Survival Analysis Math 434 Fall 2011

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1 Survival Analysis Math 434 Fall 2011 Part IV: Chap. 8,9.2,9.3,11: Semiparametric Proportional Hazards Regression Jimin Ding Math Dept. jmding/math434/fall09/index.html Basic Model Setup Introduction slide #2 Proportional Hazards Model slide #3 Properties and Interpretation slide #4 Coding Covariates slide #5 Estimation and Inference Partial Likelihood slide #6 Breslow s Estimator slide #8 Partial Likelihood More slide #9 Solving Partial Likelihood slide #10 Tests slide #11 Local Tests slide #12 More on Wald s Test slide #13 When Ties Are Present slide #14 Time-Dependent Covariates slide #16 Stratified Proportional Hazards Models slide #17 Model Diagnostic Cox-Snell Residuals slide #18 Martingale Residuals slide #20 Deviance Residuals slide #22 Check Proportional Hazard Assumption slide #24

2 Introduction Nonparametric Models: Kaplan-Meier survival function estimation, Nelson-Aalen cumulative hazard estimation. Parametric Models: Exponential, Weibull, Log-normal, Log-logistic, Gamma distributions w/o covariates. Semiparametric Models: AFT model, Proportional odds model, Proportional hazards model. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #2 Proportional Hazards Model It was first proposed in 1972 and further studied in 1975 by Cox and hence often called Cox Model. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #3

3 Properties and Interpretation If we look at two individuals with covariate values Z and Z, the ratio of their hazards is a constant: which is called as the relative risk of an individual with risk factor Z having event as compared to an individual with risk factor Z. The logarithm of the ratio of hazard rate to the baseline hazard rate is: So the coefficients {β 1,, β k } can be thought as the effect of covariates, similar as the usual linear models. The test of covariate effects is equivalent to the test of the coefficients being 0s. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #4 Coding Covariates Usually the independent variables (covariates) are known at the start of the study. They are called fixed time covariates or baseline covariates. Occasionally covariates vary with time and are called time-dependent covariates. Different methods have to be used for baseline covariates and time-dependent covariates. Here, we first discuss the fixed time covariates. Quantitative: BMI, age, blood presure,. Qualitative: Gender, race, treatment, disease type,. Dummy variable (indicators). The coefficient β k represent the difference between two categories associated with covariate Z k. When there are a categories (levels), we only need a 1 dummy variables (indicators). Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #5

4 Partial Likelihood Assuming that censoring is noninformative and there are no ties between the event times. We first derive the partial likelihood through a profile likelihood, which is discused in Johansen Denote the observed data as (T 1, δ 1, Z 1 ), (T 2, δ 2, Z 2 ),, (T n, δ n, Z n ). Let t 1 < t 2 < < t n denote the ordered event times and Z (i)k be the kth covariate associated with the individual whose failure time is t i. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #6 Partial Likelihood Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #7

5 Breslow s Estimator of Baseline Cumulative Hazard: Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #8 Partial Likelihood More from Conditional Probabilities Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #9

6 Solving Partial Likelihood Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #10 Tests Let ˆβ and I(β) denote the MLE of β and the p p information matrix evaluated at β. For large samples, ˆβ approximately follows a p dim normal distribution with mean β and variance-covariance I 1 (β). Wald s test H 0 : β = β 0. Likelihood ratio test Scores test Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #11

7 Local Tests Consider a hypothesis about a subset of β s. Let β = (β T 1, βt 2 )T, where β 1 is q 1 vector and is the interesting part of β. The remaining p q vector is denoted by β 2. Wald s test H 0 : β 1 = β 10. Likelihood ratio test Scores test Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #12 More on Wald s Test Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #13

8 When Ties Are Present Revised partial likelihood: Brewslow (1974) Efron (1977) Cox (1972) Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #14 When Ties Are Present When the ratio of the sample size to the number of ties is small, all methods have similar results. When the number of ties are small, Brewslow and Effron s methods are similar. When the number of ties is large, usually a more sophisticated and computational intensive method, exact, is preferred. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #15

9 Time-Dependent Covariates Partial likelihood: Examples 9.1 & 9.2 on page Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #16 Stratified Proportional Hazards Models Model: h j (t Z(t)) = h 0j (t)exp[β T Z(t)], j = 1,, s. The regression coefficients are assumed to be same in each stratum although the baseline hazard functions may be different. The log partial likelihood function is logpl(β) = logpl 1 (β) + logpl 2 (β) + + logpl s (β), where logpl j (β) is the log partial likelihood of the jth stratum. Examples 9.1 on page Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #17

10 Cox-Snell Residuals Facts: If a random variable X has distribution function F(x) and cumulative hazard function H(x), then Y = F(X) U[0, 1] and W = H(X) Exp(1). Cox-Snell residuals are defined as r = Ĥ(T), where Ĥ is the estimated cumulative hazard function (based on the model) and T is the observed survival time. Under proportional hazards model h(t Z) = h 0 (t) exp(β T Z), for i = 1, 2,, n observations, Under Weibull model S(t Z) = exp{ exp[ (µ+β T Z)/σ]t 1/σ }, for i = 1, 2,, n observations, Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #18 Cox-Snell Residuals If the model is exactly right, the Cox-Snell residuals should approximately follow a unit exponential distribution, hence the cumulative hazard function of the residuals should be H r (t) = t. Therefore, a plot of the Nelson-Aalen cumulative hazard estimate of residuals versus residuals should be a straight line through the origin with a slope of 1, if the model is corrected. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #19

11 Martingale Residuals Martingale residual is a slight modification of the Cox-Snell residual. In general, the martingale residual is defined as, for i = 1, 2,, n: ˆM i = N i ( ) 0 Y j (t) exp{ ˆβ T Z i (t)}dĥ0(t), where N i (t) is the counting process of having event for the ith subject and Y i (t) is the indicator that individual i is under study at a time just prior to time t (indicator of being in risk set). The ˆβ and Ĥ0(t) are regression parameter estimate and Brewslow cumulative hazard estimate. If the model is exactly right, that is, if the ˆβ and Ĥ0(t) are replaced by the true β and H 0 (t), then the martingale residuals would be martingales. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #20 Martingale Residuals Different from Cox-Snell residuals, the plot of martingale residuals usually does not only check the model assumption but also suggest the form of the covariate in the model. Suppose some covariates, which we know the proper functional form, are already in the proportional hazards model. To see how to add an additional covariate with proper form, we could plot the martingale residuals against this new covariate and fit the points using some smoothing technique. The smoothed curve suggests the proper functional form of the new covariate. Example 11.2 on page Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #21

12 Deviance Residuals The problem of martingale residuals is that they are skewed with maximum value 1 but minimum value. The deviance residual is used to obtain a more normally shaped residual. The deviance residual is defined as: D i = sign[ ˆM { i ] 2[ ˆM i + δ i log(δ i ˆM 1/2 i )]}. One may plot deviance residuals versus the risk scores ˆβ T Z i = p k=1 ˆβZ ik. When there is a light to moderate censoring, the residuals should look like a sample of normal noise. When there is heavy censoring, many values close to 0 may distort the normal approximation. In either case, potential outliers will have deviance residuals whose absolute values are too large. Example 11.2 on page Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #22 Deviance Residuals in Parametric Models Deviance residuals is defined same in parametric models and could be used to check outliers. But the martingale residuals in parametric models are simply ˆM i = δ i r i for i = 1, 2,, n, where r i is the Cox-Snell residual. Here, the martingale residuals are not martingales under the true model, but have similar properties and hence have the same name. Example 12.2 on page 416 and 420. Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #23

13 Check Proportional Hazard Assumption Besides Cox-Snell residuals, other plots, such as Anderson plot, Arjas plot, standardized score residual plot, could also be used to check the proportional hazard assumption. For more details, read chap 11.4 of the text book (page ). Jimin Ding, November 1, 2011 Survival Analysis, Fall 2011 slide #24

Cox s proportional hazards/regression model - model assessment

Cox s proportional hazards/regression model - model assessment Rasmus Waagepetersen September 27, 2017 Topics: Plots based on estimated cumulative hazards Cox-Snell residuals: overall check of fit Martingale