 21, 2015 Simulated Exponential Data To show what plots look like using randomly generated data from an exponential distribution

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96 96 January 21, 2015 Simulated Exponential Data

97 97 January 21, 2015 Exponential (treatment and age) -2 Log Likelihood = Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 treat age Scale Weibull Shape

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99 99 January 21, 2015 Weibull(treatment and age) -2 Log Likelihood = Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 treat age Scale Weibull Shape

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101 101 January 21, 2015 Log Normal(treatment and age) -2 Log Likelihood = Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 treat age Scale

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103 103 January 21, 2015 Log Logistic(treatment and age) -2 Log Likelihood = Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 treat age Scale

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105 105 January 21, 2015 Residuals (SAS Code) /* Cox-Snell */ proc lifereg data=wsda.brain; model weeks*event(0)=treat age50/d=weibull; output out=wout cres=cres sres=sres p=predm std=stdm; title 'LifeReg: Treatment & Age groups - Weibull'; run; proc lifetest data=wout plots=(ls) notable; * looking for evidence that cres is exponential using the -log(s(t)) plot; * note that censoring value is maintained from original data set; time cres*event(0); title1 'Cox-Snell Residuals - Weibull'; run; /* martingale and deviance*/ lambda=exp(-( *treat *age50)); sexp=exp(-lambda*weeks); xbexp= *treat *age50; chexp=-log(sexp); martexp=event-chexp; devexp=sign(martexp)*(-2*(martexp+event*log(event-martexp)))**1/2;

106 106 January 21, 2015 Martingale Residual Plots - Weibull Model MR Weeks event 0 1

107 107 January 21, 2015 Martingale Residual Plots - Log Logistic Model MR Weeks event 0 1

108 108 January 21, 2015 Deviance Residual Plots - Weibull Model DR Weeks event 0 1

109 109 January 21, 2015 Deviance Residual Plots - Log Logistic Model DR Weeks event 0 1

110 110 January 21, 2015 Model summaries

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116 116 January 21, 2015 Exponential Summary Exponential Exponential Exponential Exponential Exponential t S(t), h(t), Median, S(t), h(t), (weeks) Age>=50 Treatment=1 Treatment=1 Treatment=1 Treatment=0 Treatment=0 26 Yes Yes Yes Exponential Exponential Exponential t Median, Hazard Exponential Exponential Median (weeks) Treatment=0 Ratio(t) beta(treatment); Time Ratio Ratio

117 117 January 21, 2015 Weibull Summary Weibull Weibull Weibull Weibull S(t), h(t), Weibull S(t), h(t), t Treatment= Treatment= Median, Treatment= Treatment= (weeks) Age>= Treatment= Yes Yes Yes Weibull Weibull Weibull Weibull t Median, Hazard Weibull Time Median (weeks) Treatment=0 Ratio(t) beta(treatment); Ratio Ratio

118 118 January 21, 2015 Log Normal Summary Log Normal Log Normal Log Normal Log Normal S(t), h(t), Log Normal S(t), h(t), t Treatment= Treatment= Median, Treatment= Treatment= (weeks) Age>= Treatment= Yes Yes Yes Log Normal Log Normal Log Normal t Median, Hazard Log Normal Log Normal Median (weeks) Treatment=0 Ratio(t) beta(treatment); Time Ratio Ratio beta=σ*(probit(1-slnorm0) - probit(1-slnorm1)); /* log normal scale and S(t) for each group */ TR=exp(beta)

119 119 January 21, 2015 Log Logistic Summary Log Logistic Log Logistic Log Logistic Log Logistic Log Logistic t S(t), h(t), Median, S(t), h(t), (weeks) Age>=50 Treatment=1 Treatment=1 Treatment=1 Treatment=0 Treatment=0 26 Yes Yes Yes Log Logistic Log Log Odds Log Logistic Logistic Log Logistic S(t)/(1-S(t) t Median, Hazard Log Logistic Logistic Median, (weeks) Treatment=0 Ratio(t) beta(treatment); Time Ratio Ratio Treatment= Log Logistic Odds Log S(t)/(1-S(t) Log Log Logistic Log Logistic Logisitic t, Logisitic Alpha, Alpha, Alpha (weeks) Treatment=0 Odds Ratio Treatment=1 Treatment=0 Ratio

120 120 January 21, 2015 Log logistic Odds Ratio (SAS Code) hrllog=hllog1/hllog0; if sllog1^=1 then odds1=sllog1/(1-sllog1); if sllog0^=1 then odds0=sllog0/(1-sllog0); oddsratio=odds1/odds0; alpharatio=alpha0/alpha1; beta_llog=log(oddsratio)*σ; /* log logistic scale; oddsratio=exp(beta_llog/σ) */ trllog=exp(beta_llog); mrllog=mllog1/mllog0;

121 121 January 21, 2015 Odds of survival=prob(alive)/prob(died) Odds ratio (treated to not treated)= 1.47 S(t) PLOT Treated Not Treated Prob(died, treated) Prob(alive, treated) Prob(died, not treated) Prob(alive, not treated) t

122 122 January 21, 2015 Discussion Not limited to parametric models discussed today. For example (next week): Changepoint model (piecewise exponential model) Gamel-Boag model (allows for a proportion of subjects to be long term survivors) Bayesian analysis

123 123 January 21, 2015 Changepoint model When the hazard rate is constant within in time periods and changes at known timepoint For example, brain cancer hazard rate is constant for the first year of follow up but hazard rate is reduced if patient survives at least one year. 1t S( t) e t e 1 e 2 ( t ) t

124 124 January 21, 2015 SAS code to restructure data data brain2(keep=id weeks event weeks2 event2 year1); set sda.brain; id=_n_; if weeks<=52 then do; event2=event; weeks2=weeks; year1=1; output; end; else do; event2=0; weeks2=52; year1=1; output; event2=event; weeks2=weeks-52; year1=0; output; end; run;

125 125 January 21, 2015 SAS code to fit the model proc lifereg data=brain2; model weeks2*event2(0)=year1/d=exponential; title 'Piecewise Exponential'; run; data brain3; do weeks=0 to 200 by 1; /* time frame */ lambda1=exp(-( )); * = ; lambda2=exp(-(4.533)); * = ; if weeks<=52 then sexp=exp(-lambda1*weeks); else sexp=exp(-lambda1*52)*exp(-lambda2*(weeks-52)); output; end; run;

126 126 January 21, 2015 Changepoint model Standard 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept <.0001 year <.0001 Scale Weibull Shape

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128 Gamel-Boag Model Allows for a proportion of subjects to be long term survivors. Events are modeled using log-normal model. January 21, ) (0, ~ ) ln( 1 ) ( ) ( )) ( (1 ) ( ) ( ' ' ' N e e x t e e x p x t S x p x p x t S i i i i x x f

129 129 January 21, 2015 SAS code to fit log-normal model /* log normal model using proc lifereg */ proc lifereg data=sda.brain; model weeks*event(0)=age50/d=lnormal; title 'LifeReg: Survival by age group (Log normal)'; run; /* log normal model using proc nlp (SAS/OR) - maximize loglikelihood */ proc nlp data=sda.brain tech=tr cov=2 stderr; parms int gamma sig; pi= ; u=int+gamma*age50; if event=1 then logl=log(1/(sqrt(2*pi)*weeks*sig)* exp(-1/2*((lweeks-u)/sig)**2)); if event=0 then logl=log(1-probnorm((lweeks-u)/sig)); max logl; run;

130 130 January 21, 2015 SAS code to fit Gamel-Boag model /* Gamel-Boag cure model using proc nlp (SAS/OR), maximize modified loglikelihood, reference Frankel & Longmate */ proc nlp data=sda.brain tech=tr cov=2 stderr; parms intg gamma intb beta sig; pi= ; p=exp(intb+beta*age50)/(1+exp(intb+beta*age50)); /* model proportion cured by age group */ u=intg+gamma*age50; if event=1 then logl=log((1-p)*(1/(sqrt(2*pi)*weeks*sig)* exp(-1/2*((lweeks-u)/sig)**2))); if event=0 then logl=log(p+(1-p)*(1-probnorm((lweeks-u)/sig))); max logl; run;

131 131 January 21, 2015 SAS output from Proc Lifereg Analysis of Maximum Likelihood Parameter Estimates Parameter DF Estimate Standard Error 95% Confidence Limits Chi-Square Pr > ChiSq Intercept <.0001 age Scale

132 132 January 21, 2015 SAS output from Proc NLP (1) Optimization Results Parameter Estimates N Parameter Estimate Approx Std Err t Value Approx Pr > t Gradient Objective Function 1 int E gamma sig E Value of Objective Function =

133 133 January 21, 2015 SAS output from Proc NLP (2) Optimization Results Parameter Estimates N Parameter Estimate Approx Std Err t Value Approx Pr > t Gradient Objective Function 1 intg E gamma intb E beta sig E Value of Objective Function = p1 p0 or lor

134 134 Survival: Age>=50 January 21, 2015 S(t) weeks PLOT LogNormal Gamel-Boag KM

135 135 Survival: Age<50 January 21, 2015 S(t) weeks PLOT LogNormal Gamel-Boag KM

136 136 January 21, 2015 Bayesian analysis Gibbs sampling used for the location-scale models Can add priors for model parameters Can output posterior samples proc lifereg data=sda.brain; model weeks*event(0)=age50/d=weibull; bayes WeibullShapePrior=gamma seed=1254 outpost=postweibull; run;

137 137 January 21, 2015 Analysis of Maximum Likelihood Parameter Estimates Standard 95% Confidence Parameter DF Estimate Error Limits Intercept age Scale Weibull Shape Posterior Summaries Standard Percentiles Parameter N Mean Deviation 25% 50% 75% Intercept age WeibShape Posterior Intervals Parameter Alpha Equal-Tail Interval HPD Interval Intercept age WeibShape

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139 139 January 21, 2015 Discussion: Why fit parametric models? Able to describe the hazard rate AF model alternative when hazard rates are non-proportional Easier and more convenient to predict outcome for a particular outcome (see Reid (1994) conversation with D.R. Cox) If underlying hazard function is correctly specified, then parametric models give more precise estimates (K & M, p.373). Applications where parametric models are compared to Cox proportional hazard models: Chapman et al (2006). Application of log-normal model which authors conclude has a major advantage over the Cox model Nardi and Schemper (2003). Authors compare Cox and parametric models in clinical settings. Carroll (2003). Author illustrates the practical benefits of a Weibullbased analysis.

140 140 January 21, 2015 References Applied Survival Analysis, D.W. Hosmer, S. Lemeshow, S. May, Wiley 2008 Survival Analysis: Techniques for Censored and Truncated Data, J.P.Klein, M.L. Moeschberger, Springer 1997 Nardi, A. and Schemper, M. (2003), Comparing Cox and parametric models in clinical studies, Statistics in Medicine, 22, J-A. W. Chapman, H.L.A Lickley, et al. Ascertaining Prognosis for Breast Cancer in Node- Negative Patients with Innovative Survival Analysis. The Breast Journal 2006, 12(1): Moran J. L., Bersten, A.D. et al. (2008). Modelling survival in acute severe illness: Cox versus accelerated failure time models. Journal of Evaluation in Clinical Practice Carroll, K. J. (2003). On the use and utility of the Weibull model in the analysis of survival data. Controlled Clinical Trials Matthews, D. E. and Farewell, V. T. (1982). On testing for constant hazard against a changepoint alternative. Biometrics 38, Frankel, P. and Longmate, J. (2002). Parametric models for accelerated and long-term survival: a comment on proportional hazards. Statistics in Medicine Reid, N. (1994). A Conversation with Sir David Cox. Statistical Science 9(3) Cox, D.R. and Oakes, D. (1984). Analysis of Survival Data. Chapman & Hall.