Estimating survival from Gray s flexible model. Outline. I. Introduction. I. Introduction. I. Introduction

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1 Estimating survival from s flexible model Zdenek Valenta Department of Medical Informatics Institute of Computer Science Academy of Sciences of the Czech Republic I. Introduction Outline II. Semi parametric survival models III. s model introduction IV. Survival estimates based on semi-parametric models V. Estimating survival from s model (with examples) VI. Impact of misspecifying the survival model simulation study results VII. Discussion I. Introduction Let Y be a random variable capturing time to occurrence of a certain event of interest. The hazard function h(y) is at time y formally defined as follows: h(y) = lim y P (y Y < y + y Y y), () y where P (.) denotes conditional probability, that an event of interest would occur immediately after time y, given it did not prior to this time. It follows from () that the hazard function may only take non negative values. I. Introduction Let F (Y ) denote a cumulative distribution function of the random variable Y, i.e. F (Y ) = P (Y y). We assume that Y is absolutely continuous with density f(y). The expression () may be then written as: P (y Y < y + y Y y) h(y) = lim y y P (y Y < y + y) = lim y P (Y y) y d = P (Y y) dy F (y) = = f(y) S(y), f(y) P (Y y) where S(y) denotes the value of the survival function at time y. ()

2 I. Introduction We define a cumulative hazard function H(t) at time t as: H(t) = t h(y)dy (3) It follows from () that S(.) and H(.) capture equivalent information: H(t) = ln (S(t)) (4) Furthermore, it follows from (4) that we can determine the value of the survival function S(t) at time t whenever we are able to evaluate the cumulative hazard function H(t): S(t) = exp { H(t)} (5) II. Semi parametric survival models Multiplicative models: PH model: h(y Z) = h (y) exp (β Z) (6) s flexible model: h(y Z) = h (y) exp {β(y) Z} (7) Additive models: s linear model: h(y Z) = h (y) + β(y) Z (8) II. Semi parametric survival models Note: s linear model (8) may be embedded in the class of multiplicative models: s model: h(y Z) = h (y) + β(y) Z exp (h(y Z)) = exp { h (y) + β(y) Z } h (y Z) = h (y) exp { β(y) Z } The class of multiplicative models represented by the PH and s flexible model includes the whole class of models proposed by. (9) III. s model introduction Let us recall the definition of s flexible model: (7): h(y Z) = h (y) exp {β(y) Z} s model uses penalized B-splines for modelling timevarying effects β(y). B-splines allow for flexible modelling of the covariate effects β(y) and the hazard function over time. In the context of s model using piecewise-constant time-varying regression coefficients the β(y) remain constant for y [τ j, τ j ). We can thus write β(y) = β j = (β j, β j,..., β jp ), where p denotes the number of model covariates and j =,..., M + indexes the intervals on time axis. Here τ j denote the knots that allow for a change of the regression coefficients β j.

3 Piecewise-constant vs. quadratic penalized splines Piecewise-constant vs. cubic penalized splines Intervention Age Diabetes Mellitus Intervention Age Diabetes Mellitus Intervention Age 4 3 Diabetes Mellitus 5 5 Intervention Age Diabetes Mellitus IV. Survival estimates based on semi-parametric models PH model: S(t Z) = exp { t } h (y) exp (β Z) dy = exp { H (t) exp (β Z)} = [S (t)] exp(β Z), () where S (t) represents baseline survival function estimate at time t. IV. Survival estimates based on semi-parametric models s linear model: h(y Z) = h (y) + β(y) Z h(y Z) = β(y) Z, while β(y) = (h (y), β(y)) a Z = (, Z). () Survival function estimates based on s model use cumulative regression coefficients B(t), where B i (t) = t β i (y)dy. Estimating survival based on s model may then proceed as follows: { S (t Z) = exp B(t) Z } ()

4 V. Estimating survival from s model Survival function estimate based on s model a using piecewise constant penalized splines may be obtained as follows: { M+ S (t Z) = exp H j (t) exp ( β jz )}, (3) j= where Z denotes p-dimensional vector of patient s characteristics, and H j (t) = I (u t) dh (u) (4) [τ j,τ j ) represents a contribution to the cumulative baseline hazard function H (t) on the interval [τ j, τ j ), j =,..., M +. V. Estimating survival from s model Derivation of confidence limits for the survival function estimate based on s model uses the Delta method. Recall the Taylor formulae for a function f(x) of a random variable X with expectation µ: Delta method: f(x) = n k= f (k) (µ) (X µ) k + R n (5) k! Var(f(X)) Var [f(µ) + f (µ)(x µ)] = [f (µ)] Var(X) (6) a Valenta Z et al, Statistics in Medicine. V. Estimating survival from s model coxspline package in R If X is a random vector the Delta method G(X) takes the form: Var(G(X)) G (µ) Var(X) G(µ), (7) where G(µ) is a column vector of first partial derivatives of G. Confidence limits estimates a were derived for a log and log(- log) transformed survival function S(t) and are reported simultaneously in R. a Valenta Z et al, Statistics in Medicine.

5 cox.spline R-routine R-function gsurv.r cox.spline R-routine (cont.) R-function gsurv.r (cont.)

6 Example : survival estimates from s model with 95% C.L. (log-transf.) Example : survival estimates from s model with 95% C.L. (log-log-transf.) Estimated survival with 95% CL varying hazards data Increasing hazards (.,,) Decreasing hazards (,,.) Constant hazard (,,) Estimated survival with 95% CL varying hazards data Increasing hazards (.,,) Decreasing hazards (,,.) Constant hazard (,,) Follow up time (in days) Hazard change points are.6 and.8 years Follow up time (in days) Hazard change points are.6 and.8 years Example : Secondary prevention trial of CHD in Litomerice men after MI ( PH model results) Example : Secondary prevention trial of CHD in Litomerice men after MI ( s model results) Men 45 years of age Men 5 years of age Men 45 years of age Men 5 years of age 's survival probability Litomerice Study Litomerice Study 's survival probability Litomerice study Litomerice study Men 56 years of age Men 6 years of age Men 56 years of age Men 6 years of age Litomerice Study Litomerice Study Litomerice study Litomerice study Follow up time in days Follow up time in days

7 V. Estimating survival from s model Implementation of the coxspline package for R statistical system is available from the Dr. s website (Harvard University and Dana-Farber Cancer Institute, Boston, USA). Web address: Package coxspline, version.-, implements s model in R, including the survival function estimation using R-function gsurv.r. Current version of the coxspline package is compatible with the latest release of R.3.. (6-6-): VI. Impact of misspecifying the survival model a In three simulation studies we generated right-censored survival data that would satisfy exactly one of the semi-parametric survival models under consideration (i.e.,, ). The data obtained were subsequently analyzed using each of the three models considered. The performance of each model was assessed using the and Mean Square Error (MSE) of the estimated (conditional) survival distribution. a Valenta Z et al, Model misspecification effect in univariable regression models for rightcensored survival data, Proceedings of the Joint Statistical Meeting of the American Statistical Society. VI. Impact of misspecifying the survival model of the survival estimator Ŝ(t Z): ) (Ŝ(t Z) = n s Ŝ (i) (t Z) S(t Z) (8) n s Mean Square Error of the estimated survival Ŝ(t Z): i= i= ) MSE (Ŝ(t Z) = n s (Ŝ(i) ) (t Z) S(t Z) (9) n s -variance trade-off: ) MSE (Ŝ(t Z) = var(ŝ) + (Ŝ) () s model with constant β Legend Percentile of z % % 5% 5% 75% 9% 99% MSE Relative to 's model MSE Relative to 's model

8 s model with time-varying β(t) PH model Legend Legend Percentile of z % % 5% 5% 75% 9% 99% MSE Relative to 's model MSE Relative to 's model MSE Relative to 's model < median survival time....3 Percentile of z % % 5% 5% 75% 9% 99% MSE Relative to 's model < median....3 s model with time-varying β(t) VII. Discussion time > time > time >.5..5 Legend When the data satisfied s linear model, both s a s model rendered biased survival estimates. They have, however, often shown a lower MSE than the native model for the data at hand. When analyzing PH model data using the s routine, we observed no dramatic increase in bias and MSE relative to native model, while using the same criteria the survival estimates based on s model appeared to be highly distorted. MSE Relative to 's model time > MSE Relative to 's model time > censoring limit > Percentile of z % % 5% 5% 75% 9% 99% When the data followed s model with time-varying covariate effects, both s and s model rendered in terms of bias and MSE highly unreliable estimates of the conditional survival distribution. In other words, there was no alternative to using the native model in this instance

9 References [] DR: Regression Models and Life Tables (with discussion), Journal of the Royal Statistical Society, 97, Vol. 34, pp. 87. [] OO: A linear regression model for the analysis of life times, Statistics in Medicine, 989, Vol. 8, pp [3] RJ: Flexible methods for analyzing survival data using splines, with application to breast cancer prognosis, Journal of the American Statistical Association, 99, Vol. 87, pp Thank you for your attention! [4] RJ: Spline-based tests in survival analysis, Biometrics, 994, Vol. 5., pp [5] Valenta Z and Weissfeld LA: Estimation of the Survival Function for s Piecewise-Constant -Varying Coefficients Model, Statistics in Medicine,, Vol. (5), pp