Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 141Counting process approach:Used in most theoretical modern work:developing estimators, proving their large sample quantities, especiallyconsistency and asymptotic normalityWill not give a thorough account in this course; neitherdoes bookNonetheless, will outline: provides some insight intostatistical methods and notation developed later on2Counting processes: N(t), t $ 0, a stochastic process withcharacteristicsN(0) = 0N(t) < 4sample paths of N(t) are right continuous and piecewiseconstant; jumps of size +1For an individual, the process ; processjumps at failure-timeFor group, process is3history or filtration includes all past history through thattime: Ftincludes history of censoring process, baseline covariates,time-varying covariatesfiltration increases with time Fs d Ft for s # tdenote history at time just before t by Ft! dN(t) is change in the process N(t) over a short timeinterval [t,t+dt); i.e., Assuming no ties, dN(t) = 1 if death occurs at t, 0otherwiseY(t) = number of individuals with a study time T $ t4; the intensity process, is a stochastic processthat depends on the information contained in the historyprocess Ft through Y(t)Y(t) = # of subjects at risk at tleft-truncated data: if K is lefttruncation timecumulative intensity process , t $ 0Note that , because Y(t) isfixed at t! and thus 7(t) is not random given thehistory/filtration5 is the counting process martingalegiven the past, its expectation is 0Martingale is stochastic process with property that itsexpectation given the past through time s, the expectationof the margingale doesn’t change; i.e., ,or, equivalently, for all s<t.Two parts to margitingale:N(t); a nondecreasing step function7(t); a compensator of the counting process; this ispredictable in that its value at t is fixed just prior to tMartingale: mean zero noise6predictable variation process of M(t): +M,(t)compensator of M2(t)for a martingale To find if noties (Poisson process with rate 8(t))otherwise Bernoulli varianceMany statistics in survival analysis are stochasticintegrals of martingales discussed aboveLet K(t) be a stochastic process whose value is known,given Ft- ; Y(t) is such a processstochastic integral: predictable variation will use to derive estimator of the cumulative hazardfunction H(t), the Nelson-Aalen estimator7we can write ; i.e., change incounting process is decomposed into sum of predictablepart plus noisefor nonzero Y(t), we have ; i.e.,proportion failing at t is sum of predictable part (hazard)plus random processsince dM(t) is just noise, so is dM(t)/Y(t),since Y(t) is fixed just prior to t,and conditional variance of the noise isLet be indicator of whether any subjectsare at risk at t, and define 0/0=08 : integrate both sides of above equation is the Nelson-Aalen estimator of thecumulative hazard H(t)it is the sum of a smooth function ( ,the true cumulative hazard in the range in which we havedata; not in general exactly equal to H(t)) and a randomprocess , which is the a stochasticintegral of the predictable process with respect to amartingale and, hence, is also a martingale9predictable variation process: since10Martingale Central Limit Theoremheuristics for Nelson-Aalen estimatornote that Y(t)/n, N(t)/n are sample averages whichconverge in probability to particular quantities and havesmall sampling variation for large nY(t)/n converges to a deterministic function y(t)Let for large samples H*(t) is close to H(t), soThe conditional variance of the jumps in the Z processZ(n)(t) converge to h(t)/y(t):11Thus, in large samples, Z(n) has almost continuous samplepaths and a predictable variation process close toone limiting process Z(4) which is a martingale withcontinuous sample paths, deterministic predictablevariation equal to formula above12limiting process: independent increments, finite-dimensional normal distributionsindependent increments: independentfor nonoverlapping intervalslimiting process normally distributed if joint distributionof The process , has a k-variatenormal distribution with mean 0 and a covariance matrixwith entries13Confidence intervals for the cumulative hazard at a fixedtime, because has approximate normaldistribution with normal distribution with mean 0 andvariance estimate variance from , since we can estimatey(t) by Y(t)/n and h(t) by dN(t)/Y(t)14Get survival function from cumulative hazard:use “product integral”:Kaplan-Meier estimatorConstructing likelihoods: see