Survival Analysis: Counting Process and Martingale. Lu Tian and Richard Olshen Stanford University

Transcription

1 Survival Analysis: Counting Process and Martingale Lu Tian and Richard Olshen Stanford University 1

2 Lebesgue-Stieltjes Integrals G( ) is a right-continuous step function having jumps at x 1, x 2,.. b f(x)dg(x) = f(x j ){G(x j ) G(x j )) = f(x j ) {G(x j )}. a a<x j b a<x j b 2

3 One-sample problem In one-sample problem: observation (U i, δ i ), i = 1, 2,, n. N(t) = n i=1 I(U i t, δ i = 1) # observed failures by time t 0 f(t)dn(t) = K f(τ j )d j j=1 f(t) = { n i=1 I(U i t)} 1 0 f(t)dn(t) = K j=1 d j Y (τ j ), The NA estimator for the cumulative hazard function! 3

4 Two-sample problem In two-sample problem: observation (U i, δ i, Z i ), i = 1, 2,, n. N l (t) = n i=1 I(U i t, δ i = 1, Z i = l) # observed failures by time t at group l, l = 0, 1. Y l (t) = n i=1 I(U i t, Z i = l) # at risk by time t at group l, l = 0, 1. Consider the stochastic integral W = 0 Y 0 (t) Y (t) dn 1(t) 0 Y 1 (t) Y (t) dn 0(t) 4

5 Two-sample problem W = K j=1 Y 0 (τ j ) Y (τ j ) N 1(τ j ) K j=1 Y 1 (τ j ) Y (τ j ) N 0(τ j ) = K j=1 [ Y0 (τ j ) Y (τ j ) d 1j Y ] 1(τ j ) Y (τ j ) d 0j = K j=1 = j Y 0 (τ j )d 1j Y 1 (τ j )(d j d 0j ) Y (τ j ) ( d 1j d j Y1(τ ) j ) Y (τ j ) = j (O j E j ); 5

6 Regression problem In Cox regression: observations (U i, δ i, Z i ), i = 1, 2,, n. N i (t) = I(U i t, δ i = 1) The score function related to PL function is S n (β) = = n i=1 n i=1 { } δ i Z i S(1) (β, U i ) S (0) (β, U i ) { } Z i S(1) (β, s) S (0) (β, s) 0 dn i (s) 6

7 σ-algebra F is a non-empty collection of subsets in Ω. F is called σalgebra if If E F then E c F if E i F, i = 1,, n then i=1 E i F. 7

8 Probability Space A probability space (Ω, F, P ) is an abstract space Ω equipped with a σ algebra and a set function P defined on F such that 1. If P {E} 0 when E F 2. P {Ω} = 1 3. P { i=1 E i} = i=1 P {E i} when E i F, i = 1,, n and E i E j = ϕ, i j. 8

9 Random Variables and Stochastic Process (Ω, F, P ) is a probability space. A random variable X : Ω R is a real-valued function satisfying {ω : X(ω) (, b]} F for any b. A stochastic process is a family of random variables X = {X(t) : t Γ} indexed by a set Γ all defined on the same probability space (Ω, F, P ). Often times Γ = [0, ) in survival analysis. 9

10 Filtration and Adaptability A family of σ-algebra {F t : t 0} is called a filtration of a probability space (Ω, F, P ) if F t F and F s F t for s t. A filtration {F t ; t 0} is right-continuous if for any t F t + = F t where F t + = ϵ>0 F t+ϵ. A stochastic process {X(t) : t 0} is said to be adapted to {F t, t 0} if X(t) is F t measurable for each t, i.e., {ω : X(t; ω) (, b]} F t. 10

11 History of a stochastic process Let X = {X(t) : t 0} be a stochastic process, and let F t = σ{x(s) : 0 s t}, the smallest σ-algebra making all of the random variables X(s), 0 s t measurable. The filtration {F t : t 0} is called the history of X. 11

12 Conditional Expectation Suppose that Y is random variable on a probability space (Ω, F, P ) and let G be a sub-σ-algebra of F. Let X be a random variable satisfying 1. X is G-measurable 2. B Y dp = B XdP for all sets B G. The random variable X is the called the conditional expectation of Y given G and denoted by E(Y G). 12

13 Conditional Expectation If F t = {ϕ, Ω}, then E(X F t ) = E(X) E{E(X F t )} = E(X) If F s F t, then E{E(X F s ) F t } = E(X F s ) If Y is F t -measurable, then E(XY F t ) = Y E(X F t ) If X and Y are independent, then E(Y σ(x)) = E(Y ) E(aX + by F t ) = ae(x F t ) + be(y F t ) If g( ) is a convex function, then E{g(X) F t } g{e(x F t )}. E(Y σ(x t, t Γ)) = E(Y X t, t Γ) 13

14 Example of conditional expectation The probability space (Ω, F, P ) 1. Ω = [0, 1) 2. F is the Borel algebra generated by all the open interval (a, b) [0, 1] 3. The probability measure: the lebesgue measure P ((a, b)) = b a. Define the sigma fields 1. F 1 = σ([0, 1/2), [1/2, 1)) 2. F 2 = σ([0, 1/4), [1/4, 2/4), [2/4, 3/4), [3/4, 1)) 3. F k = σ([i/2 k, (i + 1)/2 k ), i = 0, 1,, 2 k 1) 14

15 Example of conditional expectation Random Variable X : (0, 1) R 1 : X(ω) = f(ω) X k = E(X F k ) X k is F k measurable X k (ω) is a step function: X k (ω) = a i, ω [i/2 k, (i + 1)/2 k ] B X kdp = B XdP For i = 0, 1,, 2 k 1, (i+1)/2 k a i = 2 k i/2 k f(ω)dω 15

16 Martingale Let X( ) = {X(t), t 0} be a right-continuous a stochastic process with left-hand limit and F t be a filtration on a common probability space. X( ) is a martingale if 1. X is adapted to {F t : t 0}. 2. E X(t) < for any t 3. E[X(t + s) F t ] = X(t) for any t, s 0. X( ) is called a sub-martingale if = is replaced by and super-martingale if = is replaced by. 16

17 Martingale In the survival analysis: F t = σ{x(s) : 0 u t}. For t, s 0 E[X(t + s) F t ] = E[X(t + s) X(u), 0 u t], where (X(u), 0 u t) is the history of the process X( ) from 0 to t. 17

18 Example of martingale Y 1, Y 2,... are i.i.d. random variables satisfying Y i = +1 w.p. 1/2 1 w.p. 1/2 Define X(0) = 0, and X(n) = n Y j n = 1, 2, (X(u) : 0 u n) = (X(1),..., X(n)) (Y 1,..., Y n ) F n = σ(x(1),..., X(n)) = σ(y 1,..., Y n ) j=1 18

19 Example of martingale Clearly, E X(n) < for each n. n+k E[X(n + k) F n ] = E Y j X(1),..., X(n) j=1 n+k = E Y j Y 1,..., Y n j=1 = Y Y n + E [Y n Y n+k Y 1,..., Y n ] = Y Y n = X(n) 19

20 Two important properties of martingale Let X be a martingale with respect to a filtration {F t : t 0} then E{X(t) F t } = X(t ), where F t = s<t F s. Let X be a martingale with respect to a filtration {F t : t 0} then E{dX(t) F t } = 0 E{X(t) X(t ϵ) F t ϵ } = X(t ϵ) X(t ϵ) = 0. 20

21 Counting Process A stochastic process N( ) = (N(t) : t 0) is called a counting process if 1. N(0) = 0 2. N(t) <, all t 3. With probability 1, N(t) is a right-continuous step function with jumps of size +1. N i (t), N(t) and N l (t) are all counting process. 21

22 Compensator Process the stochastic process A( ), defined by A(t) def = t 0 Y (u) h(u)du. E(N(t)) = E(A(t)) M(t) = N(t) A(t) is a mean zero process 22

23 Example T EXP (λ) h(t) = λ A(t) = t 0 h(u)y (u)du = λ t Y (u)du = λ min(t, U) 0 M(t) = I(U t)δ λ min(t, U) In general M(t) = I(U t)δ t 0 Y (u)h(u)du Indeed, M(t) is a martingale under independent censoring assumption! 23

24 Martingale M(t) = N(t) t 0 Y (s)h(s)ds and F t = σ{n(u), N C (u), u [0, t]}. The key is to show that E(M(t + s) M(t) F t ) = 0 a.s. Need to show that E[N(t + s) N(t) F t ] = E [ t+s t h(u)y (u)du F t ] Consider the value of Y (t) E [N(t + s) N(t) F t ] 0 if Y (t) = 0 = E [N(t + s) N(t) Y (t) = 1] if Y (t) = 1. 24

25 Martingale Since N(t+s)-N(t) is zero or one, E[N(t + s) N(t) Y (t) = 1] = P [N(t + s) N(t) = 1 Y (t) = 1] = P [t < U t + s, δ = 1 U t] = P [t < T t + s, C > T ]. P [U t] 25

26 Martingale Consider the value of Y (t) E [ t+s = t h(u)y (u)du F t ] 0 if Y (t) = 0 [ ] t+s E h(u)y (u)du Y (t) = 1 if Y (t) = 1. t When Y (t) = 1 E [ ] t+s h(u)y (u)du Y (t) = 1 t = t+s h(u)p [U u U t]du = t t+s t h(u)p (U u)du P (U t) = = t+s t h(u)e [Y (u) Y (t) = 1] du P [t < T t + s, C > T ]. P [U t] 26

27 Martingale In the last step, we argue that t+s t h(u)p (U u)du P (U t) = P [t < T t + s, C > T ] P [U t] It holds if P (u T u + ϵ T u, C u) h(u) = lim, ϵ 0 ϵ which is the noninformative censoring assumption! 27

28 Predictable process A stochastic process X( ) is called predictable with respect to a filtration F t if for very t, X(t) is measurable with respect to F t Any left-continuous adapted process is predictable. N(t) is not predictable since it is right-continuous but Y (t) = I(U t) is a predictable process. 28

29 Doob-Meyer decomposition Suppose that X( ) is a non-negative submartinagle adapted to {F t, t 0}. Then there exists a right-continuous and non-decreasing predictable process A( ) such that E(A(t)) < for all t and M( ) = X( ) A( ) is a martinagle. If A(0) = 0 a.s., then A( ) is unique! The process A( ) is the called the compensator for X( ). t 0 h(u)y (u)du is the unique compensator for the counting process N(t) = I(U t, δ = 1). 29

30 Property of Martingale associated with counting process If M( ) is a martinagle, then M 2 ( ) is a submartingale (why?) By Doob-Meyer decomposition, there is a unique predicable process < M, M > such that M 2 ( ) < M, M > ( ) is a martingale. E(M 2 (t)) = E{< M, M > (t)} If M( ) = N( ) A( ), then < M, M > ( ) = A( ) We can calculate the variance of N( ) A( )! var{n(t) A(t)} = E{A(t)} = E t 0 Y (u)h(u)du = t 0 P (U > u)h(u)du. 30