The Accelerated Gap Times Model
Rob Strawderman BSCB Department Cornell University
[email protected] http://www.bscb.cornell.edu/ Joint work with Edsel Pe˜ na, USC Stat
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Outline • Model & Intensity Formulation • Semiparametric Estimation • Asymptotics & Efficiency • Simulation Results • Bladder Tumor Data • Extensions 2
AGT Model – Motivation Consider single subject with covariates Z experiencing repeated events at times 0 < S1 < S2 < · · · . Define j th gap time as Tj = Sj − Sj−1, where j ≥ 1 & S0 = 0. Assume 0
Given Z, Tj = Vj e−θ0Z, where V1, V2, . . . iid with continuous CDF F0 independent of Z. That is: Subject-level observation is realization of a renewal process, where covariates Z directly “accelerate” or “decelerate” baseline process gap times Vj .
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Let Sn =
Pn υ j=1 Tj & define N (t) = max{n : Sn ≤ t}. Then:
• For any t, h > 0 & with Ft = σ(N υ (u), u ≤ t; Z), P {N υ (t + h) − N υ (t) = 1|Ft} = 0
0
F0(eθ0Z(t + h − SN υ (t))) − F0(eθ0Z(t − SN υ (t))) 0 1 − F0(eθ0Z(t − SN υ (t)))
.
• Dividing LHS, RHS by h and letting h → 0 yields intensity for N υ (·) (uncensored case): λ(t|Z) = λ0
µ
0Z θ e 0 R(t)
¶
0Z θ e 0 ,
d log(1 − F (t)). where R(t) = t − SN υ (t−) and λ0(t) = − dt 0
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AGT Model – Assumptions Assume n subjects, where ith observed over [0, τi] and: †
†
• Data: ({(Ni (u), Yi (u)), u ≥ 0}, Zi), i = 1 . . . n with †
†
– Ni (u) = max{n : Sn ≤ u ∧ τi} (i.e., Ni (u) = Niυ (u ∧ τi)) †
– Yi (u) = I{τi ≥ u} ¯ ¯ υ • Subjects iid, τi ∼ G(·|Zi), τi ⊥ {Ni (u), u ≥ 0}¯¯Zi. ind
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AGT Model – Specification Let Gt denote history of all subjects to time t; then, the † compensator of Ni (t) wrt Gt is † Ai (t) =
Z t 0
λ0
µ
0Z θ e 0 iR
¶
0Z † θ 0 i Y (u) du, (u) e i i
denotes backward recurrence time where Ri(u) = u−S † iN (u−) i
and λ0(·) is “baseline” intensity.
AGT Model: For i = 1 . . . n, †
†
†
Mi (·) = Ni (·) − Ai (·) is a martingale wrt {Gt, t ≥ 0}.
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• New, interesting competitor to modulated renewal process of Cox (1972), which assumes compensator RP (t) = AM i
Z t
0Z † θ λ0(Ri(u)) e 0 i Yi (u) du 0
(cf. Oakes & Cui, 1994). • Useful physical interpretation: covariates directly “shrink” or “expand” gap times. See Lin, Wei, and Ying (LWY; 1998) for marginal model formulation in terms of event times S1 < S2 < · · · (“accelerated mean model”). • ML estimation for AGT model easy for parameterized λ0(·), asymptotics follow from martingale theory. Semiparametric estimation more challenging . . .
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Semiparametric Estimation To motivate EF, start by deriving semiparametric efficient score (SES) for θ p×1: Using Jacod’s point process likelihood, easy to obtain score for θ assuming Λ0 known:
S Q (θ ) =
n Z ∞ X
i=1 0
µ
Zi Q Ri(u)e
θ0Z
i
¶
†
Mi (du)
λ00(u) u + 1 and where Q(u) = λ0(u) † † Mi (·) = Ni (·) −
Z · 0
µ
λ0 e
θ0Z
iR
¶
0Z θ i Y † (u) du. (u) e i i
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SES for θ obtained by subtracting from SQ its orthogonal projection onto score space for Λ0. Omitting details:
Seff(θ ) = where
n Z ∞ X
i=1 0
µ
Q Ri(u)e
θ0Z
i
¶µ
µ
Zi − E Ri(u)e
θ0Z
i
¶¶
†
Mi (du)
• E(·) = [E1(·), . . . , Ep(·)]T, • Ek (u) =
E [Z1k Y1(u|θ )] , k = 1...p E [Y1(u|θ )] †
• Y1(t|θ ) =
N1 (τ1 −) ½ X
I T1j e
j=1
θ0Z
1
¾
½
≥ t +I e
θ0Z
1
·
τ1 − S
†
1N1 (τ1 −)
9
¸
¾
≥t .
Using “calendar ↔ gap” time stochastic integral identities of Pe˜ na, Strawderman, & Hollander (2001 JASA), can write
Seff(θ ) ≡
n Z ∞ X
i=1 0
Q(u) (Zi − E (u)) Mi(du|θ ),
where • Mi(t|θ ) = Ni(t|θ ) − • Ni(t|θ ) =
Z ∞ 0
Z t 0
Yi(v|θ )Λ0(dv)
0Z † θ I{Ri(u)e i ≤ t}Ni (du)
Important: “Gap time” processes Mi(t|θ ), i = 1 . . . n not martingales, so asymptotics for score process Seff(θ , t) cannot be handled via direct application of MCLT. 10
Seff(θ ) involves unknown quantities: • E(u) =
E [Z1Y1(u|θ )] , E [Y1(u|θ )]
λ00(u) u + 1. • Q(u) = λ0(u) To generate useful class of EFs for θ , substitute Pn j=1 Zj Yj (u|θ ) ¯ for E(u); • E(u|θ ) = Pn j=1 Yj (u|θ )
• computable, bounded weight W (·|θ ) for Q(·). 11
We obtain: b S
W (θ ) =
n Z ∞ X
i=1 0
(τi −) n Ni X X
i=1
´
¯ (u|θ ) Ni(du|θ ) W (u|θ ) Zi − E
†
≡
³
j=1
µ
W Tij e
θ0Z
µ ¯ ¶· ¯ ¶¸ 0 ¯ i ¯θ ¯ Tij eθ Zi ¯¯θ . Zi − E
Easy to see: b (θ ) is rank-based EF for θ • S W b (θ ) reduces to weighted linear rank EF of Tsiatis • S W (1990) if at most 1 event per subject.
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Estimating θ b (θ ) non-monotone step function. E.g., not • Typically, S W guaranteed monotone for W (u|θ ) ≡ 1 (logrank). b as a minimizer of kS b (θ )k. Can use • Common: choose θ W deterministic (e.g., simplex method) or stochastic (e.g., simulated annealing, genetic algorithm) optimization. b (θ ) guaranteed monotone with Gehan-type weight • S W
W (u|θ ) = n−1
n X
Yj (u|θ );
j=1
see Fygenson & Ritov (1994). Solution set convex, fast numerical methods via LP methods (cf. LWY 1998).
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Estimating Λ0(·) For all suitable deterministic η(·), “score” for Λ0 is: n Z ∞µ X
i=1 0
η Ri(u)e
θ0Z
i
¶
†
Mi (du).
Mean zero; similarly to before, can show: n Z ∞µ X
¶
Z ∞
n X
0Z † θ η Ri(u)e i Mi (du) ≡ η (v ) Mi(dv|θ ) 0 i=1 0 i=1
where Mi(v|θ ) = Ni(v|θ ) −
Z v 0
Yi(s|θ )Λ0(ds).
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Estimator for Λ0 obtained by setting RHS = 0, solving for b . For arbitrary η(·), solution satisfies Λ 0 n h X
i=1
i b Ni(dv|θ ) − Yi(v|θ )Λ0(dv) = 0.
P b → θ 0, estimate Λ0 via Breslow-type estimator Hence: for θ
Z t Pn b) Ni(dv|θ i=1 b b . Λ0(t|θ ) = Pn b 0 i=1 Yi (v|θ )
b ) − Λ (t) more complicated than usual b (t|θ Asymptotics for Λ 0 0 b (t|θ ) not Breslow estimator; e.g., cannot employ MCLT, Λ 0 differentiable in θ , etc . . .
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b (·|θ ) & non-monotinicity of logrank EF Λ 0 †
With Ni = Ni (τi−), may write logrank EF as b S LR (θ ) ≡
n X
i=1
Z i Ni −
NX i +1 j=1
¶ 0 Z ¯¯ θ b Λ0 Xij e i ¯θ µ
for Xij = Tij , j ≤ Ni & Xi,Ni+1 = τi − Si,Ni . b • Monotonicity of S LR (θ ) fails because
isn’t monotone in θk , k = 1 . . . p.
Zi
b Λ
³
θ 0 Z i |θ X e 0 ij
b (t|θ ) isn’t monotone in θ , k = 1 . . . p. • Why: given t, Λ 0 k b (t|θ ) is monotone in t . . . • However: given θ , Λ 0
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´
∗ ∗ P e Suggests solving S LR (θ |θ ) = 0, where θ → θ 0 and
e S
∗ LR (θ |θ ) ≡
n X
i=1
Z i Ni −
Monotonicity concerns vanish. b from θ LR :
NX i +1 j=1
b Λ
µ
¶
0 Z ¯¯ ∗ θ i ¯θ . 0 Xij e
e But, solution θ LR distinct
∗ −1/2 ) e b b n1/2(θ LR − θ LR ) = Op (1) unless |θ − θ LR | = op (n
Still: useful observation, because suggests
e b • θ LR good starting value for solving SLR (θ ) = 0
• applied iteratively, produces consistent sequence of solub tions asymptotically equivalent to θ LR . 17
Asymptotics Under suitable conditions and for given weight W , •
•
√
L
b − θ ) → N (0, Σ), where Σ = D-1 Σ D-1 , Σ is score n(θ 0 0 0 0 variance, & D depends on λ0(·), λ0(·).
√
b )−Λ (·)) ⇒ G, where G a Gaussian process with b (·|θ n(Λ 0 0 covariance function
Var
ÃZ
t∧s M1 (du|θ 0 ) 0
E[Y1(u|θ 0)]
!
+ AT(t) Σ A(s)
where A(·) depends on λ0(·), λ00(·). Proof: martingale-type calculations, empirical processes 18
Efficiency Considerations for θ • Asymptotically optimal choice: W = Q for λ00(u) u + 1. Q(u) = λ0(u)
• For constant c > 0, solving ODE Q(u) = c for λ0(·) yields λ0(u) = kuc−1 for some k > 0. Implies W (u|θ ) ≡ 1 (i.e., logrank-type EF) optimal when λ0(·) in Weibull family. 0Z − θ • Results parallel AFT model. If T = V e 0 for V ∼ Weibull, then log T = log V − θ 00Z for log V ∼ Extreme
Value. Well-known that Tsiatis’s logrank EF optimal in this case (e.g., Ritov, 1990).
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Variance Estimation • Variances depend on A(·), D (i.e., on λ0(·), λ00(·)). • In similar rank-based estimation problems, LWY (1998) propose simulation-oriented approach for empirical variance estimation. Huang (2002) proposes “inverse numerb. ical differentiation” for estimating variance of θ
• Devised new approach combining small-scale simulation b and and regression to approximate A(s), D; variance for θ b (·) then directly estimated from asymptotic formulas. Λ 0
• Less computational effort than LWY; less bias, comparab ). ble variance to Huang estimator for Var(θ 20
b) Algorithm for Var(θ
For large n and θ near θ 0, asymptotic linearity of score implies . b b (θ ) = S SW (θ 0) + D (θ − θ 0) + error W b , above implies Hence, for θ ∗ near θ b S W
b (θ ∗ ) − S
³ ´ . ∗ b b W (θ ) = D θ − θ + other error
i.e., D can be viewed as slope of regression of multivariate b ) on “covariate” θ ∗ − θ b. b (θ ∗ ) − S b (θ “response” S W W Can interpret A(·) similarly . . .
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b 1. Simulate several θ ∗ in “reasonable” neighborhood of θ b) (e.g., 50 simulated values, normally distributed about θ c 2. Approximate D as indicated; call result D
3. Estimate Σ0 via P ⊗2 n b Z n i⊗2 Zj Yj (u|θ ) h X ∞ 1 j=1 b) b) b = ¯ (u|θ − Σ E N (du| θ P 0 i n b n i=1 0 j=1 Yj (u|θ ) b =D c-1 Σ b D c-1 4. Compute Σ 0
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Simulation Results
• Gap times T ∼ V e−0.5Z , τ ∼ Uniform(0, 3.5) • Exponential: V ∼ Exp(1), Z ∼ Bernoulli(0.5) • Gamma: V ∼ Gamma(0.75,0.75), Z ∼ Normal(0,1) b • Logrank weight, minimized kS LR (θ )k via simplex method b as starting value (computed via LP methods). with θ G
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Exponential n=50 n = 100
Gamma n=50 n = 100
b) Bias(θ
-0.004
0.002
0.005
-0.002
0.211
0.144
0.095
0.065
d Bias(SD)
-0.006
-0.003
0.000
0.000
-0.020
-0.009
-0.003
-0.002
d SD of (SD)
0.025
0.012
0.018
0.009
0.036
0.014
0.018
0.009
b) Emp SD(θ d Bias(Huang SD) d SD of (Huang SD)
Mean # events
2.3
3.7
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Exponential n=50 n = 100
Gamma n=50 n = 100
b (1)) Bias(Λ 0
-0.001
-0.006
-0.009
-0.004
b (2)) Bias(Λ 0
-0.008
-0.001
0.007
0.006
b (1)) Emp SE(Λ 0
0.172
0.117
0.185
0.125
d Bias(SD)
-0.005
-0.000
-0.005
0.001
b (2)) Emp SE(Λ 0
0.389
0.269
0.525
0.374
-0.016
-0.008
-0.001
-0.009
d Bias(SD)
E[Y1(1|0.5)]
0.81
0.51
E[Y1(2|0.5)]
0.19
0.07 25
VA Bladder Tumor Data (Byar, 1980)
• Analyzed in several places, including LWY (1998), who 0Z θ υ use it to illustrate MAM model: E[N (t)|Z] = µ0(te 0 ).
• Treatment: Placebo (1), Thiotepa (0) • Covariates: Initial # (1-8), largest tumor size (1 - 8 cm) • 48 placebo patients with 87 recurrences in placebo group; 38 patients with 45 recurrences in thiotepa group. Number of recurrences per patient range from 0 - 9.
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EF
LR
Ghn
Cov
AGT Model θb Est SE
MAM Model θb Est SE
Treatment
0.442
0.286
0.542
0.312
Initial Num
0.258†
0.068
0.204†
0.066
Initial Size
-0.010
0.097
-0.038
0.084
Treatment
0.433
0.257
0.657
0.314
Initial Num
0.207†
0.064
0.218†
0.086
Initial Size
-0.008
0.090
-0.022
0.101
† Highly significant at 5% level
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• AGT model suggests recurrence times are expanded on treatment (0 = thiotepa), reduced as number of initial 0Z θ υ tumors increases; MAM model E[N (t)|Z] = µ0(te 0 ) suggests mean # of recurrences exhibit same pattern. • Model correspondence? i.e., when does E[N υ (t)|Z] = µ0
0 (teθ0Z)
=
Z t 0
λ0
µ
0 eθ0ZR(u)
¶
0
eθ0Z du?
Evidently: if λ0(u) = λ0, in which case µ0(·) = Λ0(·). That is: AGT, MAM models coincide if baseline gap times Vij0 s IID exponential, covariates Z time-independent; otherwise, models distinct. • Results here, in LWY (1998) do not rule out possibility of exponential gap times for bladder tumor data. 28
Estimated Cumulative Baseline Intensity 3 Cum Haz 95% CI 95% EP CB
2.5
Λ0(t)
2
1.5
1
0.5
0
0
10
20
30 Time (in months)
40
50
60
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• MAM model more robust to model assumptions; e.g., valid for correlated and even non-iid gap times. Can summarize broad trends in total event counts, but no more; also, requires strong independent censoring conditions.
• AGT model can be extended to case where baseline gap times are correlated, have same marginal distributions (i.e., frailty-type settings); seems harder to move beyond because of “baseline” distributional assumptions.
• With AGT model, can additionally estimate recurrence time distribution, median recurrence times, etc . . . ; main expense is stronger modeling assumptions.
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Estimated Recurrence Time Distributions by Treatment (At Mean Number=2.5, Size = 2) 1 t
θ z
0.9
P(T>t|Z=z) = e−Λ0(e
0.8
Median(Placebo)=9.45
t)
Median(Thiotepa)=14.65
P( T > t | Z = z)
0.7 0.6 0.5 Thiotepa Placebo
0.4 0.3 0.2 0.1 0
0
10
20 30 40 Recurrence time T (in months)
50
60
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Some Extensions • Kalbfleisch & Prentice (2002, §9.4.3) suggest tumor recurrence intensity may depend on past recurrence history. Extension of AGT model to time-dependent covariates would (i) enable more in-depth investigation (eg, effect of recurrence pattern and timing); (ii) improve ability to predict subject-specific recurrence trajectories; and (iii) further relaxation of censoring assumptions. • Frailty-type intensity models for correlated gap times; what is appropriate EF? Estimator for Λ0? Also: is EF for θ useful under weaker marginal model assumptions? • Dependent censoring/stopping (e.g., observation ends at S ∗ ∧ τ , where S ∗ correlated with N υ (·)). 32
