# Tied survival times; estimation of survival probabilities

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1 Tied survival times; estimation of survival probabilities Patrick Breheny November 5 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/22

2 Introduction Tied survival times Introduction Breslow approximation Efron approximation Discrete model Thus far, we have worked with Cox regression under the assumption that no ties are present among the failure times, and thus, that the data can be uniquely sorted with respect to time In many data sets, however, ties are present, usually due to the fact that failure times are only reported to the nearest day Our first topic for today is how to handle tied survival times in the Cox regression model Patrick Breheny Survival Data Analysis (BIOS 7210) 2/22

3 Average partial likelihood Introduction Breslow approximation Efron approximation Discrete model Perhaps the most natural solution would be to consider all possible ways of breaking the ties as equally likely In this approach, the Cox partial likelihood would be replaced with the average of the Cox partial likelihoods over all the orderings in which the ties have been broken As a simple example, suppose subjects 2 and 3 fail at a given time, and that subject 4 is also in the risk set at that time; the likelihood contribution would be 1 2 w 2 2,3,4 w j w 3 3,4 w j w 3 w 2,3,4 w 2 j 2,4 w j Patrick Breheny Survival Data Analysis (BIOS 7210) 3/22

4 Breslow approximation: Idea Introduction Breslow approximation Efron approximation Discrete model The averaging method is intuitively reasonable, but from a mathematical and computational standpoint, very messy and time-consuming to work with An approximation that greatly simplifies the resulting calculations was proposed by Breslow (1974) To continue our simple example from the previous slide, the idea is that w j w j ( ) 2 w j w j w j 2,3,4 3,4 2,3,4 2,4 2,3,4 Patrick Breheny Survival Data Analysis (BIOS 7210) 4/22

5 Breslow approximation: Formula Introduction Breslow approximation Efron approximation Discrete model Letting s j = i D j x i denote the sum of the covariates over the set D j of subjects who fail at time j, applying the Breslow approximation yields the likelihood L(β) = J j=1 exp(s T j β) { k R j exp(x T k β)}d j The obvious advantage of this approximation is that the resulting likelihood looks almost exactly like the original Cox likelihood, and results in very minor modifications to the score, information, and algorithm that we derived last week Patrick Breheny Survival Data Analysis (BIOS 7210) 5/22

6 Efron approximation: Motivation Introduction Breslow approximation Efron approximation Discrete model An alternative approximation was proposed by Efron (1977) Clearly, there is an advantage to common denominators in the sum on slide 3, as it allows us to combine the sum into a single term It is also clear, however, that the denominator in the Breslow approximation is always larger than it should be Patrick Breheny Survival Data Analysis (BIOS 7210) 6/22

7 Efron formula Tied survival times Introduction Breslow approximation Efron approximation Discrete model As a compromise between the two, Efron suggested using the average weight among the failures at time j, w = d 1 j k D j w k : w j w j w j w j ( ) w j w j w 2,3,4 3,4 2,3,4 2,4 2,3,4 2,3,4 In general, then, the likelihood becomes L(β) = J exp(s T j β) dj 1 r=0 { k R j exp(x T k β) r w} j=1 Patrick Breheny Survival Data Analysis (BIOS 7210) 7/22

8 Remarks Tied survival times Introduction Breslow approximation Efron approximation Discrete model To summarize: Average partial likelihood: Best accuracy, hardest to work with Efron approximation: Good accuracy, moderately easy to work with Breslow approximation: Least accurate, very easy to work with When the number of ties is small, there is typically little difference between the approaches Many software programs implement the Breslow approach for its simplicity, but the survival package uses the Efron approximation Patrick Breheny Survival Data Analysis (BIOS 7210) 8/22

9 Discrete Cox model Tied survival times Introduction Breslow approximation Efron approximation Discrete model However, what if there are a lot of ties? In that case, it would make sense to treat the failure distribution as discrete, and propose a model in terms of the discrete hazards λ ij (the hazard for the ith subject at time j) as opposed to the hazard density λ i (t) The semiparametric analog to Cox regression in the discrete case is λ ij 1 λ ij = λ 0j 1 λ 0j exp(x T i β) Patrick Breheny Survival Data Analysis (BIOS 7210) 9/22

10 Conditional logistic regression Introduction Breslow approximation Efron approximation Discrete model By conditioning on the number of failures at each time j, one can construct a likelihood for β that is free of intercept terms, allowing us to avoid specifying λ j0 This model, which is also widely used in categorical data analysis, is known as conditional logistic regression, and is also referred to as the discrete Cox model in survival analysis This model is available in coxph through the ties= exact option Patrick Breheny Survival Data Analysis (BIOS 7210) 10/22

11 Motivation As we have remarked several times, the Cox model is a model only for the relative risk comparing subjects versus each other it makes no predictions as far as absolute risk In other words, we can use the Cox model to estimate coefficients, hazard ratios, etc., but we cannot use it to estimate the probability, say, that a subject will survive at least 2 years Such quantities are clearly of interest in many applications, however; is there a way we can go back and estimate a baseline hazard? Patrick Breheny Survival Data Analysis (BIOS 7210) 11/22

12 Basic approach It is certainly possible to write down the nonparametric likelihood under proportional hazards and consider the joint maximum likelihood estimation of β and the point masses λ 0j at each observed failure time More simply, however, we can just fix β at the Cox regression estimate β, then maximize the likelihood with respect to the λ 0j parameters alone Typically, there is little difference between the approaches, as the point masses themselves tend to have little impact on the relative risks Patrick Breheny Survival Data Analysis (BIOS 7210) 12/22

13 Nonparametric likelihood In our lecture on the Kaplan-Meier estimate, we showed that the nonparametric MLE of the survival function S(t) is a discrete function with point masses at the observed failure times (and no mass anywhere else) Making the appropriate modifications to allow each observation its own subject-specific hazard, we have: L(λ) = { } λ ij (1 λ ij ), j i D j i R j D j where j indexes failure times, with D j the set of individuals dying at time j and R j is the risk set Patrick Breheny Survival Data Analysis (BIOS 7210) 13/22

14 Proportional hazards on the discrete scale The proportional hazards assumption implies the following relationship between subject-specific survival and baseline survival: where w i = exp(x T i β) S i (t) = S 0 (t) w i, Since S(t) = t j t (1 λ j) for a discrete survival function, this means that λ ij = 1 (1 λ 0j ) w i Patrick Breheny Survival Data Analysis (BIOS 7210) 14/22

15 Main result Letting α j = 1 λ 0j, the nonparametric MLE of S 0 given β can be represented with L(α) = { (1 α wi j ) } α wi j, j i D j i R j D j In the case where only one failure occurs at t j, we have ˆα j = (1 π jj ) 1/w j ; if more than one failure occurs, there is no closed form solution and numerical optimization methods must be used to solve for ˆα j Patrick Breheny Survival Data Analysis (BIOS 7210) 15/22

16 Survival formulas Once we have obtained β and ˆα, our estimates of the survival function for a subject with covariate values x i is given by Ŝ i (t) = t j t ˆα exp(xt β) i j, which, like the Kaplan-Meier estimate, will be a step function with discontinuous drops at every observed failure time Patrick Breheny Survival Data Analysis (BIOS 7210) 16/22

17 survfit.coxph The survfit function from the survival package can be applied to the output of a Cox regression model in order to carry out estimation of the baseline hazard as described on the previous slides: fit <- coxph(s ~ trt + stage + hepato + bili, pbc) sfit <- survfit(fit) The methods we described previously, such as summary(sfit) and plot(sfit), work exactly as they did before The survival package also calculates a confidence interval for Ŝi(t), although in the interest of time, we won t go into the details Patrick Breheny Survival Data Analysis (BIOS 7210) 17/22

18 PBC example (baseline) 1.0 Progression free survival Time (years) By default, the curve is drawn at x = 0 for the centered model; i.e., at the mean of each covariate Patrick Breheny Survival Data Analysis (BIOS 7210) 18/22

19 Subject specific curves Like predict, survfit.coxph accepts a newdata argument, allowing the calculation of subject-specific survival curves ndat <- data.frame(trt=0, stage=1:4, hepato=0, bili=1) sfit <- survfit(fit, newdata=ndat) One can then submit plot(sfit) to plot the curves or print sfit, which will provide estimates and confidence intervals for the median survival time Patrick Breheny Survival Data Analysis (BIOS 7210) 19/22

20 PBC example: Survival by stage Stage: 1 Stage: 2 Stage: 3 Stage: Progression free survival Time (years) Patrick Breheny Survival Data Analysis (BIOS 7210) 20/22

21 GVHD example: Cox versus Kaplan-Meier MTX MTX+CSP 1.0 Progression free survival Time (days) Patrick Breheny Survival Data Analysis (BIOS 7210) 21/22

22 Remarks Although the approaches are similar, it is important to keep in mind that survival function estimation in the Cox model is restricted to obey the proportional hazards assumption, and therefore cannot capture non-proportional aspects of the data For example, in the GVHD data, there appears to be little difference between the treatment groups at early times and a more substantial difference at later times; by construction, the Cox estimates show a constant benefit over time In later lectures, we will discuss diagnostics and potential remedies for non-proportional hazards Patrick Breheny Survival Data Analysis (BIOS 7210) 22/22

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