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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

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survival analysis, bayesian analysis, data analysis, bayesian methods, bayesian statistics, statistical methods, survival data, clinical trials, regression models, bayesian inference, new york, bayesian perspective, statistical inference, introduction to bayesian statistics, bayesian survival analysis

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posted: | 1/14/2010 |

language: | English |

pages: | 209 |

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