Empirical Processes & Survival Analysis. The Functional Delta Method

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1 STAT/BMI 741 University of Wisconsin-Madison Empirical Processes & Survival Analysis Lecture 3 The Functional Delta Method Lu Mao lmao@biostat.wisc.edu 3-1

2 Objectives By the end of this lecture, you will learn the intuitive idea of functional delta method see various examples of functional derivatives be able to apply the functional delta method to survival analysis problems such as estimation of the cumulative incidence function of competing risks (Gray s estimator and tests) The Functional Delta Method 3-2

3 Contents 1.1 von-mises Calculus 1.2 Hadamard Differentiable Functions 1.3 Application: The Cumulative Incidence of Competing Risks The Functional Delta Method 3-3

4 Smooth Functionals Consider a parameter that is defined as a functional of the underlying distribution P : θ(p ). Examples: Mean: θ(p ) = P X Variance: θ(p ) = P (X P X) 2 Quantiles: θ(f ) = inf{ξ : F (ξ) p} The Functional Delta Method 3-4

5 Smooth Functionals The natural estimator is θ(p n ): Examples: Sample mean: θ(p n ) = P n X Sample variance: θ(p n ) = P n (X P n X) 2 Sample quantiles: θ( F n ) = inf{ξ : F n (ξ) p} The Functional Delta Method 3-5

6 Smooth Functionals How to derive the asymptotic distribution of θ(p n )? Note n(θ(pn ) θ(p )) = ) n( θ(p + n 1/2 G n ) θ(p ). (3.1) Now, view G n as if it were a fixed quantity H (not an entirely strange thing to do since G is stabilized over a Donsker class). Then the right hand side (3.1) can be written as for some linear operator θ. θ(p + n 1/2 H) θ(p ) n 1/2 θ(h), The Functional Delta Method 3-6

7 Smooth Functionals The linear operator θ can be computed by θ[h] = θ(p + ɛh). ɛ ɛ= A linear functional θ on the signed measures H can always be represented by θ[h] = HΦ for some function Φ. The Functional Delta Method 3-7

8 Smooth Functionals So n(θ(pn ) θ(p )) = θ[g n ] = G n Φ + o P (1). The influence function usually presents itself during the calculation of θ. The Functional Delta Method 3-8

9 Smooth Functionals Example 1: θ(p ) = P X. θ[h] = (P + ɛh)x = HX. ɛ ɛ= Hence n(pn X P X) = G n X + o p (1). This is actually a quite trivial example where the remainder is obviously zero. The Functional Delta Method 3-9

10 Smooth Functionals Example 2: θ(p ) = P (X P X) 2 =: σ 2. θ[h] = ( (P + ɛh) X (P + ɛh)x ɛ ɛ= = H(X P X) 2 2HP (X P X) = H(X P X) 2. ) 2 Hence n ( P n (X P n X) 2 σ 2) = G n (X P X) 2 + o p (1). The Functional Delta Method 3-1

11 Smooth Functionals Example 3: θ(f ) = F 1 (p) =: ξ p. Re-define θ(p ) such that P 1(X θ(p )) = p. Hence (P + ɛh)1(x θ(p + ɛh)) =. ɛ ɛ= By the chain rule, H1(X θ(p )) + f(θ(p )) θ[h] =. So, θ[h] = H1(X ξ p). f(ξ p ) The Functional Delta Method 3-11

12 Smooth Functionals So n(θ( Fn ) ξ p ) = G n 1(X ξ p ) f(ξ p ) The same result as derived in Example o P (1). The Functional Delta Method 3-12

13 Contents 1.1 von-mises Calculus 1.2 Hadamard Differentiable Functions 1.3 Application: The Cumulative Incidence of Competing Risks The Functional Delta Method 3-13

14 Hadamard Differentiable Functions In the previous section, we have treated G n as if it were fixed. If G n on a Donsker class, then it eventually ranges over a compact set. So for first-order approximation, we need the stronger condition that for every compact set K θ(p + ɛh) θ(p ) sup ɛ H K θ(h). If there exists such a linear function θ, then θ(p ) is said to be Hadamard differentiable at P with derivative θ. The Functional Delta Method 3-14

15 Hadamard Differentiable Functions More generally, let φ(η) be a Hadamard differentiable function on a (functional) parameter η, and let η n be an estimator of η, such that η n η is tight: n( ηn η ) = G n Ψ + o P (1). Let φ be the derivative (a linear operator) of φ at η. Then n(φ( ηn ) φ(η )) = G n φ[ψ] + op (1). The Functional Delta Method 3-15

16 Hadamard Differentiable Functions Many functions are Hadamard differentiable. We omit the technical proofs. Example: φ(f, G) = F dg. φ F,G [h 1, h 2 ] = (F + ɛh 1 )d(g + ɛh 2 ) ɛ ɛ= = h 1 dg + F dh 2. Application: Mann-Whitnet statistic. The Functional Delta Method 3-16

17 Hadamard Differentiable Functions Example 1.3 (Nelsen-Aalen Estimator, cont d) We have shown that the Nelsen-Aalen estimator takes the form Λ(t) = t P n dn(s) P n Y (s) The Functional Delta Method 1-17

18 Hadamard Differentiable Functions Example 1.3 (Nelsen-Aalen Estimator, cont d) Now, view Λ as a function φ of P, where Obviously, and φ(p ) = P dn(s) P Y (s). Λ( ) = φ(p n ), Λ ( ) = φ(p ). The Functional Delta Method 1-18

19 Hadamard Differentiable Functions Example 1.3 (Nelsen-Aalen Estimator, cont d) The functional derivative can be calculated by φ[h] = φ(p + ɛh) ɛ ɛ= = (P + ɛh)dn(s) ɛ ɛ= (P + ɛh)y (s) { = H = H = H dn(s) π(s) dn(s) Y (s)dλ (s) π(s) dm Λ (s). π(s) The same as derived in Example 1.3 of 1. Y (s)p dn(s) } π(s) 2 The Functional Delta Method 1-19

20 Kaplan-Meier as Product Limit Functional We consider the product limit functional mapping the space of cadlag functions on [, τ] into itself: φ(a)(t) = t (1 + da(s)) = lim s= s i s i 1 {1 + A(s i ) A(s i 1 )}, where the second equality is the definition and the limit is over partitions = s < s 1 < < s m = t. with maximum separation decreasing to zero. i The Functional Delta Method 1-2

21 Kaplan-Meier as Product Limit Functional To derive the functional derivative of φ(a), observe φ A (H)(t) = φ(a + ɛh)(t) ɛ ɛ= = lim {1 + A(s i ) A(s i 1 ) + ɛ(h(s i ) H(s i 1 ))} ɛ ɛ= s i s i 1 i = lim (H(s i ) H(s i 1 ) + A(s j ) A(s j 1 )} s i s i 1 j i{1 i = lim {1 + A(s i ) A(s i 1 )} s i s i 1 i i = φ(a)(t) t where A(s) = A(s) A(s ). {1 + A(s)} 1 dh(s), H(s i ) H(s i 1 ) 1 + A(s i ) A(s i 1 ) The Functional Delta Method 1-21

22 Kaplan-Meier as Product Limit Functional Note that the Kaplan-Meier estimator can be expressed as a product limit functional of Nelsen-Aalen estimator: Thus, Ŝ n (t) = φ( Λ n )(t) = n( Ŝ n S )(t) = G n S (t) t = G n S (t) t t (1 d Λ n (s)). s= 1 (1 Λ (s))π(s) dm Λ (s) + o P (1) 1 π(s) dm Λ (s) + o P (1). (when Λ is continuous.) The Functional Delta Method 1-22

23 Contents 1.1 von-mises Calculus 1.2 Hadamard Differentiable Functions 1.3 Application: The Cumulative Incidence of Competing Risks The Functional Delta Method 1-23

24 Competing Risks Competing risks data arise when each subject can experience one and only one of several competing causes of failure. Examples: death from cancer vs death related to treatment (e.g., chemotherapy). The Functional Delta Method 1-24

25 Competing Risks Competing risks data (T, D), T is failure time and D = 1,, J is the cause of failure. We can image (T, D) as arising from a vector of J latent competing failure times, T 1,, T J such that T = min{ T 1,, T J }, and D is the subscript of the first event time among the T j. The Functional Delta Method 1-25

26 Competing Risks However, the joint distribution of ( T 1,, T J ) cannot be identified from the observed data (T, D). Even independence is not identifiable. To make inference on the latent event times one has to make strong and usually unrealistic assumptions such as independence. The alternative is to stick with identifiable quantities based on (T, D). The Functional Delta Method 1-26

27 Competing Risks A popular quantity that is identifiable is the cause-specific hazard: dλ j (t) = Pr(t T < t + dt, D = j T t), i.e., the instantaneous rate for the jth cause of failure given survival to that point. The cause-specific hazard can be estimated by the Nelsen-Aalen estimator treating other causes as censoring. The cause-specific hazard Λ j reduces to the net hazard of T j if all other T k, k j, are independent of T j. The Functional Delta Method 1-27

28 Competing Risks Another identifiable quantity that is often of interest is the sub-distribution: F j (t) = Pr(T t, D = j), i.e., the cumulative incidence of the jth cause of failure in the presence of other causes. The sub-distribution F j (t) is not a functional of the cause-specific hazard Λ j (t); in particular, one cannot use the naive Kaplan-Meier curve (product limit of the Nelsen-Aalen estimator for the cause-specific hazard) to estimate the sub-distribution. The Functional Delta Method 1-28

29 Competing Risks Observe that df j (t) = Pr(t T < t + dt, D = j) = Pr(T t)pr(t T < t + dt, D = j T t) = S(t )dλ j (t), where S(t) = Pr(T > t). So F j (t) = t S(s )dλ j (s). The Functional Delta Method 1-29

30 Competing Risks Hence we can estimate the sub-distribution by F jn (t) = t Ŝ n (s )d Λ jn (s), where Ŝn is the Kaplan-Meier estimator for the overall survival function and Λ jn is the Nelsen-Aalen estimator fort the cause-specific hazard function of the jth cause of failure. The Functional Delta Method 1-3

31 Competing Risks Specifically, let C be the independent censoring time, then the observed data are {T I(T C), DI(T C), I(T C)}. The observed data can also be represented by N(t) = I(T t C), Y (t) = I(T C t), and N j (t) = I(T t C, D = j), j = 1,, J. Denote π(s) = PY (s), M Λ (t) = N(t) t Y (s)dλ(s), M Λj (t) = N j (t) t Y (s)dλ j(s), where Λ is the hazard function for T. The Functional Delta Method 1-31

32 Competing Risks We focus on the estimation of the cumulative incidence of the first cause of failure. We know that and n( Ŝ n S )(t) = G n S (t) t n( Λ1n Λ 1 )(t) = G n t 1 π(s) dm Λ (s) + o P (1), 1 π(s) dm Λ 1 (s) + o P (1) The Functional Delta Method 1-32

33 Competing Risks Since F 1n = φ(ŝn, Λ 1n ) and F 1 = φ(s, Λ 1 ), where φ(s, Λ 1 ) = We know that φ S,Λ 1 [H 1, H 2 ](t) = t S(s )dλ 1 (s). H 1 (s )dλ 1 (s) + t S (s )dh 2 (s). The Functional Delta Method 1-33

34 Competing Risks So, ( ) [ 1 n F1n F 1 (t) = G n φs,λ 1 S ( ) π(s) dm Λ (s), 1 ] π(s) dm Λ 1 (s) (t) + o P (1) t = G n S (u ) u 1 π(s) dm Λ (s)dλ 1 (u) t 1 + G n S (s ) π(s) dm Λ 1 (s) + o P (1). The Functional Delta Method 1-34

35 Competing Risks t To simplify the first term on the right, use integration by parts, u 1 S (u ) π(s) dm Λ (s)dλ 1 (u) = t u = F 1 (u) = t t 1 π(s) dm Λ (s)df 1 (u) 1 π(s) dm Λ (s) t u= u F 1 (s) π(s) dm Λ (s) F 1 (t) F 1 (s) dm Λ (s) π(s) The Functional Delta Method 1-35

36 Competing Risks Therefore, to conclude, ( ) t F 1 (t) F 1 (s) n F1n F 1 (t) = G n dm Λ (s) π(s) t 1 + G n S (s ) π(s) dm Λ 1 (s) + o P (1). The Functional Delta Method 1-36

37 Concluding Remarks The functional delta method is a powerful tool in semiparametric inference, particularly survival analysis. We have provided a few simple examples in this lecture. For a more formal treatment of the functional delta method and more examples refer to van der Vaart (1998, chap 2) and Andersen et al. (1993, II.8). The Functional Delta Method 1-37

38 References - Andersen, P. K., Borgan, O., Gill, R. D., & Keiding, N. (1993). Statistical models based on counting processes. Springer Science & Business Media. - van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press. The Functional Delta Method 1-38

STAT Sample Problem: General Asymptotic Results

STAT Sample Problem: General Asymptotic Results STAT331 1-Sample Problem: General Asymptotic Results In this unit we will consider the 1-sample problem and prove the consistency and asymptotic normality of the Nelson-Aalen estimator of the cumulative