Nonparametric Inference in Accelerated Failure Time Model

1. Nonparametric Inference in the Accelerated Failure Time Model Using Restricted Means Theodore Karrison, PhD Mihai Giurcanu, PhD University of Chicago

2. Outline • Accelerated Failure Time Model Louis Estimator • Restricted Means-Based Estimator • Stata Routine • Simulation Study • Example: DeCIDE Head and Neck Cancer Trial • Summary and Future Research •

3. Accelerated Failure Time Model We consider the problem of comparing two survival curves. RCT comparing an experimental treatment with a control group. The AFT model can also be framed as a model linear in log T: l?? ? = ? + ?0Z+?, where Z is an indicator variable for treatment group

4. Stata has a number of routines for parametric estimation under the AFT model, which also allow for the incorporation of covariates, using streg command: Here, however, we wish to remain nonparametric and avoid assumptions with regard to the underlying survival distribution.

5. Variance of survival estimate typically estimated by Greenwood’s formula.

6. Louis Estimator Louis (1981) derived a nonparametric estimator for the scale change parameter β by considering the efficient score from Cox’s (1972) partial likelihood function. Let ??(t) denote the number of subjects at risk at time t and ??(t) the number of events up to time t in group ?. The efficient score is given by (1) Under Cox’s proportional hazards (PH) model ?1(?) = exp(?)?0(?) and (1) becomes , independent of the underlying hazards. For AFT, however, ?1(?) = exp(−?)?0(?exp(−?)) and the score would still depend on the hazards.

7. Louis’s insight was to write (1) for β=0 “relative to a new set of data.” Replacing by and ?0? by ?0????(?) , in other words, re-scaling the group 0 survival times by ???(?) , (1) becomes ?0????(?) ?0? Louis’s estimator, denoted, ??is obtained by setting the above equation = 0.

8. Restricted Means-Based Estimator Fixed point of restriction t* Mean survival time ∞? ? ?? ? = 0 where ?(?) is the survival function for the random variable ? > 0. Due to censoring, Irwin (1949) proposed consideration of the restricted mean survival time (RMST) ?∗? ? ?? ??∗= 0 ??∗= ? ??? ?,?∗ Kaplan and Meier (1958) and Meier (1975) studied its large-sample properties. Studied further by Karrison (1985, 1997), Royston and Parmar (2013), and Zhao et al (2016), among others.

10. ? ? This provides a straightforward expression for the asymptotic variance of

11. Random point of restriction Up to this point we have assumed the point of restriction, t*, is fixed. It can be shown that, under certain regularity conditions, if t* is chosen as the minimum of the maximum observed survival times in the two treatment groups, the asymptotics still hold. In the case of a clinical trial with accrual period a and follow-up period f, the LHS = t and thus (3) holds. See also Tian et al (2020)

12. Stata Code Snippets stset stime, failure(indic) * Random point of restriction sort group stime by group: gen newvar=stime[_N] scalar max0=newvar[1] scalar max1=newvar[_N] scalar tstar=min(max0, max1) . . . . . * Obtain restricted means and variances stci if `group'==0, rmean scalar `rm0'=r(rmean) scalar `varrm0'=r(se)^2 stci if `group'==1, rmean scalar `rm1'=r(rmean) scalar `varrm1'=r(se)^2

13. * Solve estimating equation bisect auc X gvar1 gvar0 returns exp beta = 0 from 0 to -log(0.01/tstar) return scalar beta=$S_1 . . . . * Estimate variance via delta method scalar vargn=varrm1/rmean1^2 + varrm0/rmean0^2 scalar gprime=tauembeta*stauembeta/rmean0 - 1 return scalar sebeta=sqrt(vargn/gprime^2) .ado command in progress: rmaft stime indic group [if] [in], [tstar]

14. Simulation Study Therefore censoring times were uniformly distributed over [f, a+f].

15. True survival curves We evaluated bias, efficiency, and confidence interval coverage rates for the Louis, RMST-based, and parametric estimators. R=10,000 simulations were performed for each scenario.

16. Bias All estimates unbiased except when fitting wrong parametric model

18. Efficiency Efficiency of Louis greater than efficiency of RMST

19. Efficiency of RMST greater than efficiency of Louis for smaller sample sizes

20. Confidence interval coverage rates Coverage rates of Louis a little less than nominal

21. Coverage rates of Louis again a little less than nominal

22. ASE and ESE

24. Agreement between Louis and RMST-based estimators Weibull

25. Lognormal

26. Example DeCIDE was a randomized, phase III clinical trial comparing induction therapy plus chemoradiotherapy (I+CRT) vs. chemoradiotherapy alone (CRT) in patients with locally advanced head and neck cancer (Cohen et al, J Clin Oncol 2014) Recurrence-free survival

27. Inference Method Estimate 95% CI RMST-based Louis 0.593 0.595 (-0.06, 1.25) (-0.21, 1.20)

28. Model fit

29. Summary Our estimator provides similar results to the Louis estimator and neither can be strongly recommended over the other. • The RMST-based approach does have the advantage of providing a standard error without the need to estimate hazard functions or resort to a test-based CI. • Confidence interval coverage rates were close to the nominal values—a little better for RMST-based vs. Louis estimator. • Efficiencies relative to fitting the correct parametric model ranged from 75%-95%. • Fitting the wrong parametric model yielded biased estimates. • Thus we have the usual bias-variance tradeoff. •

30. Future Research Given a readily computed standard error (SE), a stratified estimate from the RMST-based approach should be straightforward: derive estimate and SE within each stratum and construct weighted average, weighted by stratum size or inverse variance. • Extend the method for other types of censoring, such as interval censoring. • Looking into incorporating covariates (analogous to Buckley-James (1979) regression model for censored data). • Thank you for your attention!

31. References Cohen EW, Karrison TG, Kocherginsky M et al. (2014). Phase III randomized trial of induction chemotherapy in patients with N2 or N3 locally advanced head and neck cancer, J Clin Onc 32:2735-2743. Cox DR (1972). Regression models and life tables (with discussion), JRSS B, 34:187-220. Irwin JO (1949). The standard error of an estimate of expectational life, Journal of Hygiene 47:188-189. Kaplan EL, Meier P (1958). Nonparametric estimation from incomplete observations, JASA 53:457-481. Karrison T (1987). Restricted mean life with adjustment for covariates, JASA 82:1169-1176. Karrison T (1997). Using Irwin’s restricted mean as an index for comparing survival in different treatment groups—Interpretation and power considerations, Controlled Clinical Trials 18:151-167. Meier P (1975). Estimation of a distribution function from incomplete observations, Perspectives in Probability and Statistics: Papers in honour of M.S. Bartlett, ed. J. Gani, New York: Academic Press, 67-87. Louis T (1981). Nonparametric analysis of an accelerated failure time model, Biometrika 68:381-390. Royston P, Parmar MKB (2013). Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Med Res Methodol 13:152. Wei LJ (1992). The accelerated failure time model: a useful alternative to the Cox regression model in survival analysis. Statistics in Medicine 11:1871-1879. Zhao L, Claggett B, Tian L, Uno H, Pfeffer MA, Solomon SD, Trippa L, Wei LJ (2016). On the restricted mean survival curve in survival analysis, Biometrics 72:215-221.