Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 17Page 18Page 19Page 201Parametric models for survival functions, hazardscan be used for estimating, comparing survival functions, hazardsCommon functions/distributions: Mathematical form, appropriateness for applied settings2Exponentialconstant hazard: h(t) = 8survival function: S(t) = exp(-8t)Mean: (mrl(0) = 1/8)Memoryless property; pr(T $ t+z | T $ t) = pr(T $z)In and of itself, not reasonable in many applications: hazards usually notconstant over time3Weibull distribution2 parameters: " - shape parameter8 - scale parameterhazard: h(t) = " 8 t "-1 Survival function S(t) = exp(-8t )"exponential distribution: special case of Weibull with " = 1second parameter makes it more flexible than exponential4look at graphs" < 1: hazardsdecrease with timehazard starts atinfinity (may beunreasonable in manyapplications)" > 1: hazardsincrease over timehazard starts at zero,increasesWhat patterns can Weibull not represent? What real situations might thisrepresent?5Weibull cannot replicate many patterns, even approximatelyhump-shaped patterns (e.g., mortality from various cancers after diagnosis)patterns with dips (e.g., total mortality, time scale since birth)Increasing hazards with time since inception of study, initial hazard > 0(e.g., observational studies of chronic diseases, where subjectsrecruited/enrolled at point of no clinical significance)In studies in which follow-up starts at significant time (or in population inwhich entry criterion is being disease- free or healthy), may bereasonable6“Shape” parameter; log hazards or log cumulative hazard plots with common "have same shape, are parallelsame vertical separation between lines at all points, so curves have same shape7Survival curves8Log-logistic distributionagain, shape, scale parameters9for t close to 0,hazards, survivalfunctions close toWeibull with sameshape, scaleparametersfor t large diverge;hazards decreaseafter some point10harder to explain shape parameter:vertical separation not constant, as with Weibull; what does shape parametermean?11horizontal separation is constant on log time scale: implies for single survival, cumulative hazard, separation of two curves with differentscale parameters 8, same shape parameter " isnot function of H (or S)in this sense, curves have same shape12Survival curves:increasing hazard for "=2 obvious from survival plotslowly decreasing hazard for "=1 not obvious13Log-normal distributionagain, 2 parameter distributionpeaked hazard functionsGompertzhazard starts above 0, increases exponentiallyform:14Gammasee bookPareto: can arise from mixture of exponential distributions with differentparameters ( distributed as gamma distribution) ; see Cox and Oakes, p. 20 (not KM)15three-parameter distributions: greater flexibilityGeneralized gamma (see book for details)Gompertz-Makehamform:16Parametric models in survival analysis used less frequently than nonparametricand semiparametric models; why relatively unpopular?17Lack of flexibility and interpretabiltyFully parametric methods will be used in some cases where there are problemswith semiparametric or nonparametric alternatives (especially regressionmodels for log lifetimes)How can one use parametric methods to approximate nonparametricdistributions/models?18Parametric approximations to nonparametric distributions/models:Piecewise exponential distribution:j 1 j-1 jJconsider a series of J ordered knots u , 0 < u < ... < u < u < ... < ujj-1 jh(t) = 8 , u #t < umaking an unrealistic smoothness assumptionnonetheless, can generally be used to approximate arbitrary (smooth) hazardfunctions well1920Discrete-time