Recent Advances in Bayesian Spatial Survival Modeling

Transcription

1 1 / 47 Recent Advances in Bayesian Spatial Survival Modeling Tim Hanson Department of Statistics University of South Carolina, U.S.A. University of Alabama Department of Mathematics April 10, 2015

2 2 / 47 Outline 1 2 3

3 3 / 47 Survival data Time to event data Functions defining lifetime distribution Can be time to any event of interest, e.g. death, leukemia remission, bankruptcy, electrical component failure, etc. Data T 1, T 2,..., T n live in R +. Called: survival data, reliability data, time to event data. Interest often focuses on relating aspects of the distribution on T i to covariates or risk factors x i, possibly time-dependent x i (t). Can be external or internal.

4 Time to event data Functions defining lifetime distribution Survival data: covariates and censoring Uncensored data: (x 1, t 1 ),..., (x n, t n ). Observe T i = t i. Right censored data: (x 1, t 1, δ 1 ),..., (x n, t n, δ n ). Observe { Ti = t i δ i = 1 T i > t i δ i = 0 }. Interval censored data: (x 1, a 1, b 1 ),..., (x n, a n, b n ). Observe T i [a i, b i ]. 4 / 47

5 5 / 47 Density and survival Time to event data Functions defining lifetime distribution Continuous T has density f (t). Survival function is S(t) = 1 F(t) = P(T > t) = Regression model: proportional odds. Probability of making it past 40 years is S(40). Odds of dying before 40 years are 1 S(40) S(40). t f (s)ds.

6 6 / 47 Quantiles Time to event data Functions defining lifetime distribution p th quantile q p for T solves P(T q p ) = p. Continuous T q p = F 1 (p). Median lifetime in the population is q 0.5 = F 1 (0.5). Regression model: accelerated failure time (proportional quantiles). Quantile regression active area of research from frequentist & Bayesian perspectives, e.g. Koenker s excellent quantreg package for R.

7 7 / 47 Residual life Time to event data Functions defining lifetime distribution Mean residual life m(t) = E{T t T > t} = t S(s)ds. S(t) Can expect to live m(40) more years given made it to 40. Regression model: proportional mean residual life. Also median residual life.

8 8 / 47 Hazard function Time to event data Functions defining lifetime distribution Hazard at t: h(t) = P(t T < t + dt T t) lim = f (t) dt 0 + dt S(t). Probability of dying tomorrow is h(40) ( 1 365) given made it to 40 years. Regression models: proportional hazards (Cox), additive hazards (Aalen), accelerated hazards, & extended hazards.

9 9 / 47 Density, survival, hazard, and MRL Time to event data Functions defining lifetime distribution

10 10 / 47 Nonparametric survival priors Time to event data Functions defining lifetime distribution Infinite-dimensional process directly defined on one of h(t), H(t), f (t), or S(t). Note that prior on one function implies prior on other three. Priors on h(t) include extended gamma, piecewise exponential, B-splines, etc. Priors on H(t) = log S(t) include gamma, beta, etc. Priors on S(t) include Dirichlet process (DP). Priors on f (t) include DP mixtures, transformed Bernstein polynomials, Polya trees, B-splines, etc. Will consider B-spline and DP mixture.

11 11 / 47 Proportional hazards Accelerated failure time Proportional odds Let s work covariates x i into the model for T i. The most common way to do this is through a semiparametric model. Why semiparametric? Splits inference into two pieces: β and S 0 (t) (or h 0 (t) or m 0 (t) or H 0 (t)). Zero subscript stands for baseline where x = 0. β = (β 1,..., β p ) succinctly summarizes effects of risk factors x on aspects of survival. Make S 0 (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.

12 12 / 47 Some semiparametric models Proportional hazards Accelerated failure time Proportional odds PH: h x (t) = exp(x β)h 0 (t). AddH: h x (t) = h 0 (t) + β x. AFT: S x (t) = S 0 {e β x t}. PO: F x (t)/s x (t) = e β x F 0 (t)/s 0 (t). PMRL: m x (t) = e β x m 0 (t). AccH: h x (t) = h 0 {e β x t}. ExtH: h x (t) = h 0 {e β x t}e γ x. Others, but this covers 99%.

13 13 / 47 Proportional hazards (PH) Proportional hazards Accelerated failure time Proportional odds Model is: h x (t) = exp(x β)h 0 (t) or S x (t) = S 0 (t) exp(x β). Stochastically orders S x1 and S x2. e β j is how risk changes when x j is increased by unity. BayesX assigns penalized B-spline prior on log h 0 (t) and allows for additive predictors, structured frailties, time-varying coefficients, etc. Free: bayesx/bayesx.html. Also R package to call BayesX. Add BAYES command in SAS PROC PHREG gives p.w. exponential. Haiming Zhou s spbayessurv has S 0 modeled as MPT.

14 14 / 47 Accelerated failure time (AFT) Proportional hazards Accelerated failure time Proportional odds Model is S x (t) = S 0 (e x β t Implies q p (x) = e x β q p (0). Stochastically orders S x1 and S x2. ), or log T x = x β + e 0. e β j how any quantile or mean changes when increasing x j by unity. Komarek s bayessurv for AFT models; spline and discrete normal mixture on error. bj() in Harrell s Design library fits Buckley-James version. Haiming Zhou s spbayessurv has S 0 modeled as MPT.

15 15 / 47 Proportional odds (PO) Proportional hazards Accelerated failure time Proportional odds Model is 1 S x (t) S x (t) = exp(x β) 1 S 0(t). S 0 (t) e β j how odds of event occuring before t changes when x j increased by unity (for any t). Attenuation of risk: Plausible in many situations. h x1 (t) lim t h x2 (t) = 1. Haiming Zhou s spbayessurv has S 0 modeled as MPT. timereg has frequentist version.

16 16 / 47 Spatial frailty survival models Proportional hazards Accelerated failure time Proportional odds Survival data often collected over region. Georeferenced includes s i = (x i, y i ), e.g. latitude & longitude. Areal includes c i {1,..., C}, e.g. the county of residence (there are C counties). Traditionally, spatial dependence induced by adding frailty (random effect) to linear predictor in semiparametric model.

17 Spatial frailty survival models Proportional hazards Accelerated failure time Proportional odds Georeferenced: Replace x i β by x i β + g i. Take g i = g(x i, y i ) where {g(s) : s S} is mean-zero stationary Gaussian process. Yields g = (g 1,..., g n ) N n (0, C θ ); C θ e.g. Matérn. Areal: Replace x i β by x i β + g c i. Define W to be adjacency matrix: w ij = 1 if counties i and j share a border, otherwise w ij = 0 (assume w ii = 0). λ CAR model assumes g j g j N(ρ g j, w j+ ) where ρ (0, 1) and g j = 1 C w j+ i=1 w ijg i. Limiting case ρ 1 called ICAR, requires C j=1 g j = 0. Both approaches provide mean-zero, smoothed spatial surface g(s) or g j over S. 17 / 47

18 18 / 47 Spatial copula in a nutshell Let T i F xi ( ) where F x c.d.f. from any survival model: parametric, semiparametric, nonparametric. U i = F xi (T i ) U(0, 1) and Y i = Φ 1 (U i ) N(0, 1). Let Y = (Y 1,..., Y n ). No spatial correlation Y N n (0, I n ). Spatial correlation Y N n (0, Γ). Here Γ n n = [γ ij ] with pairwise correlations γ ij. Li and Lin (2006) use this in PH model, term it normal transformation model. Gives marginal (population-averaged) model. Unlike frailties, can be used in models without a linear predictor.

19 19 / 47 SCCCR data set on prostate cancer survival Large dataset on prostate cancer survival that does not follow proportional hazards. n = patients from South Carolina Central Cancer Registry (SCCCR) for the period ; each recorded with county, race, marital status, grade of tumor, and SEER summary stage; 72.3% are censored. Need to allow for non-proportional hazards and accommodate correlation of survival times within county.

20 20 / 47 Extended hazards model Etezadi-Amoli and Ciampi (1987) propose ExtH model Say x = (x 1, x 2 ), then ExtH is h x (t) = h 0 (te x β )e x γ. h x (t) = h 0 (te β 1x 1 +β 2 x 2 )e γ 1x 1 +γ 2 x 2. γ 1 = β 1 x 1 has AFT interpretation; β 1 = 0 x 1 has PH interpretation; γ 1 = 0 x 1 has AccH interpretation.

21 Baseline hazard h 0 (t) Want to shrink h 0 (t) toward parametric target h θ (t): h 0 (t) = J b j B kj (t) j=1 where B k1 ( ),..., B k,j ( ) are kth order B-spline basis function over knots (s 1,..., s J+k ). Let s j = j+k j=k+1 s k/(k 1) and b j = h θ ( s j ). Schoenberg s approximation theorem (Marsden 1972) says max 0 t sj+1 h 0 (t) h θ (t) ɛ(h θ, k, J). Posterior updating: efficient MCMC with clever data augmentation. 21 / 47

22 22 / 47 Works great on simulated data Survival function Density function S 0(t) True Mean estimates 2.5%, 97.5% quantiles f 0(t) True Mean estimates 2.5%, 97.5% quantiles t t Hazard function Hazard function λ 0(t) True Mean estimates 2.5%, 97.5% quantiles λ 0(t) True Mean estimates Mean λ θ t t

23 Spatial dependence via frailties impractical PH with frailties: where g ci EH with frailties: h(t i x) = h 0 (t i )e γ x i +g ci, are county-level frailties, c i is county subject i in. h(t i x) = h 0 {t i e β x i +b ci }e γ x i +g ci, where, for our data, b 1,..., b 46 and g 1,..., g 46 are county-level frailties. Possible but impractical, and hard to interpret. 23 / 47

24 Spatial dependence via copula works great Define Y i = Φ 1 {F xi (T i )}. Under Li and Lin (2006) Y N(0, Γ). Likelihood from data {(t i, x i, δ i )} n i=1 is L(β, γ, b, θ, Γ) = How to define Γ? [ i S ] [ ] f i (t i ) f i (z i ) φ(y i ) φ(y i ) I(z i > t i ) φ(y; 0, Γ) dz i i S c i S c We consider county-level lattice data; popular correlation model is intrinsic conditional autoregressive (ICAR) prior. 24 / 47

25 25 / 47 ICAR definition α = (α 1,, α C ) is vector of correlated effects on over counties in S. ICAR prior on α is p(α) exp{ ϕα (D W)α/2} Recall ICAR prior: ( C ) α j α j, ϕ N j=1 w ijα j /w j+, 1/(ϕw j+ ). Random effects approach Ỹ = (Ỹ1,..., ỸC) = (Ỹ11,..., Ỹ1n 1,..., ỸC1,..., ỸCn C ). Ỹ ij = α i + ɛ ij, α N C (0, Ω), ɛ N n (0, Iσ 2 ). Resulting correlation matrix Γ = corr(ỹ) involves one unknown parameter ϕ.

26 26 / 47 Efficient evaluation of y Γ 1 y Γ is a n n matrix; needs to be inverted during MCMC. Elements of Γ 1 can be easily computed using SVD, Γ 1 = A 1 U 1 ( (K + σ 2 I C ) 1 σ 2 I C ) U 1 A 1 + σ 2 A 2 where A is a diagonal matrix, U 1 = (u 1,..., u C ), u i is a vector of length n with ones corresponding to county i and zero elsewhere. y Γ 1 y = z ( (K + σ 2 I C ) 1 σ 2 I C ) z + σ 2 y A 2 y where z = U 1 A 1 y.

27 27 / 47 Savage-Dickey ratio for global and per-variable tests Example of global test of PH vs. EH BF 12 = π(β = 0 D, EH) π(β = 0 EH). Example of per-variable of PH for x j vs. EH BF 12 = π(β j = 0 D, EH) π(β j = 0 EH).

28 28 / 47 SCCCR data SCCCR prostate cancer data for the period Baseline covariates are county of residence, age, race, marital status, grade of tumor differentiation, and SEER summary stage. n = patients in the dataset after excluding subjects with missing information. 72.3% of the survival times are right-censored. Goal: assess racial disparity in prostate cancer survival, adjusting for the remaining risk factors and accounting for the county the subject lives in.

29 29 / 47 SCCCR data Table: Summary characteristics of prostate cancer patients in SC from Covariate n Sample percentage Race Black White Marital status Non-married Married Grade well or moderately differentiated poorly differentiated or undifferentiated SEER summary stage Localized or regional Distant

30 30 / 47 Non-spatial EH and reduced models Table: Summary of fitting the extended hazard model EH, the reduced model, AFT, and PH; indicates LPML and DIC Covar EH Reduced AFT PH PH+additive age β = γ β = 0 β = 0 Age β (0.48,0.52) 0.48(0.46,0.50) 0.48(0.45,0.51) γ (0.42,0.49) γ 1 = β (0.62,0.68) Race β (0.15,0.21) 0.20(0.16,0.21) 0.18(0.15,0.22) γ (0.12,0.24) γ 2 = β (0.21,0.32) 0.26(0.20,0.31) Marital β (-0.11,-0.02) -0.05(-0.09,-0.00) 0.26(0.21,0.30) status γ (0.29,0.40) 0.33(0.28,0.40) 0.33(0.27,0.39) 0.31(0.26,0.37) Grade β (-0.02,0.08) β 4 = (0.22,0.32) γ (0.29,0.41) 0.37(0.31,0.43) 0.38(0.32,0.44) 0.37(0.33,0.43) SEER β (2.80,3.53) 3.27(2.79,3.57) 1.50(1.41,1.59) stage γ (0.83,1.20) 1.00(0.82,1.19) 1.56(1.47,1.64) 1.57(1.19,1.65) LPML DIC

31 31 / 47 Non-spatial EH and reduced models Table: Bayes factors for comparing EH to PH, AFT, and AH with and without spatial correlation. EH Spatial+EH Covariate PH AFT AH PH AFT AH Age > > 1000 > > 1000 Race > > 1000 > 1000 < 0.01 > 1000 Marital status 1.79 > 1000 > > 1000 > 1000 Grade 0.14 > 1000 > > 1000 > 1000 SEER stage > 1000 > 1000 > 1000 > 1000 > 1000 > 1000

32 32 / 47 Spatial EH and reduced models Table: Summary of spatial models; indicates LPML and DIC Covariates Marginal EH Marginal reduced PH+ICAR+additive age β = 0 Age β (0.47,0.52) 0.47(0.46,0.49) γ (0.43,0.49) γ 1 = β 1 Race β (0.15,0.21) 0.20(0.17,0.22) γ (0.11,0.23) γ 2 = β (0.18,0.30) Marital status β (-0.10,-0.02) -0.02(-0.05,-0.00) γ (0.28,0.41) 0.33(0.27,0.39) 0.32(0.25,0.38) Grade β (-0.01,0.07) β 4 = 0 γ (0.30,0.42) 0.38(0.32,0.43) 0.37(0.32,0.44) SEER stage β (2.86,3.34) 2.77(2.72,2.82) γ (0.94,1.26) 1.21(1.01,1.33) 1.55(1.46,1.64) ϕ 50.1(19.9,113.7) 54.6(22.7,120.8) 33.08(9.2,100.1) LPML DIC

33 < to to to to 0.33 >0.33 < to to to to 0.06 >0.06 Spatial EH and reduced models Mortality rate PH ICAR frailty Marginal reduced model random effects < to to to to 0.04 >0.04 (a) (b) (c) Figure: Map of (a) Mortality rate, (b) ICAR frailties in the PH model and (c) random effects in the marginal reduced model for SC counties. 33 / 47

34 34 / 47 Spatial EH and reduced models Hazard estimates Survival estimates Hazard PH EH Reduced AFT Survival probability PH EH Reduced AFT Time in Years Time in Years Figure: Baseline hazard (left) and survival probabilities (right) estimates.

35 35 / 47 Spatial EH and reduced models Hazard estimates Survival estimates Hazard Posterior mean for blacks Posterior mean for whites Survival probability Posterior mean for blacks Posterior mean for whites 95% CI Time in Years Time in Years Figure: Hazard and survival for black patients (solid line) and white patients; baseline covariates.

36 Interpretation for race effect Based on reduced models, white South Carolina subjects diagnosed with prostrate cancer in live 22% longer (e ) than black patients (95% CI is 18% to 25%), fixing age, stage, and SEER stage. Cox said...the physical or substantive basis for...proportional hazards models...is one of its weaknesses... and goes on to suggest that...accelerated failure time models are in many ways more appealing because of their quite direct physical interpretation. The SCCCR analysis showed that the main covariate of interest, race, is best modeled as an AFT effect. Survival probabilities for black patients are significantly lower than those for white patients when other factors are fixed at the same levels. 36 / 47

37 37 / 47 More interpretation Decreasing age by one year increases survival time by 5.4%. Hazard of dying increases 46% for poorly or undifferentiated grades vs. well or moderately differentiated, holding age, race, and SEER stage constant. SEER stage has general ExtH effects, e (AH) and e (PH). Those with distant stage are at least three times worse in one-sixteenth of the time as those with localized or regional. Marital status essentially has PH interpretation; single (including widowed or separated) subjects are e times more likely to die at any instant than married.

38 38 / 47 Extinction of mountain yellow-legged frog Frogs and other amphibians have been dying off in large numbers since the 1980s because of a deadly fungus called Batrachochytrium dendrobatidis, or Bd. Dr. Knapp has been studying the amphibian declines for the past decade at Sierra Nevada Aquatic Research Laboratory; he has hiked thousands of miles and surveyed hundreds of frog populations in Sequoia-Kings Canyon National Park collecting the data by hand. As with the SCCCR data, proportional hazards grossly violated. Instead of semiparametric, pursue nonparametric F xi ; not able to use frailties.

39 39 / 47 The Frog Data ( ) Contains 309 frog populations. Each was followed up until infection or being censored (10% censoring). Response T i is time to Bd infection. (i.e. Bd arrival year baseline year). Main covariates: x i1 {0, 1} is whether or not Bd has been found in the watershed. x i2 is straight-line distance to the nearest Bd location. Populations near each other tend to become infected at about the same time.

40 40 / 47 LDDPM model and Spatial Extension LDDPM (De Iorio et al., 2009; Jara et al., 2010): Z i = log T i given x i follows mixture model ( z x ) F xi (z) = Φ i β dg(β, σ 2 ), σ where G follows Dirichlet Process (DP) prior: G DP(α, G 0 ). Countable mixture of parametric linear models F xi = j=1 w jn(x i β j, σ 2 j ). As before, take Y i = Φ 1 {F xi (log T i )} and Y N n (0, Γ). Γ θ used for capturing spatial dependence; γ ij = θ 1 exp{ θ 2 s i s j } + (1 θ 1 )I{s i = s j }.

41 41 / 47 MCMC Overview Truncated stick-breaking representation G = N i=1 [v i j<i (1 v j)]δ βj,σj 2 where v 1,..., v N 1 iid beta(1, α), vn = 1, and (β j, σ 2 j ) iid G 0. G parameters updated based on a M-H proposal from blocked Gibbs sampler (Ishwaran and James, 2001). The latent censored t i updated via M-H sampler. Delayed rejection (Tierney and Mira, 1999) used for several parameters; helps sampler not get stuck. Correlation parameters θ are updated using adaptive M-H (Haario et al., 2001). For large n, the inversion of the n n matrix C substantially sped up using a full scale approximation (FSA) (Sang and Huang, 2012).

42 42 / 47 Frog Data: Inference on Spatial Correlation Posterior mean ˆθ 1 = Posterior mean ˆθ 2 = , indicating the correlation decays by 1 exp{ (1)} = 8% for every 1-km increase in distance and 1 exp{ (10)} = 58% for every 10-km increase in distance. Table: Posterior summary statistics for the spatial correlation parameters Par. Mean Median Std. dev. 95% HPD Interval θ (0.9879, ) θ (0.0493, )

43 43 / density survival 0.50 hazard log year year year (a) (b) (c) Figure: Fitted marginal densities, survival curves, and hazard curves w/ 90% CI for high versus low value of bddist when bdwater is equal to 0; bddist=95% and bddist=5% quantiles are solid and dashed lines.

44 44 / density 0.50 survival 0.50 hazard log year year year (a) (b) (c) Figure: Fitted marginal densities, survival curves, and hazard curves w/ 90% CI for bdwater=0 versus bdwater=1 when bddist is equal to population mean of 2.7 km; results for bdwater=0 and bdwater=1 are solid and dashed lines.

45 Frog Data: Spatial Prediction Spatial map for the transformed process z(s) = Φ 1 { F x(s) (log T (s) G) } Figure: Predictive spatial map across D. 45 / 47

46 46 / 47 Which is better, copula or frailty? LPML Model LDDPM-copula PH-copula LDDPM-independent PH-independent PH-frailty LDDPM copula model better than PH copula model. However, PH copula better than LDDPM without copula. Modeling via copula grossly improves predictive performance of the models. Frailty improves PH model only slightly.

47 47 / 47 Remarks Proposed Bayesian spatial copula approaches to estimate survival curves semiparametrically (ExtH model) and nonparametrically (LDDPM) while allowing for spatial dependence, leading to high predictive accuracy. Implementation of simpler semiparametric models such as proportional odds focus of current research, both frailty and copula. Thanks to my co-authors Li Li, Haiming Zhou, Roland Knapp, and Jiajia Zhang. Thanks for the invitation! Papers based on this work are available; if interested.