Introduction to Bayesian Survival Analysis

Document Sample
Introduction to Bayesian Survival Analysis Powered By Docstoc
					            Fundamental concepts
               Bayesian approach
           Semiparametric models
                       Examples




Introduction to Bayesian Survival Analysis

                         Tim Hanson

                     Division of Biostatistics
                 University of Minnesota, U.S.A.


                      IAP-Workshop 2009
     Modeling Association and Dependence in Complex Data
                       November 19, 2009




                                                           1 / 72
                  Fundamental concepts
                     Bayesian approach
                 Semiparametric models
                             Examples


Outline



  1   Fundamental concepts

  2   Bayesian approach

  3   Semiparametric models

  4   Examples




                                         2 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Survival data


     Can be time to any event of interest, e.g. death, leukemia
     remission, bankruptcy, electrical component failure, etc.




                                                                                  3 / 72
                    Fundamental concepts
                       Bayesian approach   Time to event data
                   Semiparametric models   Functions defining lifetime distribution
                               Examples


Survival data


     Can be time to any event of interest, e.g. death, leukemia
     remission, bankruptcy, electrical component failure, etc.
     Data T1 , T2 , . . . , Tn live in R+ .




                                                                                     3 / 72
                    Fundamental concepts
                       Bayesian approach   Time to event data
                   Semiparametric models   Functions defining lifetime distribution
                               Examples


Survival data


     Can be time to any event of interest, e.g. death, leukemia
     remission, bankruptcy, electrical component failure, etc.
     Data T1 , T2 , . . . , Tn live in R+ .
     Called: survival data, reliability data, time to event data.




                                                                                     3 / 72
                    Fundamental concepts
                       Bayesian approach   Time to event data
                   Semiparametric models   Functions defining lifetime distribution
                               Examples


Survival data


     Can be time to any event of interest, e.g. death, leukemia
     remission, bankruptcy, electrical component failure, etc.
     Data T1 , T2 , . . . , Tn live in R+ .
     Called: survival data, reliability data, time to event data.
     T1 , . . . , Tn can be iid, independent, partially exchangeable,
     dependent, etc.




                                                                                     3 / 72
                    Fundamental concepts
                       Bayesian approach   Time to event data
                   Semiparametric models   Functions defining lifetime distribution
                               Examples


Survival data


     Can be time to any event of interest, e.g. death, leukemia
     remission, bankruptcy, electrical component failure, etc.
     Data T1 , T2 , . . . , Tn live in R+ .
     Called: survival data, reliability data, time to event data.
     T1 , . . . , Tn can be iid, independent, partially exchangeable,
     dependent, etc.
     Interest often focuses on relating aspects of the distribution
     on Ti to covariates or risk factors xi , possibly
     time-dependent xi (t). Can be external or internal.



                                                                                     3 / 72
                   Fundamental concepts
                      Bayesian approach   Time to event data
                  Semiparametric models   Functions defining lifetime distribution
                              Examples


Survival data: covariates and censoring


     Uncensored data: (x1 , t1 ), . . . , (xn , tn ). Observe Ti = ti .




                                                                                    4 / 72
                    Fundamental concepts
                       Bayesian approach     Time to event data
                   Semiparametric models     Functions defining lifetime distribution
                               Examples


Survival data: covariates and censoring


     Uncensored data: (x1 , t1 ), . . . , (xn , tn ). Observe Ti = ti .
     Right censored data: (x1 , t1 , δ1 ), . . . , (xn , tn , δn ). Observe

                                  T i = ti     δi = 1
                                                              .
                                  T i > ti     δi = 0




                                                                                       4 / 72
                    Fundamental concepts
                       Bayesian approach     Time to event data
                   Semiparametric models     Functions defining lifetime distribution
                               Examples


Survival data: covariates and censoring


     Uncensored data: (x1 , t1 ), . . . , (xn , tn ). Observe Ti = ti .
     Right censored data: (x1 , t1 , δ1 ), . . . , (xn , tn , δn ). Observe

                                  T i = ti     δi = 1
                                                              .
                                  T i > ti     δi = 0

     Interval censored data: (x1 , a1 , b1 ), . . . , (xn , an , bn ).
     Observe Ti ∈ [ai , bi ].




                                                                                       4 / 72
                    Fundamental concepts
                       Bayesian approach     Time to event data
                   Semiparametric models     Functions defining lifetime distribution
                               Examples


Survival data: covariates and censoring


     Uncensored data: (x1 , t1 ), . . . , (xn , tn ). Observe Ti = ti .
     Right censored data: (x1 , t1 , δ1 ), . . . , (xn , tn , δn ). Observe

                                  T i = ti     δi = 1
                                                              .
                                  T i > ti     δi = 0

     Interval censored data: (x1 , a1 , b1 ), . . . , (xn , an , bn ).
     Observe Ti ∈ [ai , bi ].
     Not considered here: truncated data.




                                                                                       4 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.




                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.
     Discrete t has pmf. Discrete survival regression models
     include continuation ratio (hazard regression), proportional
     odds (survival odds regression), etc.




                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.
     Discrete t has pmf. Discrete survival regression models
     include continuation ratio (hazard regression), proportional
     odds (survival odds regression), etc.
     Survival function is
                                                               ∞
             S(t) = 1 − F (t) = P(T > t) =                         f (s)ds.
                                                           t




                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.
     Discrete t has pmf. Discrete survival regression models
     include continuation ratio (hazard regression), proportional
     odds (survival odds regression), etc.
     Survival function is
                                                               ∞
             S(t) = 1 − F (t) = P(T > t) =                         f (s)ds.
                                                           t

     Regression model that focuses on survival: proportional
     odds.




                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.
     Discrete t has pmf. Discrete survival regression models
     include continuation ratio (hazard regression), proportional
     odds (survival odds regression), etc.
     Survival function is
                                                               ∞
             S(t) = 1 − F (t) = P(T > t) =                         f (s)ds.
                                                           t

     Regression model that focuses on survival: proportional
     odds.
     Question: “What is probability of making it past 40 years?”
     Answer: S(40).



                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Density and survival
     Continuous T has density f (t); considered here.
     Discrete t has pmf. Discrete survival regression models
     include continuation ratio (hazard regression), proportional
     odds (survival odds regression), etc.
     Survival function is
                                                               ∞
             S(t) = 1 − F (t) = P(T > t) =                         f (s)ds.
                                                           t

     Regression model that focuses on survival: proportional
     odds.
     Question: “What is probability of making it past 40 years?”
     Answer: S(40).
     Question: “What are the odds of dying before 40?”
     Answer: 1−S(40) .
               S(40)
                                                                                  5 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Quantiles


     pth quantile qp for T is qp such that P(T ≤ qp ) = p.




                                                                                  6 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Quantiles


     pth quantile qp for T is qp such that P(T ≤ qp ) = p.
     qp = F −1 (p).




                                                                                   6 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Quantiles


     pth quantile qp for T is qp such that P(T ≤ qp ) = p.
     qp = F −1 (p).
     Question: “What is the median lifetime in the population?”
     Answer: F −1 (0.5).




                                                                                   6 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Quantiles


     pth quantile qp for T is qp such that P(T ≤ qp ) = p.
     qp = F −1 (p).
     Question: “What is the median lifetime in the population?”
     Answer: F −1 (0.5).
     Regression model that focuses on quantiles: accelerated
     failure time (proportional quantiles).




                                                                                   6 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Quantiles


     pth quantile qp for T is qp such that P(T ≤ qp ) = p.
     qp = F −1 (p).
     Question: “What is the median lifetime in the population?”
     Answer: F −1 (0.5).
     Regression model that focuses on quantiles: accelerated
     failure time (proportional quantiles).
     Quantile regression active area of research from
     frequentist and Bayesian perspective, e.g. Koenker’s
     excellent quantreg package for R.



                                                                                   6 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Residual life

      Mean residual life
                                                         ∞
                                                        t S(s)ds
                m(t) = E{T − t|T > t} =                                  .
                                                           S(t)




                                                                                   7 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Residual life

      Mean residual life
                                                         ∞
                                                        t S(s)ds
                m(t) = E{T − t|T > t} =                                  .
                                                           S(t)

      Question: “Given that I’ve made it up to 40 years, how
      much longer can I expect to live?”
      Answer: m(40).




                                                                                   7 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Residual life

      Mean residual life
                                                         ∞
                                                        t S(s)ds
                m(t) = E{T − t|T > t} =                                  .
                                                           S(t)

      Question: “Given that I’ve made it up to 40 years, how
      much longer can I expect to live?”
      Answer: m(40).
      Regression model that focuses on MRL: proportional mean
      residual life; there are others.




                                                                                   7 / 72
                  Fundamental concepts
                     Bayesian approach   Time to event data
                 Semiparametric models   Functions defining lifetime distribution
                             Examples


Residual life

      Mean residual life
                                                         ∞
                                                        t S(s)ds
                m(t) = E{T − t|T > t} =                                  .
                                                           S(t)

      Question: “Given that I’ve made it up to 40 years, how
      much longer can I expect to live?”
      Answer: m(40).
      Regression model that focuses on MRL: proportional mean
      residual life; there are others.
      Also: median (or any quantile) residual life. Much harder to
      work with in regression context.


                                                                                   7 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Hazard function


     Hazard at t:
                            P(t ≤ T < t + dt|T ≥ t)   f (t)
           h(t) = lim +                             =       .
                    dt→0              dt              S(t)




                                                                                  8 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Hazard function


     Hazard at t:
                            P(t ≤ T < t + dt|T ≥ t)   f (t)
           h(t) = lim +                             =       .
                    dt→0              dt              S(t)

     Question: “Given that I’ve made it up to 40 years, what is
     the probability I die in the next day?”
                                        1
     Answer: approximately h(40) 365 .




                                                                                  8 / 72
                 Fundamental concepts
                    Bayesian approach   Time to event data
                Semiparametric models   Functions defining lifetime distribution
                            Examples


Hazard function


     Hazard at t:
                            P(t ≤ T < t + dt|T ≥ t)   f (t)
           h(t) = lim +                             =       .
                    dt→0              dt              S(t)

     Question: “Given that I’ve made it up to 40 years, what is
     the probability I die in the next day?”
                                        1
     Answer: approximately h(40) 365 .
     Regression models that focuses on hazard function:
     proportional hazards (Cox) and additive hazards (Aalen)
     models.



                                                                                  8 / 72
                      Fundamental concepts
                         Bayesian approach    Time to event data
                     Semiparametric models    Functions defining lifetime distribution
                                 Examples


Density, survival, hazard, and MRL
                                               1
      0.25
                                              0.8
       0.2
                                              0.6
      0.15

                                              0.4
       0.1

      0.05                                    0.2


             2   4       6    8    10    12           2     4     6         8    10     12

         2                                     6
      1.75
                                               5
       1.5
                                               4
      1.25
         1                                     3
      0.75
                                               2
       0.5
                                               1
      0.25

             2   4       6    8    10   12           2     4     6      8       10    12




                                                                                             9 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Bayes modifies a likelihood
     Let θ index a probability density fθ .




                                                                           10 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Bayes modifies a likelihood
     Let θ index a probability density fθ .
     Data x = (x1 , . . . , xn ) are collected x ∼ fθ ; likelihood is
     fθ (x) as function of θ.




                                                                            10 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Bayes modifies a likelihood
     Let θ index a probability density fθ .
     Data x = (x1 , . . . , xn ) are collected x ∼ fθ ; likelihood is
     fθ (x) as function of θ.
     Frequentist might estimate θ using MLE
     θ = argmaxθ∈Θ fθ (x) and study sampling distribution of
     θ(x), often asymptotic.




                                                                            10 / 72
                  Fundamental concepts
                                          Building likelihood & posterior
                     Bayesian approach
                                          Gamma process
                 Semiparametric models
                                          Mixture of Polya trees
                             Examples


Bayes modifies a likelihood
     Let θ index a probability density fθ .
     Data x = (x1 , . . . , xn ) are collected x ∼ fθ ; likelihood is
     fθ (x) as function of θ.
     Frequentist might estimate θ using MLE
     θ = argmaxθ∈Θ fθ (x) and study sampling distribution of
     θ(x), often asymptotic.
     Bayesian places prior distribution on θ ∼ p(θ), Bayes’ rule
     gives posterior distribution:
                                           fθ (x)p(θ)
                        p(θ|x) =                        .
                                         Θ fθ (x)p(θ)dθ




                                                                            10 / 72
                  Fundamental concepts
                                          Building likelihood & posterior
                     Bayesian approach
                                          Gamma process
                 Semiparametric models
                                          Mixture of Polya trees
                             Examples


Bayes modifies a likelihood
     Let θ index a probability density fθ .
     Data x = (x1 , . . . , xn ) are collected x ∼ fθ ; likelihood is
     fθ (x) as function of θ.
     Frequentist might estimate θ using MLE
     θ = argmaxθ∈Θ fθ (x) and study sampling distribution of
     θ(x), often asymptotic.
     Bayesian places prior distribution on θ ∼ p(θ), Bayes’ rule
     gives posterior distribution:
                                           fθ (x)p(θ)
                        p(θ|x) =                        .
                                         Θ fθ (x)p(θ)dθ

     Bayes’ estimate typically posterior mean, median, or
     mode; e.g. θ = argmaxθ∈Θ fθ (x)p(θ).
                                                                            10 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Parametric survival without covariates
     Survival distribution completely defined by any of f (t), S(t),
     h(t), or m(t).




                                                                           11 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Parametric survival without covariates
     Survival distribution completely defined by any of f (t), S(t),
     h(t), or m(t).
     Each of these can be derived from one of the others.




                                                                           11 / 72
                  Fundamental concepts
                                           Building likelihood & posterior
                     Bayesian approach
                                           Gamma process
                 Semiparametric models
                                           Mixture of Polya trees
                             Examples


Parametric survival without covariates
     Survival distribution completely defined by any of f (t), S(t),
     h(t), or m(t).
     Each of these can be derived from one of the others.
     Simplest case: iid with (noninformative) right censoring
     gives
                         n                               n
             L(S) =                δi
                              f (ti ) S(ti )1−δi
                                                   =         S(ti )h(ti )δi .
                        i=1                            i=1




                                                                                11 / 72
                  Fundamental concepts
                                            Building likelihood & posterior
                     Bayesian approach
                                            Gamma process
                 Semiparametric models
                                            Mixture of Polya trees
                             Examples


Parametric survival without covariates
     Survival distribution completely defined by any of f (t), S(t),
     h(t), or m(t).
     Each of these can be derived from one of the others.
     Simplest case: iid with (noninformative) right censoring
     gives
                         n                                n
             L(S) =                 δi
                               f (ti ) S(ti )1−δi
                                                    =         S(ti )h(ti )δi .
                         i=1                            i=1

     If S(t) is parametric, e.g. Sθ (t) = exp −(t/θ2 )θ1 , then
     likelihood is finite-dimensional:
                     n
                                                                                 δi
          L(θ) =         exp −(t/θ2 )θ1              (θ1 /θ2 )(ti /θ2 )θ1             .
                   i=1


                                                                                          11 / 72
                  Fundamental concepts
                                            Building likelihood & posterior
                     Bayesian approach
                                            Gamma process
                 Semiparametric models
                                            Mixture of Polya trees
                             Examples


Parametric survival without covariates
     Survival distribution completely defined by any of f (t), S(t),
     h(t), or m(t).
     Each of these can be derived from one of the others.
     Simplest case: iid with (noninformative) right censoring
     gives
                         n                                n
             L(S) =                 δi
                               f (ti ) S(ti )1−δi
                                                    =         S(ti )h(ti )δi .
                         i=1                            i=1

     If S(t) is parametric, e.g. Sθ (t) = exp −(t/θ2 )θ1 , then
     likelihood is finite-dimensional:
                     n
                                                                                 δi
          L(θ) =         exp −(t/θ2 )θ1              (θ1 /θ2 )(ti /θ2 )θ1             .
                   i=1
     Bayesian further places prior on θ, e.g.
     θ1 ∼ Γ(7.3, 2.4) ⊥ θ2 ∼ exp(0.74).                                                   11 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).




                                                                          12 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).
     Priors on h(t) include extended gamma, piecewise
     exponential, correlated processes, etc.




                                                                          12 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).
     Priors on h(t) include extended gamma, piecewise
     exponential, correlated processes, etc.
                          t
     Priors on H(t) =     0   h(s)ds include gamma, beta, etc.




                                                                          12 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).
     Priors on h(t) include extended gamma, piecewise
     exponential, correlated processes, etc.
                          t
     Priors on H(t) =     0   h(s)ds include gamma, beta, etc.
     Priors on S(t) include Dirichlet process (DP).




                                                                          12 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).
     Priors on h(t) include extended gamma, piecewise
     exponential, correlated processes, etc.
                          t
     Priors on H(t) =     0   h(s)ds include gamma, beta, etc.
     Priors on S(t) include Dirichlet process (DP).
     Priors on f (t) include DP mixtures, more general
     nonparametric mixtures, finite mixtures, Polya trees,
     log-splines, etc.




                                                                          12 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


Nonparametric survival without covariates

     Infinite-dimensional process directly defined on one of h(t),
     H(t), f (t), or S(t).
     Priors on h(t) include extended gamma, piecewise
     exponential, correlated processes, etc.
                          t
     Priors on H(t) =     0   h(s)ds include gamma, beta, etc.
     Priors on S(t) include Dirichlet process (DP).
     Priors on f (t) include DP mixtures, more general
     nonparametric mixtures, finite mixtures, Polya trees,
     log-splines, etc.
     H(t) ∼ GP(c, Hθ ) and S(t) ∼ PT (c, ρ, Sθ ) described
     below...

                                                                          12 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Gamma process prior on H(t)



     Let Hθ (t) be increasing on t > 0, left-continuous,
     Hθ (0) = 0.




                                                                           13 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Gamma process prior on H(t)



     Let Hθ (t) be increasing on t > 0, left-continuous,
     Hθ (0) = 0.
     H(t) ∼ GP(c, Hθ ) if
         H(0) = 0.
         H(t) has independent increments in disjoint intervals.
         t > s implies H(t) − H(s) ∼ Γ(c(Hθ (t) − Hθ (s)), c).




                                                                           13 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Gamma process prior on H(t)



     Let Hθ (t) be increasing on t > 0, left-continuous,
     Hθ (0) = 0.
     H(t) ∼ GP(c, Hθ ) if
         H(0) = 0.
         H(t) has independent increments in disjoint intervals.
         t > s implies H(t) − H(s) ∼ Γ(c(Hθ (t) − Hθ (s)), c).
     Note that E{H(t)} = Hθ (t) and var{H(t)} = Hθ (t)/c. Also
     H(t) increasing.




                                                                           13 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Piecewise exponential approximates gamma process


     Let R+ = [0, a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [aJ−1 , ∞) be fixed, known.




                                                                            14 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Piecewise exponential approximates gamma process


     Let R+ = [0, a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [aJ−1 , ∞) be fixed, known.
     If H(t) ∼ GP(c, Hθ ), then
                                 ind.
     λj = H(aj ) − H(aj−1 ) ∼ Γ(c(Hθ (aj ) − Hθ (aj−1 )), c).




                                                                            14 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Piecewise exponential approximates gamma process


     Let R+ = [0, a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [aJ−1 , ∞) be fixed, known.
     If H(t) ∼ GP(c, Hθ ), then
                                 ind.
     λj = H(aj ) − H(aj−1 ) ∼ Γ(c(Hθ (aj ) − Hθ (aj−1 )), c).
     Take partition to be a fine mesh and assume hazard is
     constant with value λj over interval [aj−1 , aj ) ⇒
     approximates the gamma process.




                                                                            14 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Piecewise exponential approximates gamma process


     Let R+ = [0, a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [aJ−1 , ∞) be fixed, known.
     If H(t) ∼ GP(c, Hθ ), then
                                 ind.
     λj = H(aj ) − H(aj−1 ) ∼ Γ(c(Hθ (aj ) − Hθ (aj−1 )), c).
     Take partition to be a fine mesh and assume hazard is
     constant with value λj over interval [aj−1 , aj ) ⇒
     approximates the gamma process.
     Finite dimensional. Easy to fit.




                                                                            14 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Piecewise exponential approximates gamma process


     Let R+ = [0, a1 ) ∪ [a1 , a2 ) ∪ · · · ∪ [aJ−1 , ∞) be fixed, known.
     If H(t) ∼ GP(c, Hθ ), then
                                 ind.
     λj = H(aj ) − H(aj−1 ) ∼ Γ(c(Hθ (aj ) − Hθ (aj−1 )), c).
     Take partition to be a fine mesh and assume hazard is
     constant with value λj over interval [aj−1 , aj ) ⇒
     approximates the gamma process.
     Finite dimensional. Easy to fit.
     How to pick a1 < a2 < · · · < aJ−1 ?




                                                                            14 / 72
                Fundamental concepts
                                       Building likelihood & posterior
                   Bayesian approach
                                       Gamma process
               Semiparametric models
                                       Mixture of Polya trees
                           Examples


Piecewise constant hazard: h(t)


         0.8




         0.6




         0.4




         0.2




                   2            4          6              8              10




                                                                              15 / 72
               Fundamental concepts
                                      Building likelihood & posterior
                  Bayesian approach
                                      Gamma process
              Semiparametric models
                                      Mixture of Polya trees
                          Examples
                                                          t
Piecewise constant hazard: H(t) =                        0    h(s)ds



          3




        2.5




          2




        1.5




         1




        0.5




                  2            4          6              8              10




                                                                             16 / 72
                Fundamental concepts
                                       Building likelihood & posterior
                   Bayesian approach
                                       Gamma process
               Semiparametric models
                                       Mixture of Polya trees
                           Examples


Piecewise constant hazard: S(t) = exp{−H(t)}


          1




        0.8




        0.6




         0.4




        0.2




                    2           4          6             8               10




                                                                              17 / 72
                Fundamental concepts
                                       Building likelihood & posterior
                   Bayesian approach
                                       Gamma process
               Semiparametric models
                                       Mixture of Polya trees
                           Examples


Piecewise constant hazard: f (t) = h(t)S(t)



         0.5




         0.4




         0.3




         0.2




         0.1




                   2            4          6              8              10




                                                                              18 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Polya tree = partition + beta conditional probabilities



      Notation: S ∼ PT (c, ρ(·), Sθ ). S is random probability
      measure centered at Sθ , parametric on R.




                                                                            19 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Polya tree = partition + beta conditional probabilities



      Notation: S ∼ PT (c, ρ(·), Sθ ). S is random probability
      measure centered at Sθ , parametric on R.
      Polya tree prior on S defined through nested partitions of
      R, say Πθ , and associated conditional probabilities Yj at
               j
      level j.




                                                                            19 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Polya tree = partition + beta conditional probabilities



      Notation: S ∼ PT (c, ρ(·), Sθ ). S is random probability
      measure centered at Sθ , parametric on R.
      Polya tree prior on S defined through nested partitions of
      R, say Πθ , and associated conditional probabilities Yj at
               j
      level j.
      Partition Πθ at level j splits R into 2j pieces of equal
                 j
      probability under Sθ . Sets denoted Bθ ( ) where is binary.




                                                                            19 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Polya tree = partition + beta conditional probabilities



      Notation: S ∼ PT (c, ρ(·), Sθ ). S is random probability
      measure centered at Sθ , parametric on R.
      Polya tree prior on S defined through nested partitions of
      R, say Πθ , and associated conditional probabilities Yj at
               j
      level j.
      Partition Πθ at level j splits R into 2j pieces of equal
                 j
      probability under Sθ . Sets denoted Bθ ( ) where is binary.
      Next slide shows Π1 , Π2 , and Π3 for Sθ = N(0, 1).




                                                                            19 / 72
                    Fundamental concepts
                                                Building likelihood & posterior
                       Bayesian approach
                                                Gamma process
                   Semiparametric models
                                                Mixture of Polya trees
                               Examples


Polya tree sets for Sθ = N(0, 1)




              -2      000        001 010 011 100 001 110                   111    2

                            00             01          10             11
                                    0                        1


         Figure: First 3 partitions of R generated by N(0, 1).

                                                                                      20 / 72
                  Fundamental concepts
                                         Building likelihood & posterior
                     Bayesian approach
                                         Gamma process
                 Semiparametric models
                                         Mixture of Polya trees
                             Examples


Additional parameters give flexibility



      Parametric Sθ gives partition.




                                                                           21 / 72
                     Fundamental concepts
                                            Building likelihood & posterior
                        Bayesian approach
                                            Gamma process
                    Semiparametric models
                                            Mixture of Polya trees
                                Examples


Additional parameters give flexibility



      Parametric Sθ gives partition.
      Add Y1 = {Y0 , Y1 }, Y2 = {Y00 , Y01 , Y10 , Y11 },
      Y3 = {Y000 , Y001 , Y010 , Y011 , Y100 , Y101 , Y110 , Y111 }, etc. to
      refine density shape. Let Y = {Y1 , Y2 , . . . , YJ }.




                                                                               21 / 72
                      Fundamental concepts
                                                 Building likelihood & posterior
                         Bayesian approach
                                                 Gamma process
                     Semiparametric models
                                                 Mixture of Polya trees
                                 Examples


Additional parameters give flexibility



      Parametric Sθ gives partition.
      Add Y1 = {Y0 , Y1 }, Y2 = {Y00 , Y01 , Y10 , Y11 },
      Y3 = {Y000 , Y001 , Y010 , Y011 , Y100 , Y101 , Y110 , Y111 }, etc. to
      refine density shape. Let Y = {Y1 , Y2 , . . . , YJ }.
      Y   0   = S{Bθ ( 0)|Bθ ( )}. Y         1   = 1 − Y 0.




                                                                                   21 / 72
                      Fundamental concepts
                                                 Building likelihood & posterior
                         Bayesian approach
                                                 Gamma process
                     Semiparametric models
                                                 Mixture of Polya trees
                                 Examples


Additional parameters give flexibility



      Parametric Sθ gives partition.
      Add Y1 = {Y0 , Y1 }, Y2 = {Y00 , Y01 , Y10 , Y11 },
      Y3 = {Y000 , Y001 , Y010 , Y011 , Y100 , Y101 , Y110 , Y111 }, etc. to
      refine density shape. Let Y = {Y1 , Y2 , . . . , YJ }.
      Y   0   = S{Bθ ( 0)|Bθ ( )}. Y         1   = 1 − Y 0.
      Next slides take Sθ to be N(0, 1) and fix values of Y1 , Y2 ,
      and Y3 .




                                                                                   21 / 72
      Fundamental concepts
                                   Building likelihood & posterior
         Bayesian approach
                                   Gamma process
     Semiparametric models
                                   Mixture of Polya trees
                 Examples




-2      0.5         0.5 0.5 0.5 0.5 0.5 0.5                   0.5    2

              0.5            0.5         0.5            0.5
                      0.5                      0.5


     Figure: All pairs (Y 0 , Y 1 ) are 0.5.


                                                                         22 / 72
         Fundamental concepts
                                      Building likelihood & posterior
            Bayesian approach
                                      Gamma process
        Semiparametric models
                                      Mixture of Polya trees
                    Examples




   -2      0.5         0.5 0.5 0.5 0.5 0.5 0.5                   0.5    2

                 0.5            0.5         0.5            0.5
                         0.45                   0.55


Figure: Pair of level j = 1 probabilities (Y0 , Y1 ).


                                                                            23 / 72
          Fundamental concepts
                                       Building likelihood & posterior
             Bayesian approach
                                       Gamma process
         Semiparametric models
                                       Mixture of Polya trees
                     Examples




    -2      0.5         0.5 0.5 0.5 0.5 0.5 0.5                   0.5    2

                  0.7            0.3         0.5            0.5
                          0.45                   0.55


Figure: Pair of level j = 2 probabilities (Y00 , Y01 ).


                                                                             24 / 72
          Fundamental concepts
                                       Building likelihood & posterior
             Bayesian approach
                                       Gamma process
         Semiparametric models
                                       Mixture of Polya trees
                     Examples




    -2      0.5         0.5 0.5 0.5 0.5 0.5 0.5                   0.5    2

                  0.7            0.3         0.6            0.4
                          0.45                   0.55


Figure: Pair of level j = 2 probabilities (Y10 , Y11 ).


                                                                             25 / 72
           Fundamental concepts
                                        Building likelihood & posterior
              Bayesian approach
                                        Gamma process
          Semiparametric models
                                        Mixture of Polya trees
                      Examples




     -2      0.8         0.2 0.5 0.5 0.5 0.5 0.5                   0.5    2

                   0.7            0.3         0.6            0.4
                           0.45                   0.55


Figure: Pair of level j = 3 probabilities (Y000 , Y001 ).


                                                                              26 / 72
           Fundamental concepts
                                        Building likelihood & posterior
              Bayesian approach
                                        Gamma process
          Semiparametric models
                                        Mixture of Polya trees
                      Examples




     -2      0.8         0.2 0.7 0.3 0.5 0.5 0.5                   0.5    2

                   0.7            0.3         0.6            0.4
                           0.45                   0.55


Figure: Pair of level j = 3 probabilities (Y010 , Y011 ).


                                                                              27 / 72
           Fundamental concepts
                                        Building likelihood & posterior
              Bayesian approach
                                        Gamma process
          Semiparametric models
                                        Mixture of Polya trees
                      Examples




     -2      0.8         0.2 0.7 0.3 0.4 0.6 0.5                   0.5    2

                   0.7            0.3         0.6            0.4
                           0.45                   0.55


Figure: Pair of level j = 3 probabilities (Y100 , Y101 ).


                                                                              28 / 72
           Fundamental concepts
                                        Building likelihood & posterior
              Bayesian approach
                                        Gamma process
          Semiparametric models
                                        Mixture of Polya trees
                      Examples




     -2      0.8         0.2 0.7 0.3 0.4 0.6 0.55                 0.45    2

                   0.7            0.3         0.6            0.4
                           0.45                   0.55


Figure: Pair of level j = 3 probabilities (Y110 , Y111 ).


                                                                              29 / 72
      Fundamental concepts
                                   Building likelihood & posterior
         Bayesian approach
                                   Gamma process
     Semiparametric models
                                   Mixture of Polya trees
                 Examples




-2      0.8         0.2 0.7 0.3 0.4 0.6 0.55                 0.45    2

              0.7            0.3         0.6            0.4
                      0.45                   0.55


 Figure: Mixture of Finite Polya trees.


                                                                         30 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Prior on (Y 0 , Y 1 )



       Want E(Y 0 ) = 0.5 to center S at Sθ . Take

                           Y   0   ∼ beta(cρ(j), cρ(j)).




                                                                            31 / 72
                   Fundamental concepts
                                          Building likelihood & posterior
                      Bayesian approach
                                          Gamma process
                  Semiparametric models
                                          Mixture of Polya trees
                              Examples


Prior on (Y 0 , Y 1 )



       Want E(Y 0 ) = 0.5 to center S at Sθ . Take

                           Y   0   ∼ beta(cρ(j), cρ(j)).

       Conjugate beta distribution gives ‘Polya tree’ – other
       distributions give a tailfree prior.




                                                                            31 / 72
                    Fundamental concepts
                                           Building likelihood & posterior
                       Bayesian approach
                                           Gamma process
                   Semiparametric models
                                           Mixture of Polya trees
                               Examples


Prior on (Y 0 , Y 1 )



       Want E(Y 0 ) = 0.5 to center S at Sθ . Take

                            Y   0   ∼ beta(cρ(j), cρ(j)).

       Conjugate beta distribution gives ‘Polya tree’ – other
       distributions give a tailfree prior.
       c and ρ(j) affect how quickly data “take over” Sθ .




                                                                             31 / 72
                    Fundamental concepts
                                           Building likelihood & posterior
                       Bayesian approach
                                           Gamma process
                   Semiparametric models
                                           Mixture of Polya trees
                               Examples


Prior on (Y 0 , Y 1 )



       Want E(Y 0 ) = 0.5 to center S at Sθ . Take

                            Y   0   ∼ beta(cρ(j), cρ(j)).

       Conjugate beta distribution gives ‘Polya tree’ – other
       distributions give a tailfree prior.
       c and ρ(j) affect how quickly data “take over” Sθ .
       c is weight, ρ(j) affects “clumpiness.”




                                                                             31 / 72
                            Fundamental concepts
                                                            Building likelihood & posterior
                               Bayesian approach
                                                            Gamma process
                           Semiparametric models
                                                            Mixture of Polya trees
                                       Examples


“Standard” parameterization ρ(j) = j 2

                                                         R
                              B0                                                  B1
                                              (Y0 , Y1 ) ∼ Dir(c, c)
                  B00                      B01                      B10                      B11
                   (Y00 , Y01 ) ∼ Dir(4c, 4c)                        (Y10 , Y11 ) ∼ Dir(4c, 4c)
          B000        B001        B010         B011        B100         B101         B110        B111
           (Y000 , Y001 ) ∼         (Y010 , Y011 ) ∼         (Y100 , Y101 ) ∼         (Y110 , Y111 ) ∼
              Dir(9c, 9c)              Dir(9c, 9c)              Dir(9c, 9c)              Dir(9c, 9c)


                            Π1 = {B0 , B1 }, Y1 = {Y0 , Y1 }.
         Π2 = {B00 , B01 , B10 , B11 }, Y2 = {Y00 , Y01 , Y10 , Y11 }.
          Π3 = {B000 , B001 , B010 , B011 , B100 , B101 , B110 , B111 }
          Y3 = {Y000 , Y001 , Y010 , Y011 , Y100 , Y101 , Y110 , Y111 }
  Adds 7 free parameters
  Y = {Y0 , Y00 , Y10 , Y000 , Y010 , Y100 , Y110 }.

                                                                                                         32 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


What do random densities look like?



     MPT prior S ∼ PT5 (1, ρ, exp(θ))
     where θ ∼ Γ(10, 10) so E(θ) = 1. So overall centering
     distribution is exp(1).




                                                                          33 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


What do random densities look like?



     MPT prior S ∼ PT5 (1, ρ, exp(θ))
     where θ ∼ Γ(10, 10) so E(θ) = 1. So overall centering
     distribution is exp(1).
     Take J = 5, and c = 1.




                                                                          33 / 72
                 Fundamental concepts
                                        Building likelihood & posterior
                    Bayesian approach
                                        Gamma process
                Semiparametric models
                                        Mixture of Polya trees
                            Examples


What do random densities look like?



     MPT prior S ∼ PT5 (1, ρ, exp(θ))
     where θ ∼ Γ(10, 10) so E(θ) = 1. So overall centering
     distribution is exp(1).
     Take J = 5, and c = 1.
     Look at 10 random f (t)’s from MPT prior. That is, 10
     random Y. The densities are averaged over θ ∼ Γ(10, 10).




                                                                          33 / 72
            Fundamental concepts
                                     Building likelihood & posterior
               Bayesian approach
                                     Gamma process
           Semiparametric models
                                     Mixture of Polya trees
                       Examples


                                 MPT
3




2




1




                     1                        2                        3
                           iid
    Figure: f1 , . . . , f10 ∼     PT5 (1, ρ, exp(θ))P(dθ).

                                                                           34 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why semiparametric?


     Splits inference into two pieces: β and S0 (t).




                                                                    35 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why semiparametric?


     Splits inference into two pieces: β and S0 (t).
     Ideally, β succinctly summarizes effects of risk factors x on
     aspects of survival. Make S0 (t) as flexible as possible.




                                                                     35 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why semiparametric?


     Splits inference into two pieces: β and S0 (t).
     Ideally, β succinctly summarizes effects of risk factors x on
     aspects of survival. Make S0 (t) as flexible as possible.
     Can make easily digestible statements concerning the
     population, e.g. “Median life on those receiving treatment A
     is 1.7 times those receiving B, adjusting for other factors.”




                                                                     35 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why semiparametric?


     Splits inference into two pieces: β and S0 (t).
     Ideally, β succinctly summarizes effects of risk factors x on
     aspects of survival. Make S0 (t) as flexible as possible.
     Can make easily digestible statements concerning the
     population, e.g. “Median life on those receiving treatment A
     is 1.7 times those receiving B, adjusting for other factors.”
     Good starting place for fully nonparametric models (e.g.
     additive models, varying coefficient models, dependent
     process models, MARS).




                                                                     35 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why semiparametric?


     Splits inference into two pieces: β and S0 (t).
     Ideally, β succinctly summarizes effects of risk factors x on
     aspects of survival. Make S0 (t) as flexible as possible.
     Can make easily digestible statements concerning the
     population, e.g. “Median life on those receiving treatment A
     is 1.7 times those receiving B, adjusting for other factors.”
     Good starting place for fully nonparametric models (e.g.
     additive models, varying coefficient models, dependent
     process models, MARS).
     I will use mixtures of Polya trees priors on S0 in examples.


                                                                     35 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).




                                                                   36 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).
    AH: hx (t) = h0 (t) + β x.




                                                                   36 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).
    AH: hx (t) = h0 (t) + β x.
    AFT: Sx (t) = S0 {eβ x t}.




                                                                   36 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).
    AH: hx (t) = h0 (t) + β x.
    AFT: Sx (t) = S0 {eβ x t}.
    PO: Fx (t)/Sx (t) = eβ x F0 (t)/S0 (t).




                                                                   36 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).
    AH: hx (t) = h0 (t) + β x.
    AFT: Sx (t) = S0 {eβ x t}.
    PO: Fx (t)/Sx (t) = eβ x F0 (t)/S0 (t).
    PMRL mx (t) = eβ x m0 (t).




                                                                   36 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Some models



    PH: hx (t) = exp(x β)h0 (t).
    AH: hx (t) = h0 (t) + β x.
    AFT: Sx (t) = S0 {eβ x t}.
    PO: Fx (t)/Sx (t) = eβ x F0 (t)/S0 (t).
    PMRL mx (t) = eβ x m0 (t).
    Others, but this is a nice start...




                                                                   36 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .




                                                                    37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).




                                                                    37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).
     Stochastically orders Sx1 and Sx2 .




                                                                    37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).
     Stochastically orders Sx1 and Sx2 .
     eβj is how risk changes when xj is increased by unity.




                                                                    37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).
     Stochastically orders Sx1 and Sx2 .
     eβj is how risk changes when xj is increased by unity.
     Priors placed on β and one of h0 (t), H0 (t), or S0 (t).




                                                                    37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

           hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).
     Stochastically orders Sx1 and Sx2 .
     eβj is how risk changes when xj is increased by unity.
     Priors placed on β and one of h0 (t), H0 (t), or S0 (t).
     Cox (1972) is second most cited paper in statistics. (First is
     Kaplan and Meier, 1958).


                                                                      37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards (PH)

     Model is:

            hx (t) = exp(x β)h0 (t) or Sx (t) = S0 (t)exp(x β) .

     Extended to time dependent covariates via
     hx (t) = exp(x(t) β)h0 (t).
     Stochastically orders Sx1 and Sx2 .
     eβj is how risk changes when xj is increased by unity.
     Priors placed on β and one of h0 (t), H0 (t), or S0 (t).
     Cox (1972) is second most cited paper in statistics. (First is
     Kaplan and Meier, 1958).
     Why?

                                                                      37 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards the “default”...

  Therneau and Grambsch (2000) Modeling Survival Data:
  Extending the Cox Model discuss the Cox model including
  many generalizations. When proportional hazards fails they
  recommend:
      Stratification within the Cox model.




                                                                    38 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards the “default”...

  Therneau and Grambsch (2000) Modeling Survival Data:
  Extending the Cox Model discuss the Cox model including
  many generalizations. When proportional hazards fails they
  recommend:
      Stratification within the Cox model.
      PH may hold over short time periods, so partition the time
      axis within the Cox model.




                                                                    38 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards the “default”...

  Therneau and Grambsch (2000) Modeling Survival Data:
  Extending the Cox Model discuss the Cox model including
  many generalizations. When proportional hazards fails they
  recommend:
      Stratification within the Cox model.
      PH may hold over short time periods, so partition the time
      axis within the Cox model.
      Time varying effects β(t) within the Cox model.




                                                                    38 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards the “default”...

  Therneau and Grambsch (2000) Modeling Survival Data:
  Extending the Cox Model discuss the Cox model including
  many generalizations. When proportional hazards fails they
  recommend:
      Stratification within the Cox model.
      PH may hold over short time periods, so partition the time
      axis within the Cox model.
      Time varying effects β(t) within the Cox model.
      Only as a last resort consider other models, e.g.
      accelerated failure time or additive hazards.




                                                                    38 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Proportional hazards the “default”...

  Therneau and Grambsch (2000) Modeling Survival Data:
  Extending the Cox Model discuss the Cox model including
  many generalizations. When proportional hazards fails they
  recommend:
      Stratification within the Cox model.
      PH may hold over short time periods, so partition the time
      axis within the Cox model.
      Time varying effects β(t) within the Cox model.
      Only as a last resort consider other models, e.g.
      accelerated failure time or additive hazards.
      Why the reluctance to explore other semiparametric
      models?

                                                                    38 / 72
                   Fundamental concepts   Proportional hazards
                      Bayesian approach   Accelerated failure time
                  Semiparametric models   Proportional odds
                              Examples    Other models


Why is proportional hazards the “default?”




  If you have a hammer, every problem looks like a nail.

                                                                     39 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.




                                                                    40 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.
     SAS PHREG, other software provided momentum.




                                                                    40 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.
     SAS PHREG, other software provided momentum.
     Naturally generalized to time dependent covariates,
     time-varying effects, frailties, etc.




                                                                    40 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.
     SAS PHREG, other software provided momentum.
     Naturally generalized to time dependent covariates,
     time-varying effects, frailties, etc.
     Highly interpretable.




                                                                    40 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.
     SAS PHREG, other software provided momentum.
     Naturally generalized to time dependent covariates,
     time-varying effects, frailties, etc.
     Highly interpretable.
     But...with today’s computing power other semiparametric
     models may provide vastly improved fit over PH or
     generalizations of PH.




                                                                    40 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Why is proportional hazards the “default?”


     Initially, partial likelihood made relatively fitting easy.
     SAS PHREG, other software provided momentum.
     Naturally generalized to time dependent covariates,
     time-varying effects, frailties, etc.
     Highly interpretable.
     But...with today’s computing power other semiparametric
     models may provide vastly improved fit over PH or
     generalizations of PH.
     Having said that, there are a number of excellent packages
     available for fitting Bayesian PH models...


                                                                    40 / 72
               Fundamental concepts   Proportional hazards
                  Bayesian approach   Accelerated failure time
              Semiparametric models   Proportional odds
                          Examples    Other models


Fitting Bayesian PH in packages
     SAS: BAYES command in PROC PHREG gives piecewise
     exponential.




                                                                 41 / 72
               Fundamental concepts   Proportional hazards
                  Bayesian approach   Accelerated failure time
              Semiparametric models   Proportional odds
                          Examples    Other models


Fitting Bayesian PH in packages
     SAS: BAYES command in PROC PHREG gives piecewise
     exponential.
     SAS: PROC MCMC.




                                                                 41 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Fitting Bayesian PH in packages
     SAS: BAYES command in PROC PHREG gives piecewise
     exponential.
     SAS: PROC MCMC.
     Belitz, Brezger, Kneib, and Lang’s BayesX assigns
     penalized B-spline prior on log h0 (t) and allows for additive
     predictors, structured frailties, time-varying coefficients,
     etc. Free:
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.




                                                                      41 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Fitting Bayesian PH in packages
     SAS: BAYES command in PROC PHREG gives piecewise
     exponential.
     SAS: PROC MCMC.
     Belitz, Brezger, Kneib, and Lang’s BayesX assigns
     penalized B-spline prior on log h0 (t) and allows for additive
     predictors, structured frailties, time-varying coefficients,
     etc. Free:
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Spiegelhalter, Thomas, Best, and Lunn’s WinBUGS has
     example of counting process likelihood that can be easily
     modified to piecewise exponential. Also parametric
     example with frailties.


                                                                      41 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Fitting Bayesian PH in packages
     SAS: BAYES command in PROC PHREG gives piecewise
     exponential.
     SAS: PROC MCMC.
     Belitz, Brezger, Kneib, and Lang’s BayesX assigns
     penalized B-spline prior on log h0 (t) and allows for additive
     predictors, structured frailties, time-varying coefficients,
     etc. Free:
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Spiegelhalter, Thomas, Best, and Lunn’s WinBUGS has
     example of counting process likelihood that can be easily
     modified to piecewise exponential. Also parametric
     example with frailties.
     Alejandro Jara’s DPpackage for R can fit PH with
     piecewise constaint h0 (t) and nonparametric frailties.
                                                                      41 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .




                                                                   42 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Accelerated failure time (AFT)
     Model is
             Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).




                                                                    42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).
     Stochastically orders Sx1 and Sx2 .




                                                                   42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).
     Stochastically orders Sx1 and Sx2 .
     eβj how any quantile – or mean – changes when
     increasing xj by unity.




                                                                   42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).
     Stochastically orders Sx1 and Sx2 .
     eβj how any quantile – or mean – changes when
     increasing xj by unity.
     Priors can be placed on S0 (t) or equivalently e0 . Prior
     elicitation in Bedrick, Christensen, and Johnson (2000).




                                                                   42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).
     Stochastically orders Sx1 and Sx2 .
     eβj how any quantile – or mean – changes when
     increasing xj by unity.
     Priors can be placed on S0 (t) or equivalently e0 . Prior
     elicitation in Bedrick, Christensen, and Johnson (2000).
     Komarek’s bayesSurv for AFT models, spline and
     discrete normal mixture on error. Versions can be fit in
     WinBUGS.


                                                                   42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Accelerated failure time (AFT)
     Model is
            Sx (t) = S0 e−x β t , or log Tx = x β + e0 .

     Implies qp (x) = ex β qp (0).
     Stochastically orders Sx1 and Sx2 .
     eβj how any quantile – or mean – changes when
     increasing xj by unity.
     Priors can be placed on S0 (t) or equivalently e0 . Prior
     elicitation in Bedrick, Christensen, and Johnson (2000).
     Komarek’s bayesSurv for AFT models, spline and
     discrete normal mixture on error. Versions can be fit in
     WinBUGS.
     bj() in Harrell’s Design library fits Buckley-James
     version.
                                                                   42 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Proportional odds (PO)

     Model is
                   1 − Sx (t)            1 − S0 (t)
                              = exp(x β)            .
                     Sx (t)                S0 (t)




                                                                   43 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Proportional odds (PO)

     Model is
                   1 − Sx (t)            1 − S0 (t)
                              = exp(x β)            .
                     Sx (t)                S0 (t)
     βj how odds of event occuring before t changes when xj
     increased by unity (for any t).




                                                                   43 / 72
                  Fundamental concepts     Proportional hazards
                     Bayesian approach     Accelerated failure time
                 Semiparametric models     Proportional odds
                             Examples      Other models


Proportional odds (PO)

     Model is
                    1 − Sx (t)            1 − S0 (t)
                               = exp(x β)            .
                      Sx (t)                S0 (t)
     βj how odds of event occuring before t changes when xj
     increased by unity (for any t).
     Attenuation of risk:
                                         hx1 (t)
                                 lim             = 1.
                                t→∞      hx2 (t)

     Plausible in many situations.




                                                                      43 / 72
                  Fundamental concepts     Proportional hazards
                     Bayesian approach     Accelerated failure time
                 Semiparametric models     Proportional odds
                             Examples      Other models


Proportional odds (PO)

     Model is
                    1 − Sx (t)            1 − S0 (t)
                               = exp(x β)            .
                      Sx (t)                S0 (t)
     βj how odds of event occuring before t changes when xj
     increased by unity (for any t).
     Attenuation of risk:
                                         hx1 (t)
                                 lim             = 1.
                                t→∞      hx2 (t)

     Plausible in many situations.
     No ready software for fitting Bayes version. timereg has
     frequentist version. (My code in FORTRAN.)

                                                                      43 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Additive hazards (AH)

     Model is
                          hx (t) = h0 (t) + x(t) β.




                                                                   44 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Additive hazards (AH)

     Model is
                          hx (t) = h0 (t) + x(t) β.
     βj is how risk changes when increasing xj by unity.




                                                                   44 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Additive hazards (AH)

     Model is
                          hx (t) = h0 (t) + x(t) β.
     βj is how risk changes when increasing xj by unity.
     Can be estimated in standard software using empirical
     Bayes approach with gamma process prior on H0 (t) (Sinha
     et al., 2009).




                                                                   44 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Additive hazards (AH)

     Model is
                          hx (t) = h0 (t) + x(t) β.
     βj is how risk changes when increasing xj by unity.
     Can be estimated in standard software using empirical
     Bayes approach with gamma process prior on H0 (t) (Sinha
     et al., 2009).
     Other approaches require elaborate model specification to
     incorporate awkward constraints (Yin and Ibrahim, 2005;
     Dunson and Herring, 2005).




                                                                   44 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Additive hazards (AH)

     Model is
                          hx (t) = h0 (t) + x(t) β.
     βj is how risk changes when increasing xj by unity.
     Can be estimated in standard software using empirical
     Bayes approach with gamma process prior on H0 (t) (Sinha
     et al., 2009).
     Other approaches require elaborate model specification to
     incorporate awkward constraints (Yin and Ibrahim, 2005;
     Dunson and Herring, 2005).
     Non-Bayesian approach nicely implemented in
     Martinussen and Scheike (2006) timereg package.

                                                                   44 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Proportional mean residual life (PMRL)



     Model is
                         mx (t) = exp(x β)m0 (t).




                                                                   45 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Proportional mean residual life (PMRL)



     Model is
                         mx (t) = exp(x β)m0 (t).
     eβj how expected lifetime from current timepoint t
     increases when xj increased by unity, for any t. Very nice
     interpretation. Often what patients want to know.




                                                                   45 / 72
                 Fundamental concepts   Proportional hazards
                    Bayesian approach   Accelerated failure time
                Semiparametric models   Proportional odds
                            Examples    Other models


Proportional mean residual life (PMRL)



     Model is
                         mx (t) = exp(x β)m0 (t).
     eβj how expected lifetime from current timepoint t
     increases when xj increased by unity, for any t. Very nice
     interpretation. Often what patients want to know.
     Very hard to fit. Frequentist approaches but “real” Bayesian
     approach not developed yet...




                                                                   45 / 72
              Fundamental concepts   Proportional hazards
                 Bayesian approach   Accelerated failure time
             Semiparametric models   Proportional odds
                         Examples    Other models


Super models!!!




                                                                46 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Faster than a speeding bullet...
      Generalized odds-rate model (Scharfstein et al., 1998):
                     qρ {Sx (t)} = −x β + qρ {S0 (t)}
      where qρ (s) = log{ρsρ /(1 − sρ )}. ρ = 1 gives PO and
      ρ → 0+ PH. Special case of transformation model.




                                                                    47 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Faster than a speeding bullet...
      Generalized odds-rate model (Scharfstein et al., 1998):
                     qρ {Sx (t)} = −x β + qρ {S0 (t)}
      where qρ (s) = log{ρsρ /(1 − sρ )}. ρ = 1 gives PO and
      ρ → 0+ PH. Special case of transformation model.
      Chen and Jewell (2001): h(t) = h0 (tex β1 )ex β2 .
      β 2 = 0 gives PH and β 1 = β 2 gives AFT.




                                                                    47 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Faster than a speeding bullet...
      Generalized odds-rate model (Scharfstein et al., 1998):
                     qρ {Sx (t)} = −x β + qρ {S0 (t)}
      where qρ (s) = log{ρsρ /(1 − sρ )}. ρ = 1 gives PO and
      ρ → 0+ PH. Special case of transformation model.
      Chen and Jewell (2001): h(t) = h0 (tex β1 )ex β2 .
      β 2 = 0 gives PH and β 1 = β 2 gives AFT.
      Yin and Ibrahim (2005):
                   hx (t)ρ − 1   h0 (t)ρ − 1
                               =             + β x(t).
                        ρ             ρ
      ρ = 1 gives AH model, ρ → 0 gives PH. Authors treat ρ as
      known when fitting.

                                                                    47 / 72
                  Fundamental concepts   Proportional hazards
                     Bayesian approach   Accelerated failure time
                 Semiparametric models   Proportional odds
                             Examples    Other models


Faster than a speeding bullet...
      Generalized odds-rate model (Scharfstein et al., 1998):
                     qρ {Sx (t)} = −x β + qρ {S0 (t)}
      where qρ (s) = log{ρsρ /(1 − sρ )}. ρ = 1 gives PO and
      ρ → 0+ PH. Special case of transformation model.
      Chen and Jewell (2001): h(t) = h0 (tex β1 )ex β2 .
      β 2 = 0 gives PH and β 1 = β 2 gives AFT.
      Yin and Ibrahim (2005):
                   hx (t)ρ − 1   h0 (t)ρ − 1
                               =             + β x(t).
                        ρ             ρ
      ρ = 1 gives AH model, ρ → 0 gives PH. Authors treat ρ as
      known when fitting.
      β loses interpretability; estimation of ρ problematic.
                                                                    47 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)
     Joint longitudinal/survival models. yi (t) = xi (t) + ei (t).




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)
     Joint longitudinal/survival models. yi (t) = xi (t) + ei (t).
     Recurrent events.




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)
     Joint longitudinal/survival models. yi (t) = xi (t) + ei (t).
     Recurrent events.
     Completely nonparametric approaches.




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)
     Joint longitudinal/survival models. yi (t) = xi (t) + ei (t).
     Recurrent events.
     Completely nonparametric approaches.
     Multistate models.




                                                                      48 / 72
                    Fundamental concepts   Proportional hazards
                       Bayesian approach   Accelerated failure time
                   Semiparametric models   Proportional odds
                               Examples    Other models


Other generalizations


     Frailties. hij (t) = exij β+γi h0 (t).
     Cure rate. P(T = ∞) > 0.
     Time dependent covariates. hx (t) = ex(t) β h0 (t)
     Time varying effects. hx (t) = ex β(t) h0 (t)
     Joint longitudinal/survival models. yi (t) = xi (t) + ei (t).
     Recurrent events.
     Completely nonparametric approaches.
     Multistate models.
     Competing risks.


                                                                      48 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Lung cancer data

     Treatment of limited-stage small cell lung cancer in n = 121
     patients, data presented in Maksymiuk et al. (1993).




                                                                    49 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Lung cancer data

     Treatment of limited-stage small cell lung cancer in n = 121
     patients, data presented in Maksymiuk et al. (1993).
     Used in median-regression models (which have the AFT
     property) by Ying et al. (1995), Walker and Mallick (1999),
     Yang (1999), Kottas and Gelfand (2001), Hanson (2006).




                                                                    49 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Lung cancer data

     Treatment of limited-stage small cell lung cancer in n = 121
     patients, data presented in Maksymiuk et al. (1993).
     Used in median-regression models (which have the AFT
     property) by Ying et al. (1995), Walker and Mallick (1999),
     Yang (1999), Kottas and Gelfand (2001), Hanson (2006).
     Of interest: which sequence of cisplaten and etoposide
     increased the lifetime from time of diagnosis, adjusted for
     patient age.




                                                                    49 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Lung cancer data

     Treatment of limited-stage small cell lung cancer in n = 121
     patients, data presented in Maksymiuk et al. (1993).
     Used in median-regression models (which have the AFT
     property) by Ying et al. (1995), Walker and Mallick (1999),
     Yang (1999), Kottas and Gelfand (2001), Hanson (2006).
     Of interest: which sequence of cisplaten and etoposide
     increased the lifetime from time of diagnosis, adjusted for
     patient age.
     Treatment A: cisplaten followed by etoposide, B is
     vice-versa.




                                                                    49 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Lung cancer data

     Treatment of limited-stage small cell lung cancer in n = 121
     patients, data presented in Maksymiuk et al. (1993).
     Used in median-regression models (which have the AFT
     property) by Ying et al. (1995), Walker and Mallick (1999),
     Yang (1999), Kottas and Gelfand (2001), Hanson (2006).
     Of interest: which sequence of cisplaten and etoposide
     increased the lifetime from time of diagnosis, adjusted for
     patient age.
     Treatment A: cisplaten followed by etoposide, B is
     vice-versa.
     Treatment A administered to 62 patients, treatment B
     administered to 59 patients; 23 patients right-censored.

                                                                    49 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.




                                                                   50 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.
     In three semiparametric models,
     S0 ∼ PT5 (1, ρ, Sθ )dP(θ).




                                                                   50 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.
     In three semiparametric models,
     S0 ∼ PT5 (1, ρ, Sθ )dP(θ).
     For PH and AFT models S0 centered at the Weibull
                      α
     {Sθ (t) = e−(t/λ) : α > 0, λ > 0}.




                                                                   50 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.
     In three semiparametric models,
     S0 ∼ PT5 (1, ρ, Sθ )dP(θ).
     For PH and AFT models S0 centered at the Weibull
                      α
     {Sθ (t) = e−(t/λ) : α > 0, λ > 0}.
     PO model centered at log-logistic
     {Sθ (t) = (1 + λt α )−1 : α > 0, λ > 0}.




                                                                   50 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.
     In three semiparametric models,
     S0 ∼ PT5 (1, ρ, Sθ )dP(θ).
     For PH and AFT models S0 centered at the Weibull
                      α
     {Sθ (t) = e−(t/λ) : α > 0, λ > 0}.
     PO model centered at log-logistic
     {Sθ (t) = (1 + λt α )−1 : α > 0, λ > 0}.
     Parametric Weibull and log-logistic models also fit.




                                                                   50 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Comparing AFT, PO, PH


     Patient covariates are xi = (xi1 , xi2 ) age and treatment.
     In three semiparametric models,
     S0 ∼ PT5 (1, ρ, Sθ )dP(θ).
     For PH and AFT models S0 centered at the Weibull
                      α
     {Sθ (t) = e−(t/λ) : α > 0, λ > 0}.
     PO model centered at log-logistic
     {Sθ (t) = (1 + λt α )−1 : α > 0, λ > 0}.
     Parametric Weibull and log-logistic models also fit.
     p(α, λ) ∝ 1. p(β) flat, but calibrated to place models on
     “equal ground” using Weibull baseline.


                                                                   50 / 72
           Fundamental concepts
              Bayesian approach    Lung cancer I
          Semiparametric models    Lung cancer II
                      Examples




        Weibull     log-logistic       PO            PH    AFT
LPML:
        −747           −735           −734          −737   −734




                                                                  51 / 72
            Fundamental concepts
               Bayesian approach    Lung cancer I
           Semiparametric models    Lung cancer II
                       Examples




        Weibull      log-logistic       PO            PH    AFT
LPML:
        −747            −735           −734          −737   −734
Little predictive difference among the AFT, PO, and
log-logistic models.




                                                                   51 / 72
             Fundamental concepts
                Bayesian approach    Lung cancer I
            Semiparametric models    Lung cancer II
                        Examples




        Weibull       log-logistic       PO            PH    AFT
LPML:
        −747             −735           −734          −737   −734
Little predictive difference among the AFT, PO, and
log-logistic models.
Weibull model clearly inferior.




                                                                    51 / 72
             Fundamental concepts
                Bayesian approach    Lung cancer I
            Semiparametric models    Lung cancer II
                        Examples




        Weibull       log-logistic       PO            PH    AFT
LPML:
        −747             −735           −734          −737   −734
Little predictive difference among the AFT, PO, and
log-logistic models.
Weibull model clearly inferior.
AFT and PO models have a pseudo Bayes factor of about
10 relative to the PH model.




                                                                    51 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Integrated Cox-Snell residual plots


              Weibull                                         AFT
         3                                   3
         2                                   2
         1                                  1

             1       2       3                            1         2   3

                   PH                                         PO
         3                                   3
         2                                   2
         1                                  1

              1          2       3                        1         2   3



                                                                            52 / 72
                    Fundamental concepts
                       Bayesian approach       Lung cancer I
                   Semiparametric models       Lung cancer II
                               Examples




    Par.            MPT AFT                      MPT PO                MPT PH
 β1 (age)     0.007 (−0.004,0.036)         0.034 (−0.001,0.071)   0.028 (0.003,0.054)
β2 (A or B)    0.345 (0.157,0.533)          0.930 (0.292,1.568)   0.533 (0.130,0.926)


    Posterior regression effects.




                                                                                        53 / 72
                    Fundamental concepts
                       Bayesian approach       Lung cancer I
                   Semiparametric models       Lung cancer II
                               Examples




    Par.            MPT AFT                      MPT PO                MPT PH
 β1 (age)     0.007 (−0.004,0.036)         0.034 (−0.001,0.071)   0.028 (0.003,0.054)
β2 (A or B)    0.345 (0.157,0.533)          0.930 (0.292,1.568)   0.533 (0.130,0.926)


    Posterior regression effects.
    Holding age fixed, patients typically survive e0.345 ≈ 1.4
    times longer under treatment A versus B under the AFT
    assumption.




                                                                                        53 / 72
                    Fundamental concepts
                       Bayesian approach       Lung cancer I
                   Semiparametric models       Lung cancer II
                               Examples




    Par.            MPT AFT                      MPT PO                MPT PH
 β1 (age)     0.007 (−0.004,0.036)         0.034 (−0.001,0.071)   0.028 (0.003,0.054)
β2 (A or B)    0.345 (0.157,0.533)          0.930 (0.292,1.568)   0.533 (0.130,0.926)


    Posterior regression effects.
    Holding age fixed, patients typically survive e0.345 ≈ 1.4
    times longer under treatment A versus B under the AFT
    assumption.
    The PO model indicates odds of surviving past any time t
    is e0.93 ≈ 2.5 greater for treatment A versus B.



                                                                                        53 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


MPT AFT comparing treatments




       0.0012


       0.0008


       0.0004


                        500                 1500          2500


                Figure: Treatment A solid, B dashed.


                                                                 54 / 72
                Fundamental concepts
                   Bayesian approach   Lung cancer I
               Semiparametric models   Lung cancer II
                           Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.




                                                             55 / 72
                Fundamental concepts
                   Bayesian approach   Lung cancer I
               Semiparametric models   Lung cancer II
                           Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.
     It is available for download at
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.




                                                             55 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.
     It is available for download at
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Primarily written by Christiane Belitz, Andreas Brezger,
     Thomas Kneib, and Stefan Lang.




                                                                55 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.
     It is available for download at
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Primarily written by Christiane Belitz, Andreas Brezger,
     Thomas Kneib, and Stefan Lang.
     Models can include spatial random effects (frailties), both
     areal and point referenced (but not nonparametric).




                                                                   55 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.
     It is available for download at
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Primarily written by Christiane Belitz, Andreas Brezger,
     Thomas Kneib, and Stefan Lang.
     Models can include spatial random effects (frailties), both
     areal and point referenced (but not nonparametric).
     Additive effects carried out primarily through penalized
     B-splines.




                                                                   55 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Proportional hazards in BayesX

     BayesX is a free, amazing Windows-based program to fit
     Bayesian generalized additive mixed models.
     It is available for download at
     http://www.stat.uni-muenchen.de/∼bayesx/bayesx.html.
     Primarily written by Christiane Belitz, Andreas Brezger,
     Thomas Kneib, and Stefan Lang.
     Models can include spatial random effects (frailties), both
     areal and point referenced (but not nonparametric).
     Additive effects carried out primarily through penalized
     B-splines.
     Fitting of the proportional hazards (& important extensions)
     model is easy; log-baseline hazard is modeled as a
     penalized B-spline.
                                                                    55 / 72
                        Fundamental concepts
                           Bayesian approach   Lung cancer I
                       Semiparametric models   Lung cancer II
                                   Examples


BayesX code using default priors
  delimiter = ;
  %%%%%%%%%%%%%%%%%%%%%%
  % input data
  %%%%%%%%%%%%%%%%%%%%%%
  dataset surv;
  surv.infile age time delta group, maxobs=5000
   using c:\some_folder\cancer.txt;
  %%%%%%%%%%%%%%%%%%%%%%
  % linear predictor
  %%%%%%%%%%%%%%%%%%%%%%
  bayesreg lp;
  lp.outfile = c:\some_folder\lp;
  lp.regress delta = time(baseline) + group + age,
  iterations=12000 burnin=2000 step=10 family=cox using surv;
  %%%%%%%%%%%%%%%%%%%%%%
  % additive age
  %%%%%%%%%%%%%%%%%%%%%%
  bayesreg aa;
  aa.outfile = c:\some_folder\BayesX\aa;
  aa.regress delta = time(baseline) + group + age(psplinerw2),
  iterations=12000 burnin=2000 step=10 family=cox using surv;
  %%%%%%%%%%%%%%%%%%%%%%
  % varying coefficients
  %%%%%%%%%%%%%%%%%%%%%%
  bayesreg vc;
  vc.outfile = c:\some_folder\BayesX\vc;
  vc.regress delta = time(baseline) + group*time(baseline) + age*time(baseline),
  iterations=12000 burnin=2000 step=10 family=cox using surv;
                                                                                   56 / 72
                         Fundamental concepts
                            Bayesian approach      Lung cancer I
                        Semiparametric models      Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg gi + βa ai }



  ESTIMATION RESULTS:

    FixedEffects1

    Acceptance rate:     75.19 %

    Variable   mean           Std. Dev.         2.5% quant.         median        97.5% quant.
      const      -10.2569       1.43149           -13.4               -10.1214      -7.90685
      group      0.535453       0.198478          0.150767            0.546233      0.915852
      age        0.0278298      0.0132894         0.00240202          0.0282356     0.0534999



  Treatment B increases hazard of death by e0.535 ≈ 1.7 times.




                                                                                                 57 / 72
                         Fundamental concepts
                            Bayesian approach      Lung cancer I
                        Semiparametric models      Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg gi + βa ai }

                                       Effect of time

           5.07


           -1.71


           -8.49


           -15.3


           -22.1

                   83            557        1032         1506       1980
                                            time




                                       Figure: f0 (t)

                                                                           58 / 72
                         Fundamental concepts
                            Bayesian approach      Lung cancer I
                        Semiparametric models      Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg gi + fa (ai )}

                                       Effect of time

           4.51


           -1.55


           -7.61


           -13.7


           -19.7

                   83            557        1032         1506       1980
                                            time




                                       Figure: f0 (t)

                                                                           59 / 72
                         Fundamental concepts
                            Bayesian approach        Lung cancer I
                        Semiparametric models        Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg gi + fa (ai )}

                                         Effect of age

           1.63


           0.89


           0.15


           -0.59


           -1.33

                   36            46.8         57.5          68.3      79
                                              age




                                        Figure: fa (age)

                                                                           60 / 72
                         Fundamental concepts
                            Bayesian approach      Lung cancer I
                        Semiparametric models      Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg (t)gi + βa (t)ai }

                                       Effect of time

           8.68


           -0.5


           -9.67


           -18.8


           -28

                   83            557        1032         1506       1980
                                            time




                                       Figure: f0 (t)

                                                                           61 / 72
                         Fundamental concepts
                            Bayesian approach       Lung cancer I
                        Semiparametric models       Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg (t)gi + βa (t)ai }

                                       Effect of group

           22.2


           15.2


           8.23


           1.22


           -5.78

                   83            557         1032         1506       1980
                                            group




                                       Figure: βg (t)

                                                                            62 / 72
                         Fundamental concepts
                            Bayesian approach      Lung cancer I
                        Semiparametric models      Lung cancer II
                                    Examples


BayesX: λi (t) = exp{β0 + f0 (t) + βg (t)gi + βa (t)ai }

                                       Effect of age

           0.31


           -0.06


           -0.43


           -0.8


           -1.17

                   83            557        1032         1506       1980
                                             age




                                       Figure: βa (t)

                                                                           63 / 72
                       Fundamental concepts
                          Bayesian approach        Lung cancer I
                      Semiparametric models        Lung cancer II
                                  Examples


BayesX: option predict gives DIC



         λi (t)                                       barD          pD        DIC
         exp{β0 + f0 (t) + βg gi + βa ai }            1449.2        8.3       1465.8
         exp{β0 + f0 (t) + βg gi + fa (ai )}          1447.2        9.4       1466.0
         exp{β0 + f0 (t) + βg (t)gi + βa (t)ai }      2638.7        -1199.8   239.2

  Something not quite right with varying coefficient model’s DIC...
  In richly parameterized models, posterior mean may not be
  ideal to evaluate deviance at.




                                                                                       64 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Lung cancer II
     Veterans Administration (VA) Lung Cancer data introduced
     by Prentice (1973).




                                                                65 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Lung cancer II
     Veterans Administration (VA) Lung Cancer data introduced
     by Prentice (1973).
     Semiparametric PO model (Cheng et al., 1997; Murphy et
     al, 1997; Yang and Prentice, 1999) & parametric models
     (Farewell and Prentice, 1977; Bennett, 1983).




                                                                65 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Lung cancer II
     Veterans Administration (VA) Lung Cancer data introduced
     by Prentice (1973).
     Semiparametric PO model (Cheng et al., 1997; Murphy et
     al, 1997; Yang and Prentice, 1999) & parametric models
     (Farewell and Prentice, 1977; Bennett, 1983).
     Survival in days of men with advanced inoperable lung
     cancer.




                                                                65 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Lung cancer II
     Veterans Administration (VA) Lung Cancer data introduced
     by Prentice (1973).
     Semiparametric PO model (Cheng et al., 1997; Murphy et
     al, 1997; Yang and Prentice, 1999) & parametric models
     (Farewell and Prentice, 1977; Bennett, 1983).
     Survival in days of men with advanced inoperable lung
     cancer.
     Predictors of survival well established: tumor type (large,
     adeno, small, squamous) & fitness performance score
     ranging from 10 (completely hospitalized) to 90 (able to
     take care of oneself).



                                                                   65 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Lung cancer II
     Veterans Administration (VA) Lung Cancer data introduced
     by Prentice (1973).
     Semiparametric PO model (Cheng et al., 1997; Murphy et
     al, 1997; Yang and Prentice, 1999) & parametric models
     (Farewell and Prentice, 1977; Bennett, 1983).
     Survival in days of men with advanced inoperable lung
     cancer.
     Predictors of survival well established: tumor type (large,
     adeno, small, squamous) & fitness performance score
     ranging from 10 (completely hospitalized) to 90 (able to
     take care of oneself).
     Following others, consider a subgroup of n = 97 patients
     with no prior therapy. Six of the 97 survival times are
     censored.
                                                                   65 / 72
                   Fundamental concepts
                      Bayesian approach   Lung cancer I
                  Semiparametric models   Lung cancer II
                              Examples


Comparing fits

    Parameter                      MPT              MPLE              MDF
    Score                −0.055 (0.010)     −0.055 (0.010)   −0.034 (0.007)
    Adeno vs. large       1.303 (0.559)      1.339 (0.556)    1.411 (0.674)
    Small vs. large       1.362 (0.527)      1.440 (0.525)    1.353 (0.506)
    Squamous vs. large   −0.173 (0.580)     −0.217 (0.589)    0.165 (0.653)

     MPT PO model with J = 5 and c = 1; maximum profile
     likelihood estimator (MPLE) of Murphy et al. (1997); one
     minimum distance estimator (MDF) of Yang and Prentice
     (1999).




                                                                              66 / 72
                   Fundamental concepts
                      Bayesian approach   Lung cancer I
                  Semiparametric models   Lung cancer II
                              Examples


Comparing fits

    Parameter                      MPT              MPLE              MDF
    Score                −0.055 (0.010)     −0.055 (0.010)   −0.034 (0.007)
    Adeno vs. large       1.303 (0.559)      1.339 (0.556)    1.411 (0.674)
    Small vs. large       1.362 (0.527)      1.440 (0.525)    1.353 (0.506)
    Squamous vs. large   −0.173 (0.580)     −0.217 (0.589)    0.165 (0.653)

     MPT PO model with J = 5 and c = 1; maximum profile
     likelihood estimator (MPLE) of Murphy et al. (1997); one
     minimum distance estimator (MDF) of Yang and Prentice
     (1999).
     Posterior medians and standard deviations obtained under
     the MPT model are very close to the MPLE estimates.



                                                                              66 / 72
                   Fundamental concepts
                      Bayesian approach   Lung cancer I
                  Semiparametric models   Lung cancer II
                              Examples


Comparing fits

    Parameter                      MPT              MPLE              MDF
    Score                −0.055 (0.010)     −0.055 (0.010)   −0.034 (0.007)
    Adeno vs. large       1.303 (0.559)      1.339 (0.556)    1.411 (0.674)
    Small vs. large       1.362 (0.527)      1.440 (0.525)    1.353 (0.506)
    Squamous vs. large   −0.173 (0.580)     −0.217 (0.589)    0.165 (0.653)

     MPT PO model with J = 5 and c = 1; maximum profile
     likelihood estimator (MPLE) of Murphy et al. (1997); one
     minimum distance estimator (MDF) of Yang and Prentice
     (1999).
     Posterior medians and standard deviations obtained under
     the MPT model are very close to the MPLE estimates.
     Increasing performance score by 20 increases the odds of
     surviving past any fixed time point by about 200%,
     e(−20)(−0.055) ≈ 3.
                                                                              66 / 72
                    Fundamental concepts
                       Bayesian approach        Lung cancer I
                   Semiparametric models        Lung cancer II
                               Examples


Comparing PO, AFT, PH, and parametric


             log-logistic    MPT PO        MPT gen. odd rate      MPT AFT       MPT PH
                            centered at      centered at         centered at   centered at
     LPML:                  log-logistic      log-logistic         Weibull       Weibull
               −508           −508              −511               −514          −516




                                                                                             67 / 72
                    Fundamental concepts
                       Bayesian approach        Lung cancer I
                   Semiparametric models        Lung cancer II
                               Examples


Comparing PO, AFT, PH, and parametric


             log-logistic    MPT PO        MPT gen. odd rate      MPT AFT       MPT PH
                            centered at      centered at         centered at   centered at
     LPML:                  log-logistic      log-logistic         Weibull       Weibull
               −508           −508              −511               −514          −516

     Little predictive difference among PO and log-logistic
     models. (log-logistic has PO property).




                                                                                             67 / 72
                    Fundamental concepts
                       Bayesian approach        Lung cancer I
                   Semiparametric models        Lung cancer II
                               Examples


Comparing PO, AFT, PH, and parametric


             log-logistic    MPT PO        MPT gen. odd rate      MPT AFT       MPT PH
                            centered at      centered at         centered at   centered at
     LPML:                  log-logistic      log-logistic         Weibull       Weibull
               −508           −508              −511               −514          −516

     Little predictive difference among PO and log-logistic
     models. (log-logistic has PO property).
     Weibull model clearly inferior. Pseudo Bayes factor of 3000
     in favor of PO over PH model.




                                                                                             67 / 72
                    Fundamental concepts
                       Bayesian approach        Lung cancer I
                   Semiparametric models        Lung cancer II
                               Examples


Comparing PO, AFT, PH, and parametric


             log-logistic    MPT PO        MPT gen. odd rate      MPT AFT       MPT PH
                            centered at      centered at         centered at   centered at
     LPML:                  log-logistic      log-logistic         Weibull       Weibull
               −508           −508              −511               −514          −516

     Little predictive difference among PO and log-logistic
     models. (log-logistic has PO property).
     Weibull model clearly inferior. Pseudo Bayes factor of 3000
     in favor of PO over PH model.
     Proportional odds implies attenuation of risk as time goes
     on.




                                                                                             67 / 72
                 Fundamental concepts
                    Bayesian approach   Lung cancer I
                Semiparametric models   Lung cancer II
                            Examples


Integrated Cox-Snell residual plots


                    PO                                   AFT
         6                                 6
         4                                 4
         2                                 2

                2        4     6                     2         4   6

             Log logistic                                PH
         6                                 6
         4                                 4
         2                                 2

                2        4     6                     2         4   6



                                                                       68 / 72
                    Fundamental concepts
                       Bayesian approach   Lung cancer I
                   Semiparametric models   Lung cancer II
                               Examples


MPT PO comparing treatments



            0.01     PS=40




           0.005
                                PS=60

                                             PS=80

                         100                   300          500

  Figure: Predictive densities, squamous, MPT with c = 1; survival is in
  days.


                                                                           69 / 72
                    Fundamental concepts
                       Bayesian approach   Lung cancer I
                   Semiparametric models   Lung cancer II
                               Examples


MPT PO comparing treatments

           0.015

                     adeno, small
            0.01



           0.005               large, squamous



                         100                   300          500

  Figure: Predictive densities, performance status = 60, MPT with
  c = 1.


                                                                    70 / 72
                    Fundamental concepts
                       Bayesian approach   Lung cancer I
                   Semiparametric models   Lung cancer II
                               Examples


MPT PO median survival

            days

           400


           300


           200


           100


                                                                 score
                    40       50       60     70        80   90

  Figure: Median survival with 95% CI versus score for squamous,
  c = 1.


                                                                         71 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Discussion...


      Bayesian approach allowed comparison of parametric and
      semiparametric survival regression models using standard
      model selection criterion.




                                                                 72 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Discussion...


      Bayesian approach allowed comparison of parametric and
      semiparametric survival regression models using standard
      model selection criterion.
      All fitting via MCMC routines.




                                                                 72 / 72
                  Fundamental concepts
                     Bayesian approach   Lung cancer I
                 Semiparametric models   Lung cancer II
                             Examples


Discussion...


      Bayesian approach allowed comparison of parametric and
      semiparametric survival regression models using standard
      model selection criterion.
      All fitting via MCMC routines.
      Non-asymptotic inference. Everything exact up to MCMC
      error.




                                                                 72 / 72
                   Fundamental concepts
                      Bayesian approach   Lung cancer I
                  Semiparametric models   Lung cancer II
                              Examples


Discussion...


      Bayesian approach allowed comparison of parametric and
      semiparametric survival regression models using standard
      model selection criterion.
      All fitting via MCMC routines.
      Non-asymptotic inference. Everything exact up to MCMC
      error.
      Able to get inferences for hazard ratios, quantiles, etc.




                                                                  72 / 72
                   Fundamental concepts
                      Bayesian approach   Lung cancer I
                  Semiparametric models   Lung cancer II
                              Examples


Discussion...


      Bayesian approach allowed comparison of parametric and
      semiparametric survival regression models using standard
      model selection criterion.
      All fitting via MCMC routines.
      Non-asymptotic inference. Everything exact up to MCMC
      error.
      Able to get inferences for hazard ratios, quantiles, etc.
      Differences minor here, but have seen very marked
      differences across surivival models (e.g. PH vs. PO).



                                                                  72 / 72