An augmented inverse probability weighted survival function estimator

Transcription

1 An augmented inverse probability weighted survival function estimator Sundarraman Subramanian & Dipankar Bandyopadhyay Abstract We analyze an augmented inverse probability of non-missingness weighted estimator of a survival function for a missing censoring indicator model, in the absence and presence of left truncation. The estimator improves upon its precursor but is still not the best in terms of achieving minimal asymptotic variance. KEY WORDS: Asymptotic variance; Dikta semiparametric estimator; Empirical estimator; Gaussian process; Gradient vector; Missing at random. 1 Introduction In this article, we focus on a missing censoring indicator MCI) model, with observed data {X i, ξ i, σ i ) 1 i n }, where X i = mint i, C i ), T is a lifetime of interest, C is an independent censoring variable, ξ i = 1 when δ i = IT i C i ) is observed and is otherwise, and σ i = ξ i δ i. The censoring indicators δ i are assumed to be missing at random MAR), which implies that P ξ = 1 X = x, δ = d) = P ξ = 1 X = x) = πx) van der Laan and McKeague, 1998). Writing px) = P δ = 1 X = x), MAR means that ξ and δ are conditionally independent given X: P σ = 1 X = x) = πx)px). We investigate semiparametric estimation of St), the survival function of T, using an augmented inverse probability weighted AIPW) scheme. Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 712, United States Department of Biostatistics, Bioinformatics and Epidemiology, Medical University of South Carolina, Charleston, SC 29425, United States 1

2 Inverse probability weighting IPW) has gained much currency of late. Horvitz and Thompson 1952) introduced IPW in the context of sample survey, while Koul, Susarla, and van Ryzin 1981) first employed the weighting scheme for censored regression. More recently, Robins and Rotnitzky 1992), Satten and Datta 21, 24), and Rotnitzky and Robins 25), among others, have also introduced IPW-based analysis in survival and related settings. However, the IPW estimators are not consistent under misspecification of the weighting probability, leading to the investigation of AIPW estimators Scharfstein, Rotnitzky and Robins 1999). The key element of such AIPW estimators is that they are doubly robust in the sense that they remain consistent if any one of px) and πx) is specified correctly. Robins, Rotnitzky and van der Laan 2), among others, have studied double robust estimators. See also Tsiatis 26) and van der Laan and Robins 23). 2 Estimation We first introduce some preliminary notation and present a brief review of two semiparametric estimators of St) when there are MCIs. These are the Dikta estimator Subramanian, 24a) and the IPW estimator. The latter was investigated recently for a left truncated version of the MCI model Subramanian and Bandyopadhyay, 28). 2.1 Preliminaries Let Ht) denote the distribution function of X and Ĥt) denote its empirical estimator. We focus on estimation over [, τ], where Hτ) < 1. Let Qt) = P X t, δ = 1) denote the subdistribution function corresponding to uncensored failures. Its empirical estimator, however, is inapplicable when there are MCIs. To address this, specify a model for pt). 2

3 That is, take pt) = pt, θ), where θ is a k dimensional unknown Euclidean parameter. The Dikta semiparametric estimator of Qt) is given by ˆQ D t) = ˆps)dĤs), 1) where ˆps) = ps, ˆθ), and ˆθ is the maximum likelihood estimate MLE) of θ. The MLE ˆθ can be obtained by maximizing a likelihood based on the complete cases ξ i = 1, see, for example, Subramanian 24a). Let θ denote the true value of θ and define p s) = ps, θ ). For r = 1,..., k, write p r s, θ ) = ps, θ)/ θ r θ=θ and let P s) = [p 1 s, θ ),..., p k s, θ )] T denote the gradient vector evaluated at θ. Let I = E[πX)P X)P T X)/p X)1 p X)))]. Also, let αs, t) = P T s)i 1 P t), and define Bt) 1 n B i t) = 1 n ξ i δ i p X i )) p X i )1 p X i )) αt, X i). 2) Let M be such that Ms)dHs) <. From Lemma 3.5 of Dikta 1998), we have the following representation, assuming that the model for pt) is correctly specified: n 1/2 {ˆpt) p t)} = n 1/2 Bt) + Rn t), 3) where, uniformly for t τ, the term n 1/2 R n t) is bounded above by Mt) ˆθ r θ,r )ˆθ s θ,s ) = Mt) O p n 1/2 ) + o p n 1/2 ) ) 2 = Mt)Op n 1 ). 1 r,s k Under the regularity conditions given by Dikta 1998), which we denote henceforth by R, it can be shown that Subramanian, 24a) n 1/2 ˆQ D t) Qt)) is asymptotically linear with influence function given by At) + Bt), where At) = p X)IX t) Qt), 4) Bt) = Bs)dHs) = ξ δ p X)) p X)1 p X)) αs, X)dHs). 5) 3

4 It follows that the process n 1/2 ˆQ D t) Qt)) converges weakly to a zero-mean Gaussian process whose variance at t [the second moment of At)+Bt)], the benchmark for comparing competing estimators of Qt) in this article, is given by V D t) = p s)dqs) Q 2 t) + αu, v)dhu)dhv). 6) Alternatively, specify πt) = πt, γ) parametrically, where γ is an unknown m dimensional parameter. Set ˆπs) = πs, ˆγ) and ˆγ is the MLE of γ. Let Ŵ t) denote the empirical estimator of W t) = P X t, ξ = 1, σ = 1). From an IPW representation for Qt) e.g., Subramanian, 26), the IPW estimator takes the form ˆQ I t) = 1 dŵ s). 7) ˆπs) Denote the gradient vector for this scenario by P s, θ), and by P s) when it is evaluated at the true value γ. Write βs, t) = P T s)j 1 E[ P X) P T X)/π X)1 π X)))]. Define P t), where the information matrix J = V t) 1 n V i t) = 1 n ξ i π X i )) π X i )1 π X i ) βt, X i). 8) Analogous to Eq. 3), assuming that the model for πt) is correctly specified, we have that n 1/2 ˆπt) π t)) = n 1/2 V t) + Rn t). 9) Assuming that πt) is bounded away from, uniformly for t [, τ], Eq. 8) and Eq. 9) and some basic analysis imply that n 1/2 ˆQ I t) Qt)) is asymptotically linear with influence function given by Ăt) + Bt), where IX t, ξ = 1, σ = 1) Ăt) = Qt), 1) π X) V s) Bt) = π s) dqs) = ξ π X)) t βs, X) dqs). 11) π X)1 π X)) π s) 4

5 It follows that the process n 1/2 ˆQ I t) Qt)) converges weakly to a zero-mean Gaussian process whose variance at t [the second moment of Ăt) + Bt)] is given by V I t) = 1 π s) dqs) Q2 t) βu, v) dqu)dqv). 12) π u)π v) Furthermore, it can be shown that V I t) V D t), see, for example, Subramanian and Bandyopadhyay 28), where an analogous result is derived for a left truncated version. Both estimators ˆQ D t) and ˆQ I t) work under the tacit assumption that the parametric models for pt) and πt) respectively are correctly specified, and are inconsistent when the putative models are ill-specified. The AIPW estimator of Qt) that we analyze in the next subsection is a double robust estimator Scharfstein et. al., 1999, Tsiatis, 26), in that it retains consistency when either or both of pt) and πt) is specified correctly. 2.2 AIPW subdistribution function estimator The AIPW estimator of Qt) takes the form e.g., Tsiatis, 26) ˆQ AI t) = 1 n { ξi δ i ˆπX i ) + 1 ξ ) } i ˆpX i ) IX i t). 13) ˆπX i ) We analyze ˆQ AI t) for three different scenarios. Scenario 1 represents the case only pt) correctly specified, Scenario 2 the case only πt) correctly specified, and Scenario 3 represents both pt) and πt) correctly specified. Clearly, ˆQAI t) = ˆQ D t) + T n t), where T n t) = 1 n { } ξi ˆπX i ) δ i ˆpX i )) IX i t), 14) and ˆQ D t) is defined by Eq. 1). We analyze this version of ˆQ AI t) for Scenario 1. On the other hand, we can also write ˆQ AI t) = ˆQ I t) + T n t), where ˆQ I t) is defined by Eq. 7) and T n t) = 1 n 1 ξ ) i ˆpX i )IX i t), 15) ˆπX i ) 5

6 which is the version that we will analyze for Scenario 2. We now state some basic assumptions, introduced by Kouassi and Singh 1997). We denote them henceforth by RP. When the model for pt) is misspecified Scenario 2), we assume that there exists θ IR k such that, with p t) = pt, θ ), the following hold: E ˆθ θ ) = o n 1/2), 16) ˆθ θ = O p n 1/2 ), 17) p t) p t) <. 18) Eq. 16) and Eq. 17) are introduced to ensure that the MLEs ˆθ and ˆpt) converge at the standard n 1/2 rates even when the parametric model is misspecified. Eq. 18) requires that the bias of ˆpt) is bounded when pt, θ) does not hold. Analogous conditions for the MLEs ˆγ, ˆπt), and π t) = πt, γ ) are also assumed denoted by RPI henceforth). Also, it will be assumed that ˆpt) P p t) and ˆπt) P π t) uniformly for t [, τ]. The last conditions of uniformity which are also included in RP or RPI) can be enforced through appropriate boundedness assumptions, see Eq. 3) and the equation following it. For Scenario 1, ˆpt) P p t) and ˆQ D t) P Qt), uniformly for t [, τ], see Eq. 3) or Dikta 1998). We can write [cf. Eq. 14)] T n t) = T n,1 t) + T n,2 t) + T n,3 t) where { ξi T n,1 t) = 1 } n π X i ) δ i p X i )) IX i t) 19) T n,2 t) = 1 { } ξi n π X i ) ˆpX i) p X i )) IX i t) 2) T n,3 t) = 1 { } ξi ˆπX i ) π X i )) δ n ˆπX i )π i ˆpX i )) IX i t). 21) X i ) By the strong law large numbers T n,1 t) a.s.. Next, T n,2 t) = O p sup t τ ˆpt) p t) ), 6

7 and therefore T n,2 t) P. Also, T n,3 t) = O p sup t τ ˆπt) π t) ) P. Therefore, when only pt) is specified correctly, it is clear that ˆQ AI t) = ˆQ D t)+t n t) retains consistency. ) ) Note, however, that n 1/2 ˆQAI t) Qt) and n 1/2 ˆQD t) Qt) do not have the same limiting distribution, since n 1/2 T n t) is not o p 1), see Theorem 1. For Scenario 2, we can show that ˆπt) P π t) and ˆQ I t) P Qt), uniformly for t [, τ]. We can write [cf. Eq. 15)] T n t) = T n,1 t) + T n,2 t) + T n,3 t) where T n,1 t) = 1 n T n,2 t) = 1 n T n,3 t) = 1 n 1 ξ i π X i ) { ξi ˆπX i ) π X i )) ˆπX i )π X i ) 1 ξ i π X i ) ) p X i )IX i t) 22) } ˆpX i ) IX i t) 23) ) ˆpX i ) p X i )) IX i t). 24) As before, we can show that T n t). P Therefore, for this case also, ˆQ AI t) = ˆQ I t) + T n t) ) ) retains consistency. Again, it may be noted that n 1/2 ˆQAI t) Qt) and n 1/2 ˆQI t) Qt) do not have the same limiting distribution, since n 1/2 Tn t) is not o p 1), see Theorem 2. Let πt) = 1 π t)/π t), which equals when πt) is correctly specified. Along with Eq. 4) and Eq. 5), we also define two other quantities Ct) = { } ξ π X) δ p X)) IX t), 25) Dt) = ξ δ p X)) p X)1 p X)) as well as the following variance function V 1 t): π s) αs, X)dHs), 26) π s) V 1 t) = p s)dqs) Q 2 t) + + πu)αu, v)dhu)dhv) π u) π u)) 2 1 p u))dqu). 27) The following theorem describes the asymptotic distribution of ˆQ AI t) for Scenario 1. 7

8 Theorem 1 Suppose that the regularity conditions R and RPI hold and that π t) is bounded ) away from, uniformly for t [, τ]. The process n 1/2 ˆQAI t) Qt) converges weakly in D[, τ] to a zero-mean Gaussian process Z 1 with variance function V 1 t) defined by Eq. 27). Proof We know that n 1/2 ˆQ D t) Qt)) is asymptotically linear with influence function At) + Bt). From Eq. 19), n 1/2 T n,1 t) is asymptotically linear with influence function Ct). Let ˆ W t) denote the empirical estimator of W t) = P X t, ξ = 1). For Eq. 2), analogous to calculations in Subramanian and Bean 28), we can show that, uniformly for t [, τ], T n,2 t) = ˆps) p s) d ˆ W s) = π s) ˆps) p s) d π s) W s) + o p n 1/2 ). It readily follows, from Eq. 3) and then Eq. 2), that n 1/2 T n,2 t) is also asymptotically linear with influence function Dt). Next, note that dw s) = p s)d W s). It can be shown that, uniformly for t [, τ], T n,3 t) defined by Eq. 21) is asymptotically negligible: T n,3 t) = = ˆπs) π s) ˆπs)π s) {dŵ s) ˆps)d ˆ W s) } ˆπs) π s) { dw s) p π s)) 2 s)d W s) } + o p n 1/2 ) = o p n 1/2 ). It follows that n 1/2 ˆQ AI t) Qt)) is asymptotically linear with influence function At) + Bt)+Ct)+Dt) [cf. Eqs. 4), 5), 25), and 26)]. After some routine variance calculations, it can be shown that the asymptotic variance is given by V 1 t) defined by Eq. 27). We next focus on Scenario 2. Let d p t) = p t) p t), which is when pt) is correctly specified. Along with Eq. 1) and Eq. 11), we also define two other quantities: Ct) = Dt) = 1 ξ ) p X)IX t) 28) π X) ξ π X) π X)1 π X)) 8 p s) βs, X)dHs), 29) π s)

9 as well as the following variance function V 2 t): V 2 t) = 1 π u) dqu) Q2 t) 1 π u) p u)dhu) π u) d p u) d p v) βu, v)dhv)dhu). 3) π u) π v) The following theorem describes the asymptotic distribution of ˆQ AI t) for Scenario 2. Theorem 2 Suppose that the regularity conditions R and RP hold and that π t) is bounded ) away from, uniformly for t [, τ]. The process n 1/2 ˆQAI t) Qt) converges weakly in D[, τ] to a zero-mean Gaussian process Z 2 with variance function V 2 t) defined by Eq. 3). Proof We know that n 1/2 ˆQ I t) Qt)) is asymptotically linear with influence function Ăt) + Bt). From Eq. 22), n 1/2 Tn,1 t) is asymptotically linear with influence function Ct). For T n,2 t) defined by Eq. 23), we can show that, uniformly for t [, τ], T n,2 t) = ˆπs) π s)) ˆps)d ˆ W s) = ˆπs)π s) ˆπs) π s)) p s)d πs) W s) + o 2 p n 1/2 ). It follows, from Eq. 9) and then Eq. 8), that n 1/2 Tn,2 t) is also asymptotically linear with influence function Dt). Furthermore, it can be shown that T n,3 t), defined by Eq. 24) is o p n 1/2 ). It follows that n 1/2 ˆQ AI t) Qt)) is asymptotically linear with influence function Ăt)+ Bt)+ Ct)+ Dt) [cf. Eqs. 1), 11), 28), and 29)]. After some variance calculations, it can be shown that the asymptotic variance is given by V 2 t) defined by Eq. 3). It is difficult to compare the variances V 1 t) and V 2 t) with V D t) [Eq. 6)], or for that matter V I t) [Eq. 12)]. However, for Scenario 3, a definitive conclusion can be made. Define V t) = Qt)1 Qt)) + 1 p u)) 1 π u) π u) dqu). 31) The following theorem describes the asymptotic distribution of ˆQ AI t) for Scenario 3. 9

10 Theorem 3 Suppose that the regularity condition R holds and that π t) is bounded away ) from, uniformly for t [, τ]. The process n 1/2 ˆQAI t) Qt) converges weakly in D[, τ] to a zero-mean Gaussian process Z with variance function V t) defined by Eq. 31). Proof Follow proof of Th. 1 with π t) = π t), or follow proof of Th. 2 with p t) = p t). In each case, the asymptotic variance reduces to V t), since πt) = and d p t) =. Note that when π and p are both correctly specified Scenario 3), the variance of the AIPW estimator equals the information bound for estimating Qt) Subramanian 24a), which was in fact derived from the efficient influence function for estimating St) van der Laan and McKeague, 1998). However, V t) V D t), see Subramanian 24a). In particular, we have the order relation V D t) V t) V I t). Thus, although the AIPW estimator ˆQ AI t) improves upon its IPW precursor, it is still inferior to the Dikta estimator ˆQ D t). 2.3 AIPW survival function estimator We now briefly describe the AIPW estimator of St). This estimator results from employing a series of compactly differentiable mappings of the basic estimators ˆQ AI t) and 1 Ĥt) Gill and Johansen, 199). Specifically, the estimator takes the form Ŝ AI t) = s t π 1 d ˆQs) ) 1 Ĥs ). 32) The weak convergence of ŜAIt) can be derived in a straightforward way through the functional delta method, see p of Gill and Johansen 199). Considering only Scenario 3, n 1/2 { ˆQ AI t) Qt), 1 Ĥt ) 1 Ht)} D Z, Z H ), in D[, τ] D [, τ], ), as n, where Z is the limiting Gaussian process defined in the statement of Th. 3. Here, is the supremum norm, and is the max supremum norm. We then have by 1

11 the functional delta method that n 1/2 {ŜAIt) St)} converges weakly in D[, τ], ) to W t) = St) dw s)/1 Hs)) as n, where W t) = Zt) Z Hs)dΛs). We can then show that the variance function V t) of W t) is given by V t) = St) 2 [ dλu) 1 Hu) + 1 p u)) 1 π u) π u) dλu) 1 Hu) ], 33) which is the asymptotic variance of the reduced data NPMLE van der Laan and McKeague, 1998), and of two competing estimators Subramanian 24b, 26) as well. However, V t) VD t) Subramanian, 24a), where V Dt) = St) 2 [ p s) 1 Hs) dλs) + ] αu, v) 1 Hu))1 Hv)) dhu)dhv) 34) is the asymptotic variance of the Dikta semiparametric estimator ŜDt). Note that V D t) can also be derived from Eq. 6) and the functional delta method described above. 2.4 Extension to a left truncated version The three scenarios investigated above can be analyzed in an analogous way for a left truncated version of the MCI model. For this version, we observe X, Z, ξ, σ) whenever X Z, where Z is a left truncation variable, ξ = 1 when δ is observed and otherwise, and σ = ξδ Subramanian and Bandyopadhyay, 28). When X < Z, however, nothing is observed. The observed data are {X i, Z i, ξ i, σ i } 1 i n, an i.i.d. sample from the distribution of X, Z, ξ, σ) which is observed i.e., when Z i X i ). Let W t) = P Z t X Z X) and W 1 t) = P X t, δ = 1 Z X). The cumulative hazard takes the form Λt) = a W dw 1 u)/ W u), where W denotes the d.f. of X and a W = inf{x : W x) > }. The 11

12 asymptotic variance of the AIPW estimator of St) for Scenario 3 is [compare with Eq. 33)] V T t) = St) 2 [ a W dλu) W u) + 1 p u)) 1 π u) a W π u) ] dλu), W u) where p t) = P δ = 1 X = t, Z X), and π t) = P ξ = 1 X = t, Z X). The Dikta semiparametric estimator for this version of the MCI model has asymptotic variance V T t) = a W p u) W u) dλu) + a W αu, v) a W p u)p v) dλu)dλv), see Subramanian and Bandyopadhyay 28). Moreover, we can show that V T t) V T t). References Dikta, G., On semiparametric random censorship models. J. Statist. Plann. Inference 66, Gill, R. D., Johansen, S., 199. A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18, Horvitz, D.G., Thompson, D.J., A generalization of sampling without replacement from a finite universe, J. Ameri. Statist. Assoc. 47, Kouassi, D. A., Singh, J., A semiparametric approach to hazard estimation with randomly censored observations. J. Ameri. Statist. Assoc. 92, Koul, H., Susarla, V., Van Ryzin, J., Regression analysis of randomly right censored data. Ann. Statist. 9, van der Laan, M. J., McKeague, I. W., Efficient estimation from right-censored data when failure indicators are missing at random. Ann. Statist. 26,

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