Assume the model:

1. Borgan and Henderson: Event History Methodology Lancaster, September 2006Session 2: Nelson-Aalen, Kaplan-Meier and Aalen-Johansen estimators

2. Nelson-Aalen estimator Assume the model: at risk indicator common hazard/intensity (non-negative function) Aggregated counting process N(t) has intensity process number at risk

3. Will estimate the cumulative hazard/intensity Have the decomposition: Estimating equation (when Y(t) > 0)

4. Thus (when Y(t) > 0 ): This motivates the Nelson-Aalen estimator: with Nelson-Aalen estimator is a sum over the observed event times (assuming no tied event times)

5. Nelson-Aalen estimator: Examples Males: Females:

6. Stochastic integrals and properties of the Nelson-Aalen estimator We may write:

7. Thus we have the decomposition: systematic part random part The random part is the stochastic integral: where is a predictable process We may use properties of stochastic integrals to study the Nelson-Aalen estimator

8. The stochastic integral is a martingale. In particular Thus The Nelson-Aalen estimator is approximately unbiased.

9. Predictable variation of a martingale In order to study the variability of the Nelson-Aalen estimator (and a number of other estimators and test statistics), we need a concept of variability of a martingale M(t). Such a concept of variability is the predictable variation process given by:

10. One may in general prove that is a martingale. In particular therefore and it follows that

11. For a counting process martingale we have This motivates the important result:

12. Predictable variation of a stochastic integral We also need the predictable variation of the stochastic integral We have, since H(t) is predictable:

13. This motivates the key result In particular for a counting process martingale we have

14. Variance of the Nelson-Aalen estimator For the Nelson-Aalen estimator we have Thus For estimation we replace by and obtain

15. Martingale central limit theorem In order to derive the large sample distribution of the Nelson-Aalen estimator (and a number of other estimators and test statistics), we use the martingale central limit theorem (MCLT) The classical CLT describes how sums of random variables (properly normalized) becomes approximately normally distributed as n increases. In a similar manner the MCLT describes how martingales (properly normalized) become approximately distributed as Gaussian martingales (transformed Brownian motions)

16. A Gaussian martingale is a stochastic process X(t) in continuous time satisfying: • its increment X(t) – X(s) over an interval (s,t] is normally distributed with mean zero and variance V(t) – V(s) for a continuous strictly increasing function V(t)(the variance function) • its increments over non-overlapping intervals are independent • its sample paths (realizations) are continuous For the special case V(t) = t we get the classical Wiener process (or Brownian motion); cf next slide.

17. Simulation of the sample path of a Wiener process:

18. A counting process and its cumulative intensity process (left) and the corresponding counting process martingale (right). (Based on n =10 simulated censored survival times)

19. Counting process martingales based on n =10, 50, 250 and 1250 simulated censored survival times:

20. Consider a stochastic integral where is a counting process martingale based on observing n individuals Assume: (convergence in probability) (1) implies that the predictable variation process of converges to the deterministic function , while (2) states that its jumps disappear in the limit.

21. Under assumptions (1) and (2) (plus some regularity conditions) the process converges in distribution to a Gaussian martingale with variance function In particular for given t0 the random variable is approximately normally distributed with mean 0 and variance

22. Large sample properties of Nelson-Aalen We have May use the MCLT with Assume that Then:

23. Thus (1) and (2) hold and the MCLT gives that converges in distribution to a Gaussian martingale with variance function In particular for given t0 the random variable is approximately normally distributed around with a variance that can be estimated as described earlier.

24. Survival functions, cumulative hazards, and product integrals: the general case Uncensored survival time T Survival function: For the absolute continuous case, the hazard function is given by: Cumulative hazard function:

25. We have the relations: For a general distribution the hazard rate is not defined, but we may define the cumulative hazard rate as (generalizing the first relation above): How can the second relation be generalized?

26. Need product-integrals to achieve this generalization. Partition [0,t] into small time intervals: si-1 si 0 t This is a product-integral.

27. For the continuous case we have: For the discrete case we have: where is the increment of the cumulative hazard (a step function) at s. For the general case we have a mixture of the two.

28. The Kaplan-Meier estimator For right censored survival data we observe: Model: the uncensored survival times Ti are i.i.d. with hazard Counting and intensity processes:

29. Aggregated counting process: Intensity process: with the number at risk just before timet

30. Nelson-Aalen estimator: (a step function) Plug this into the product-integral expression for the survival function: (a finite product) This is the Kaplan-Meier estimator

31. Kaplan-Meier estimator: Examples Males: Females:

32. Kaplan-Meier estimator: Properties May show that(this is Duhamel's equation) Asymptotically:

33. Thus: The statistical properties for Kaplan-Meier may be derived from those of Nelson-Aalen:

34. Usually the variance is estimated by Greenwood's formula: Only minor difference between the two variance estimators

35. Pointwise confidence intervalsfor S(t) Linear: Log-log-transformed: The default confidence interval in R is based on the log-transformation, and that is a bad choice for Kaplan-Meier!

36. The Aalen-Johansen estimator A multivariate version of the Kaplan-Meier estimator applies to Markov processes. Consider Markov process with states 0, 1, …, K. transition intensities Phj(s,t)transition probabilities Transition probability matrix:

37. Aalen-Johansen estimator: Here is the matrix of Nelson-Aalen estimators with If e.g. K=2 and a 1–>2 transition is observed at tj The statistical properties of the estimator may be derived in a similar manner as for Kaplan-Meier.

38. Aalen-Johansen estimator: Examples Causes of death in Norway. Estimates of 1) Cancer 2) Cardiovascular disease 3) Other medical 4) Alcohol abuse, violence, accidents

39. 374 female diabetes patients in Denmark with disease onset before age 10 yrs Use illness-death model with diabetic nephropathy as "disease state" (state 1). Estimate of P01(5,t): (Markov assumption is dubious)