Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths

Citation preview

Biostatistics (2005), 6, 2, pp. 241–258 doi:10.1093/biostatistics/kxi006

Directly parameterized regression conditioning on being alive: analysis of longitudinal data truncated by deaths BRENDA F. KURLAND∗ National Alzheimer’s Coordinating Center, University of Washington, 4311 11th Avenue NE #300, Seattle, WA 98105, USA [email protected] PATRICK J. HEAGERTY Department of Biostatistics, University of Washington, Seattle, WA 98195, USA

S UMMARY For observational longitudinal studies of geriatric populations, outcomes such as disability or cognitive functioning are often censored by death. Statistical analysis of such data may explicitly condition on either vital status or survival time when summarizing the longitudinal response. For example a patternmixture model characterizes the mean response at time t conditional on death at time S = s (for s > t), and thus uses future status as a predictor for the time t response. As an alternative, we define regression conditioning on being alive as a regression model that conditions on survival status, rather than a specific survival time. Such models may be referred to as partly conditional since the mean at time t is specified conditional on being alive (S > t), rather than using finer stratification (S = s for s > t). We show that naive use of standard likelihood-based longitudinal methods and generalized estimating equations with non-independence weights may lead to biased estimation of the partly conditional mean model. We develop a taxonomy for accommodation of both dropout and death, and describe estimation for binary longitudinal data that applies selection weights to estimating equations with independence working correlation. Simulation studies and an analysis of monthly disability status illustrate potential bias in regression methods that do not explicitly condition on survival. Keywords: Binary longitudinal data; Censoring; Dropout; Missing data.

1. I NTRODUCTION The precipitating events project (PEP) is a prospective cohort study of 754 community-living persons, aged 70 years or older. The PEP is designed to evaluate the epidemiology of disability, defined as needing assistance in any of four key activities of daily living (ADL) (bathing, dressing, walking within the home or getting in and out of a chair). No participants were disabled at baseline, and disability was assessed by self-report in a monthly telephone interview (Gill et al., 2001). By 24 months after baseline, 64 of 754 subjects (8.5%) had died, and 19 (2.5%) had dropped out of the study. Two of the decedents dropped out of the study prior to death. ∗ To whom correspondence should be addressed.

c The Author 2005. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. 

242

B. F. K URLAND AND P. J. H EAGERTY

The focus of the PEP study is disability. However, survival and disability are almost certainly related, with the disabled at greater risk of death. Studies of geriatric populations need to consider how to incorporate deaths into the analysis of other outcomes. Deaths are often viewed as a type of missing data. However, in many applications there is a fundamental distinction between response values not measured due to participant dropout and scheduled measurements unavailable due to death. When a subject drops out, missing random variables can be considered as existing yet not collected. When a subject dies, subsequent planned responses either do not exist or require explicit coding (i.e. a zero value). Feasibility of such coding depends on the response variable. For example it could be argued that analyses of medical costs can naturally assign a value to deceased patients: no further costs. The PEP data might be analyzed with deceased subjects classified as disabled. However, coding schemes for biological or clinical measurements such as blood pressure are not clear-cut, and using the lowest value for deceased subjects’ quality of life measures may invalidate psychometric scaling (Diehr et al., 2001). Several possible mean models could form the basis for longitudinal regression analysis in the presence of survival information. Responses for deceased subjects could be given standardized codes, as discussed above. Some researchers impute data to recover response values as if the subjects were still alive (Revicki et al., 2001), attempting to characterize a longitudinal trajectory for a population where no subjects died. Other authors treat death as an absorbing state separate from monotone dropout (Miller et al., 2001) or condition on the specific time of death (Pauler et al., 2003). Finally, response estimates for a given time point can condition on being alive at that time (Brayne et al., 1999). The primary contribution of this paper is to explore regression models conditioning on being alive as a valid target of inference. To place this approach in the context of other models for data subject to death and dropout, we first present a careful classification of the candidate mean models (Section 2). We then describe methods of estimation and inference for regression models conditioning on being alive (Section 3). Section 4 examines how to extend these models to accommodate monotone dropout, as well as deaths. Simulations and analysis of the PEP data are used in Sections 5 and 6 (respectively) to illustrate how various regression models fitted to available data estimate different targets of inference.

2. M ODELS FOR SURVIVAL AND LONGITUDINAL RESPONSE DATA In this section, we discuss targets of inference for regression models involving survival (S) and a longitudinal response vector (Y). The joint distribution f (S, Y) has two natural factorizations: f (Y|S) f (S) and f (S|Y) f (Y). Models for Y are characterized as unconditional, fully conditional or partly conditional based on how, or whether, the longitudinal response model conditions on S. Analogous models for the joint distribution of dropout and response, f (R, Y), are discussed to highlight differences between dropout and death, and because the dropout literature is well-developed for these factorizations. Unconditional, fully conditional and partly conditional mean models are illustrated using four sample 24-month records of PEP data (Table 1). Measurement may be terminated by death, dropout or the end of follow-up. The monthly entry is 1 if a participant reports any of four ADL disabilities, and 0 if no disability is reported. Symbol ‘.’ denotes months with missing data, and ‘x’ appears each month after the study participant has died. 2.1

Targets of inference

Unconditional mean model. First consider data missing due to dropout, where Yit takes the value 1 if subject i is disabled t months after baseline (0 if not disabled), and Rit is a dichotomous variable for retention (Rit = 1 if Yit is observed, 0 if missing). For modeling disability status as a function of time,

Directly parameterized regression conditioning on being alive

243

Table 1. Sample data records for PEP study (x = deceased, . = missing) ID 2452 1054 2049 2663

ADL months 1–24 20 10 0000010000 00xxxxxxx 0000000001 1....xxxx 0000000000 000000001 0000000000 .1.111101

xxxxx xxxxx ..... 110.0

the unconditional target of inference is µU it = E(Yit ) = probability that subject i is disabled t months after baseline. This unconditional expectation is contrasted with an expectation which conditions on retention: E(Yit |Rit = 1) = probability of disability given that the disability status was observed. The unconditional probability refers to the original target population, rather than the subset of protocol compliers. It is well known that analysis conditioning on Rit = 1 may be invalid, due to selection bias (Little and Rubin, 1987). Accordingly, the selection model factorization for longitudinal data with monotone dropout (Diggle and Kenward, 1994), f (R, Y) = f (R|Y) f (Y), results in an unconditional model for Y. If f (R|Y) does not depend on current or future values of Y [that is if data are missing at random (MAR)], likelihood-based methods such as the marginalized transition model (MTM, Heagerty, 2002) can provide valid estimation for parameters characterizing f (Y) without modeling f (R|Y). To illustrate unconditional models for Y in the presence of both deaths and dropout, consider the sample PEP data in Table 1. Ten months after baseline, the unconditional probability of disability is 1/4. At month 11, disability status is missing for one subject. The unconditional expectation of disability will be a mixture of means for missing and non-missing data: µU it = E(Yit ) = E(Yit |Rit = 1) · P(Rit = 1) + E(Yit |Rit = 0) · P(Rit = 0) 1 3 1 = · + ? · . 3 4 4 Implicit or explicit imputation of E(Yit |Rit = 0), the expected value for missing responses, is necessary to determine the unconditional mean. By 19 months after baseline, two subjects are deceased and two are disabled. An unconditional longitudinal model of disability, based on following the entire cohort of four subjects, would need to assign response values to the deceased subjects. The mixture in the presence of deaths (but no dropout) is µU it = E(Yit ) = E(Yit |Ait = 1) · P(Ait = 1) + E(Yit |Ait = 0) · P(Ait = 0), where Ait indicates whether subject i is alive t months after baseline. If the PEP study defines death as akin to extreme disability, the unconditional disability rate at month 19 is 4/4. As discussed in Section 1, valid codes for Yit may not always exist when Ait = 0. In addition to the selection model factorization, unconditional Y appears in the analysis of a multivariate response (Y, S). An example is the joint assessment of treatment effects for survival and longitudinal quality of life (Gray and Brookmeyer, 2000). However, if survival (S) and longitudinal response (Y) are related and Y is undefined for deceased subjects, an unconditional model f (Y) will probably not be of great scientific interest.

244

B. F. K URLAND AND P. J. H EAGERTY

Fully conditional mean model. A fully conditional model for Y given S comprises part of the patternmixture factorization f (Y, S) = f (Y|S) f (S). The target of inference for a fully conditional model is the longitudinal trajectory stratified by the specific time of death (Ribaudo et al., 2001; Pauler et al., 2003), µitS = E(Yit |Si = s) = probability that subject i is disabled t months after baseline, given that the subject dies at time s. For example evaluation of whether a biomarker elevates prior to disease onset or death may focus on E(Yit |Si = s) as a function of the lead time (S − t). In Table 1, the month 10 disability rate is 0/2 for subjects who will survive at least 1 year past the 10th month, and is 1/2 for subjects who will die within a year. These rates are perhaps more informative than the unconditional rate (1/4), but would not be useful for a predictive model used to guide medical decisions at month 10, since time of death is not known in advance. Partly conditional mean model: regression conditioning on being alive. For a partly conditional model (see Pepe et al., 1999), µitA = E(Yit |Ait = 1) = E(Yit |Si > t) = probability that subject i is disabled t months after baseline, given that subject i is alive t months after baseline, we can avoid conditioning on future events by specifying only that a subject is alive at the time point being modeled. The partly conditional model directly parameterizes what we refer to as regression conditioning on being alive (RCA). The RCA model will be of scientific interest when survival is related to the longitudinal response, but the longitudinal response is of primary scientific interest. For the PEP data, some non-RCA uses may include examination of how disability predicts death, or computing whether women or men are more likely to live to the age of 85. A distinct scientific question, though, is whether 85-year-old men or women are more prone to disability, given that they are alive. This last question is addressed directly by RCA. In the sample PEP data (Table 1), the month 10 RCA disability rate is 1/4. Since all four subjects are alive at month 10, it is the same as the unconditional rate. At 19 months after baseline, the disability rate is 2/2 among the surviving subjects. The RCA approach specifies a longitudinal model of disability in which a different number of subjects are in the risk set at different time points. This concept is similar to use of hazard rates in survival analysis, but is perhaps unusual for a prospective longitudinal study that follows a cohort over a short time period. 2.2

Functional form of mean models

Although unconditional, fully conditional and partly conditional models may be derived from a single joint probability distribution f (Y, S), the different mean models may not share the same functional form for the regression relationship between covariates and Y. Figure 1 shows four hypothetical trajectories for a continuous measure of disability. The fully conditional means are piecewise linear, with an increase in slope starting 1.5 years before death, marked by X: E(Yi j |Si ) = β0 + β1 ti j + β2 (ti j + 1.5 − Si )

[Si
(ti j + 1.5)],

where (β1 , β2 ) = (2, 2). The solid line shows the partly conditional trajectory E(Yi j |Ai j = 1), and the dashed line shows unconditional E(Yi j ) averaged over time. These curves are lowess smooths to the four expected trajectories of the fully conditional mean model, with death equally likely at 2.5, 3.5, 4.5 or more than 5 years after baseline.

Directly parameterized regression conditioning on being alive

245

Fig. 1. Functional form example: four piecewise linear individual trajectories (E(Yi j |Si = s), fully conditional on time of death, marked by X), and lowess curves for unconditional model E(Yi j ) and partly conditional model E(Yi j |Ai j = 1) (RCA).

The fully conditional means shown in Figure 1 reflect the functional form of individual trajectories. The unconditional model characterizes the population where deaths do not occur, while the partly conditional model describes the dynamic population of survivors. Cross-sectional data may therefore follow the functional form of a partly conditional model. Although cross-sectional data might permit valid inference for some RCA models, longitudinal data are relatively free of cohort effects, and may be used to fit additional (non-RCA) models.

3. E STIMATION FOR PARTLY CONDITIONAL MEAN MODELS The goal of this section is to identify estimation methods for the partly conditional target of inference in Section 2.1, defined as the expected value for longitudinal response Yi j at time t j conditional upon being alive at that time: µiAj = E(Yi j |Xi j , Si > t j ) = E(Yi j |Xi j , Ai j = 1), where Xi j is a ( p × 1) covariate vector corresponding to the measurement time t j (a more general time scale than for the PEP data). Random variable Si denotes the time of death for subject i. For now, we

246

B. F. K URLAND AND P. J. H EAGERTY

assume that no response or survival time data are missing or censored. The length of the response vector Yi , n i , therefore depends only on the study design and survival time Si . Values for the partly conditional target of inference µiAj may be obtained from fully conditional and joint models (Section 3.1). However, only the partly conditional regression model, E(Yi j |Xi j , Si > t j ), directly parameterizes µiAj using a parameter vector β A . This section shows that likelihood-based methods and generalized estimating equations (GEE) with non-independence working correlation are better suited to unconditional and fully conditional regression models, for which modeling of partly conditional targets of inference is indirect. Estimation with independence working correlation, however, may yield valid direct estimation of partly conditional regression models. 3.1 Indirectly parameterized estimation of µiAj We first estimate RCA summaries using a full likelihood approach for longitudinal response and survival. A full likelihood will parameterize the entire joint distribution for response vector Yi and survival time Si for the i-th subject, f (Yi , Si ), so we will start with a fully conditional model: f (Yi , Si |θ) = f (Si |η) f (Yi |Si , β S , α S ).

(3.1)

Equation (3.1) specifies µiSj = E(Yi j |Si = s) via parameter vector β S . Vector α S defines association among elements of Yi . Target of inference µiAj = E(Yi j |Si > t j ) is recovered from the pattern-mixture model f (Yi |Si , β S , α S ) f (Si |η) by summing over the survival distribution. In discrete time,
S S k> j E(Yi j |Si = tk , β , α )P(Si = tk |η) A
. (3.2) E(Yi j |Si > t j ) = µi j = k> j P(Si = tk |η)

The expected value µiAj can be computed by modeling survival time explicitly and summing expected values over the support restricted to Si > t j . This summation yields valid estimates of E(Yi j |Si > t j ), but only if f (Yi |Si , β S , α S ) and f (Si |η) are both specified correctly. Directly parameterized RCA coefficients β A cannot be recovered from (3.2), unless strong assumptions are made about modeling relationships among β A and (β S , α S , η). This situation is similar to the marginalization of pattern-mixture models for dropout discussed by Fitzmaurice and Laird (2000). A confidence interval can be constructed for each µiAj point estimate using the delta method, but comparisons such as treatment main effects are not directly parameterized. The selection model factorization of f (Yi , Si ), f (Yi , Si |θ) = f (Si |Yi , η) f (Yi |β U , α U ), may also be used to develop a mixing distribution to induce µiAj . Using Bayes’ theorem (for binary response and discrete-time survival),
U U k> j P(Si = tk |Yi j = 1, η)P(Yi j = 1|β , α ) A
E(Yi j |Si > t j ) = µi j =
. (3.3) U U k> j m=0,1 P(Si = tk |Yi j = m, η)P(Yi j = m|β , α )

Two features of (3.3) make it impractical for use in estimation of partly conditional regression models. First, standard selection models do not specify P(Si = tk |Yi j = 1) when time point tk is after time t j , since discrete hazard models condition on the full response history prior to time tk while (3.3) requires a lag from Yi j at t j to time tk . Second, the selection model has imposed a structure for E(Yi j |Ai j = 0). As discussed in the Unconditional Mean Model section, fitting a selection model to available data (for which subjects

Directly parameterized regression conditioning on being alive

247

are living) does not result in an RCA model, but to an unconditional model for f (Yi ) that implicitly gives response values to deceased subjects. The models in this section are not limited to likelihood-based estimation methods. f (Yi |Si ) and f (Si ), or f (Si |Yi ) and f (Yi ) can be estimated using semiparametric or non-parametric methods, and then input to the mixing equations ((3.2) or (3.3)). In general, any joint model can be used to derive RCA point estimates. 3.2

Direct estimation of µiAj : moment-based approach

The likelihood-based approaches discussed above do not directly parameterize partly conditional means. Therefore, we turn to semiparametric methods for estimation of regression models for µiAj = E(Yi j |Xi j , Ai j = 1). Regression models may be fit using moment-based GEE (Liang and Zeger, 1986). We present quasi-score equations for binary response data, but the method can be applied to any generalized linear model. For a logistic-linear model, unconditional mean µU i j = E(Yi j |Xi j ), and independence working correlation, an unbiased, linear quasi-score equation for regression parameter vector β U is U (β U ) =

ni N   i=1 j=1

Xi j (Yi j − µU i j ).

To change the target of inference to when subjects are alive, we explicitly restrict quasi-score contributions using the indicator Ai j : ni N   U (β A ) = Xi j Ai j (Yi j − µiAj ), i=1 j=1

where Ai j equals 1 when Si > t j and 0 otherwise. This approach will yield valid statistical inference for µiAj , using a directly parameterized partly conditional regression model, logit(µiAj ) = Xi j β A . The essential requirements for consistent estimation of µiAj are that the regression model is correctly specified and that information increases with the number of clusters (Crowder, 1986). Note that fitting a direct, partly conditional model using independence estimating equations (IEE) does not require modeling of the survival distribution. Unfortunately, GEE with non-independence working correlation applied to observations with Ai j = 1 will generally not yield valid inference for RCA models. If subjects die during a study, the analysis cluster size n i is determined by survival time Si = s, as well as by the study design. (This dependence is suppressed in the notation above, since it does not affect values of Xi j , Yi j or Ai j .) When the GEE [n (s)] working correlation is non-diagonal, variance inverse terms, wi jki , differ according to the cluster size n i (s), and the quasi-score for β A would be A

U (β ) = [n (s)]

N n i (s) n i (s)  i=1 j=1 k=1

[n (s)]

Xik wi jki

Ai j (Yi j − µiAj ).

depends on the specific value of survival time Si , the expectation of the quasi-score under a Since wi jki regression model for (Yi j |Xi j , Ai j = 1) will not result in unbiased estimation of µiAj .

248

B. F. K URLAND AND P. J. H EAGERTY

In summary, likelihood-based methods and GEE with non-independence correlation can be used to estimate either fully conditional or marginal means. When combined with a mixing distribution these methods also fit partly conditional means, but the parameterization of µiAj = E(Yi j |Xi j , Ai j = 1) is not direct. If IEE is used, direct and partly conditional regression is achieved. Furthermore, the survival distribution is ignorable in that (unlike for the mixing distributions) it does not have to be correctly specified for unbiased estimation of µiAj .

4. RCA

WITH BOTH DROPOUT AND DEATH

We now consider longitudinal data with both missing data and deaths, describing when RCA models can be fitted directly, and when dropout and/or survival distributions are ignorable. Inverse probability of censoring-weighted GEE (IPCW-GEE, Robins et al., 1995) is adapted to RCA models. Using principles of sampling design (Horvitz and Thompson, 1952), IPCW-GEE requires modeling of the dropout pattern, usually parametrically. The inverse probability of observing each longitudinal response weights the observed data in the GEE quasi-score, shown here for a working independence correlation model and logistic-linear modeling of binary data: U

U (β ) =

ni N   i=1 j=1

Xi j

Ri j πiUj

(Yi j − µU i j ).

Indicator Ri j reflects missingness status and πiUj = P(Ri j = 1|Di ), the probability of retention. Missingness model matrix Di can include covariates and observed (past) responses, which permits accommodation of MAR dropout. IPCW-GEE provides consistent estimation of µU i j = E(Yi j |Xi j ) when 

  Ri j Yi j  E Xi j = E(Yi j |Xi j ), πU  ij

which holds if inverse probability of censoring (IPC) weights πiUj are modeled correctly. To apply IPCW-GEE for estimation of µiAj = E(Yi j |Xi j , Ai j = 1), weights πiAj must be estimated so that    Ri j Yi j  Xi j , Si > t j = E(Yi j |Xi j , Si > t j ). E πiAj 

We will show that f (Si ) may be ignorable with respect to the regression model, even when IPC weights may require finer conditioning on survival status. As an illustration using PEP data (Table 1, ignoring intermittent missing data and assuming a common dropout process), the month 20 response for subject 2663 should be weighted to represent the missing response of 2049 but not responses for 2452 or 1054, who died before month 20. Table 2 describes a hierarchy of missing data, as defined for partly conditional RCA. Like the MCAR/ MAR/NI (nonignorable) missing data hierarchy for unconditional regression (see Laird, 1988), the categories in Table 2 define the situations when missingness is ignorable, and how estimation methods may accommodate the missing data process. We first consider data missing completely at random (MCAR). Suppressing dependence on fixed (Y) (Y) covariates, MCAR is defined as P(Ri j = 1|Yi j , Hi j , Si ) = P(Ri j = 1|Si > t j ), where Hi j = (Yi1 , . . . , Yi j−1 ) is the history for Yi through time ( j − 1). Returning to the quasi-score for direct

Directly parameterized regression conditioning on being alive

249

Table 2. Monotone dropout hierarchy and required IPC weights: directly parameterized RCA models Dropout pattern

Dropout pattern: P(Ri j = 1|Yi , Si = s) reduces to

IPC weights πi j for IEE quasi-score

MCAR MCAR-S

P(Ri j = 1|Si > t j ) P(Ri j = 1|Si = s)

None: f (Ri ) is ignorable πiSj = P(Ri j = 1|Si = s, Si > t j )

(Y)

(Y)

MAR

P(Ri j = 1|Hi j , Si > t j )

πiAj = P(Ri j = 1|Hi j , Si > t j )

MAR-S

P(Ri j = 1|Hi j , Si = s)

πiSj = P(Ri j = 1|Hi j , Si = s, Si > t j )

(Y)

(Y)

Notation response vector for subject i Yi Ri dropout vector for subject i (Ri j = 1 if Yi j observed, 0 if not observed) Si survival time (time of death) for subject i (Y) Yi1 , Yi2 , . . . , Yi j−1 (history vector for Yi j ) Hi j

parameterization of µiAj using IEE (Section 3.2, adding an indicator Ri j for missingness):

U (β A ) =

ni N   i=1 j=1

Xi j Ai j Ri j (Yi j − µiAj ),

we take the expected value under f (Y, R|A) to evaluate whether estimates of β A will be consistent:

EY,R|A [U (β A )]

ni N  

Xi j Ai j EY|A [(Yi j − µiAj )ER|Y,A (Ri j )]

=

ni N  

Xi j Ai j EY|A (Yi j − µiAj )P(Ri j = 1|Ai j = 1)

=

0,

if µiAj = E(Yi j |Xi j , Ai j = 1).

= MCAR

i=1 j=1

i=1 j=1

As for data with no dropout, estimation of β A using IEE is consistent under MCAR if the regression model is properly specified. Weights based on MCAR dropout may affect efficiency, but not consistency, of regression estimators β A . Both f (Ri |φ A ) and f (Si |η) are ignorable, if the regression parameter vector β A is distinct from dropout and survival parameters φ A and η. (Y)

We now consider MAR dropout, for which P(Ri j = 1|Yi , Si ) = P(Ri j = 1|Hi j , Si > t j ). By analogy to non-RCA IPCW-GEE, we weight the quasi-score by the inverse of censoring weights πiAj = (Y)

P(Ri j = 1|Hi j , Si > t j ):

U (β A ) =

ni N   i=1 j=1

Xi j Ai j

Ri j πiAj

(Yi j − µiAj ).

250

B. F. K URLAND AND P. J. H EAGERTY

Taking the expectation over the observed data distribution, ni N  

A

EY,R|A [U (β )] =



Xi j Ai j EY|A ER|Y,A

i=1 j=1

MAR

=



= 0,

Xi j Ai j EY|A



πiAj πiAj



Ri j



πiAj 

(Yi j − µiAj )



(Yi j − µiAj )

if µiAj = E(Yi j |Xi j , Si > t j ).

Dropout process f (Ri |φ A ) is not ignorable, since dropout must be modeled correctly to obtain valid inference for µiAj . However, the survival distribution f (Si |η) is ignorable under IEE for MAR data. Parametric modeling of weights πiAj for monotone dropout may be conducted as a telescoping product, with each element of R conditioning on past values:

P(Ri j =

(Y) 1|Hi j , Si

> tj) =

j 

k=2

(R)

(Y)

P(Rik = 1|Hik ≡ 1, Hik , Si > t j ),

assuming that the first observation in the cluster is never missing (Robins et al., 1995). Each probability in the product conditions on the same survival time (indexed by j). To illustrate this, Table 3 shows hypothetical missingness information for five subjects in a study with four time points. Observed responses are √ denoted by until the subjects die (×). Symbol ‘·’ shows missing data due to dropout. Assuming that the subjects are similar with respect to relevant covariates, IPC weights for the fourth response of subject 1 should represent all three living subjects (subjects 1, 3 and 5). However, if the conditioning on survival A would be P(R = 1|S > 4) = P(R = 1|R = changes as the dropout model telescopes, the weight π14 4 4 3 1, S > 4)·P(R3 = 1|R2 = 1, S > 3)·P(R2 = 1|S > 2) = 14 , rather than 13 . (R)

(Y)

Let u ik( j) = P(Rik = 1|Hik , Hik , Si > t j ) where 2
k
j. When dropout time is td and survival time ts , (R)

(Y)

1−rik ik P(Rik = rik |Hik ≡ 1, Hik , Si > t j ) = u rik( j) (1 − u ik( j) )

and u ik( j)

⎧ ⎪ ⎨

pik( j) : k
d and k < s, 0 : k > d and k < s, = ⎪ ⎩undefined : k  s (deceased),

Table 3. Data records to illustrate weighting schemes under dropout and death ( response missing, × = deceased) ID 1 2 3 4 5

Time 1 √ √ √ √ √

Time 2 √

Time 3 √

Time 4 √

· √

× · × √

× · × ·

× √

√

= observed, · =

Directly parameterized regression conditioning on being alive (R)

(Y)

251 (Y)

where pik( j) = P(Rik = 1|Hik ≡ 1, Hik , Si > t j ) = P(Rik = 1|Rik−1 = 1, Hik , Si > t j ) under monotone dropout. So (Y)

P(Ri j = ri j |Hi j , Si > t j ) =

(d−1)∧(s−1)∧ j  k=2

⎛



rik I (d
j t j ) = β0A + β1A · groupi + β2A · timei j + β3A · groupi · timei j ,

where β A = (−2.19, 0.5, 0.1, −0.025). In this response trajectory, women (group = 0) are less likely to show the binary response (disability) among survivors at 65 years. By 85 years (time = 20), both men and women are equally likely to be disabled, given that they are alive. Data are generated using a probit model for the joint distribution of survival time and longitudinal responses. The binary response Yi j is an indicator for (Yi∗j > 0) where (Yi∗ , Si ) follow a multivariate ∗ normal distribution with mean (µU i , E(Si )). Correlation among Y is autoregressive with correlation 0.7. The mean survival time is 85 years for women (group = 0) and 80 years for men (group = 1), with standard deviation of 5 years for both groups. The covariance of S and Y∗ is −0.4 for women and −0.3 for men. Unconditional means µU i j = E(Yi j ) are found through the following correspondence: µiAj

= E(Yi j |Si > t j ) =

P(Yi∗j

> 0|Si > t j ) =

P(Yi∗j > 0, Si > t j ) P(Si > t j )

.

(5.1)

Mean response conditioning on survival (µiAj ) and the unconditional survival distribution for Si are known, as are all parameters of the joint cumulative density (Yi∗ , Si ) except for unconditional mean µU i j . We find U A µi j by bisection, to an arbitrary precision tolerance (0.0001). Equation (5.1) also shows that µi j , which we define as logistic-linear in covariates group and time, is not linearly related to the joint density of Yi∗ and Si . Once the joint distribution f (Yi∗ , Si ) is specified, Si and Yi∗ are generated by the function rmvnorm( ) in package mvtnorm of R version 1.5.1. One thousand data sets are simulated with 1000 clusters each. Monotone MCAR-S (Table 2) dropout is simulated according to the following model:   logit P(Ri j = 1|Ri j−1 = 1, Xi j , Si ) = φ0S + φ1S (Si − timei j )

where missingness parameter vector φ S = (−0.5, 0.15). The odds of not dropping out 15 years before death are exp(10 · φ1S ) ≈ 4.5 times the odds of not dropping out 5 years before death. No data are missing due to dropout at time = 0, but about 33% of responses (Yi j at times when the subject is alive) are missing due to dropout overall. Four models are fitted, each with a logistic-linear regression model for an intercept, group, time and a group by time interaction: 1. IEE: GEE with independent correlation structure (a generalized linear model). Although the target of inference is (correctly) the RCA model, and GEE parameter estimates are usually consistent for MCAR dropout, MCAR-S dropout depends on random variable Si and bias is expected in β A . 2. IPCW-GEE: GEE with autoregressive correlation structure and correctly specified IPC weights. Estimates of RCA parameter vector β A are not expected to be consistent (Section 3.2). 3. MTM: Second-order MTM (Heagerty, 2002) fitted to available data. The MTM is a likelihoodbased method, and will fit an unconditional mean model to available data. The resulting regression estimates are not expected to be consistent for a partly conditional mean model. 4. IPCW-IEE: IEE with correctly specified IPC weights. This model should yield consistent estimates of β A , the RCA parameters. Bias as a percentage of the true β A value is computed for the average of the 1000 parameter vectors.

Directly parameterized regression conditioning on being alive

253

Table 4. Percent bias for logistic regression (IEE), inverse probability of censoring-weighted GEE (IPCW-IEE and IPCW-GEE) and MTM fitted to RCA regression with MCAR-S dropout, for 1000 simulated data sets with 1000 subjects

IEE IPCW-GEE MTM IPCW-IEE

Intercept (β0A = −2.19)

Group (β1A = 0.50)

Time (β2A = 0.10)

Group:Time (β3A = −0.025)

2 2 2 0

2 4 4 1

−11 9 0 1

−12 −1 −16 0

Fig. 2. Fitted trajectories: bias in binary regression model with MCAR-S dropout, conditioning on survival.

Table 4 shows percent bias in regression parameters for models fitted to binary data with deaths and with dropout that is survival-dependent MCAR (MCAR-S). Figure 2 shows regression trajectories using average fitted regression parameters. The IEE model without IPC weights underestimates event probabilities for both groups. People closer to death are more likely to drop out, so by not giving weight to those dropouts, we underestimate the probability of disability in the target population of survivors (both dropouts and non-dropouts). In contrast, because of implicit imputation of decedents’ responses, GEE with censoring weights (IPCW-GEE) overestimates event probabilities compared to µiAj . The MTM, like GEE without IPC weights (not shown), displays 16% negative bias in group:time but small (
5%) bias in time and other regression parameters. Predicted response probabilities are slightly smaller for

254

B. F. K URLAND AND P. J. H EAGERTY

women and slightly larger for men, compared to µiAj . Similar results (bias in time and group:time) were observed in simulated binary data with deaths but no dropout, and in linear models. Although IPCW-GEE and MTM fit unconditional models for responses, the trajectories for these models do not fit µU i j exactly (Figure 2). The IPCW-GEE and MTM trajectories result from logit-linear regression on covariates group and time. The unconditional mean, µU i j , does not have a directly specified functional form, but is generated from the probit model and the logit-linear RCA model. This lack of correspondence between µU i j and MTM or IPCW-GEE illustrates the functional form issue discussed in Section 2.2. For binary longitudinal data with moderate correlation among responses and survival time, fitted values for µiAj = E(Yi j |Si > t j ) may be quite different from the unconditional mean µU i j = E(Yi j ), which implicitly imputes responses after death. Fitting GEE or maximum likelihood models without µiAj as the target of inference results in both positive and negative bias for within-cluster regression parameters of magnitude in the low teens as a percentage of true β A values. 6. E XAMPLE Introduced in Section 1, the PEP explores monthly patterns of disability in senior citizens. This section illustrates the impact of analysis method on targets of inference for regression models. Using 24 months of PEP data, we model a quadratic time trend, with linear effects differing by risk group. Absence or presence of ADL disability for subject i at month (t j − 12) is denoted by Yi j . Enrollment in the study was stratified by low, intermediate and high risk for ADL disability, using a model developed in an earlier study (Gill et al., 1999). We model risk of disability over time for the three risk groups: logit[P(Yi j = 1|Xi j , Si > t j )] = β0A + β1A · MEDi + β2A · HIGHi + β3A · monthi j + β4A · monthi2j +β5A · MEDi · monthi j + β6A · HIGHi · monthi j .

Five approaches are used to fit this regression model: 1. IEE: GEE with independent correlation structure. If data are not MCAR, RCA estimators may be biased. 2. IPCW-GEE: IPCW-GEE with autoregressive correlation structure. Section 3.2 shows that this model is not expected to be consistent for estimation of RCA parameter vector β A . 3. MTM: Second-order MTM. The target of estimation for this maximum likelihood method is µU ij, not µiAj , so we expect bias in RCA parameter estimates (Section 3.1). 4. IPCW-IEE: IPCW-IEE. If the dropout weights and regression model are specified correctly, we expect this model to be unbiased for β A (Section 4). 5. IEE (pattern-mixture): A fully conditional model is not feasible, since only 64 subjects die at any of the 24 time points. We fit a partly conditional model stratified by decedent status at the time of analysis (Si > 12 or Si > 24 months after baseline). Of the 64 decedents, 44 survive for at least 12 months of observation. Since the decedent sample is small (N = 44) with few dropouts (N = 2), dropout is not modeled, and the interaction of risk group and month (parameters β5A and β6A ) are not included in the regression model. A small number of responses are missing. Of 17 401 anticipated months of response data, 219 are missing due to intermittent dropout, and 224 are missing due to the monotone dropout of 19 subjects. Intermittent missing responses are imputed using a non-parametric approximate Bayesian bootstrap

Directly parameterized regression conditioning on being alive

255

Fig. 3. Fitted trajectories and cross-sectional means for PEP data (N = 754): different analysis methods yield different targets of inference. The pattern-mixture model excludes 20 subjects who die before 12 months of follow-up.

(Rubin, 1987). Missing data are matched to non-missing data by decedent status, risk group, sex, mean number of months disabled, 6-month interval within the follow-up period and ADL function in adjacent months (Gill and Kurland, 2003). The selection model at time t j for the two IPCW-GEE models is MAR with the covariates from the regression model, plus additional main effects: sex, ADL disability status at time t j−1 and an indicator for depression at baseline. The dropout model is very dense for describing a small amount of missing data: the fitted probability of retention is at least 98% for all data points. However, an underspecified dropout model may result in regression model bias (Robins et al., 1995). Figure 3 shows fitted regression trajectories for four models fitted to the PEP data. (The IEE model differs only slightly from IPCW-IEE since very few data are missing due to dropout.) Cross-sectional means for living subjects are shown (◦) for a data set in which both intermittent missing data and data missing due to dropout are imputed using the single imputation method described above. The fitted trajectory for the decedents is truncated to avoid predicting beyond the range of the data. The patternmixture model shows that people who will die soon are more likely to experience disability than those who will not die soon. The model would only be useful for predictions if it were known when death was imminent. However, many analyses in the medical literature stratify results by decedent status at time of analysis. The IPCW-IEE trajectory (solid line) is closer to the raw data than IPCW-GEE (dotted line) or the MTM (dashed line). Implicit imputation of data missing due to dropout and death causes the MTM fitted

256

B. F. K URLAND AND P. J. H EAGERTY

probabilities to be higher than cross-sectional means, especially for subjects at high risk of disability. This implicit imputation is desirable if data are missing due to dropout, but show that an unconditional target of inference is modeled by the MTM, rather than the desired partly conditional target. By extending the trajectory of deceased subjects, both the MTM and IPCW-GEE models overestimate the probability of ADL disability. The impact of dropout and death is unlikely to be large in the PEP data set, since very few subjects drop out (n = 19, 2.5%) and only 64 (8.5%) die in the first 2 years of the study. Thirty of 432 low-risk group subjects die (7%), and 15/214 (7%) and 19/108 (18%) die in the medium- and high-risk disability groups, respectively. Although inference about parameter estimates does not differ greatly between models, choice of model does have a noticeable impact on regression parameters and fitted values. For example the fitted odds ratio comparing odds of disability in the high- and low-risk groups (ignoring the non-significant group-by-time interaction) is 8.95 under the IPCW-IEE model, and 10.00 for the second-order MTM fitted to available data. In summary, this modest example illustrates that IEE is a promising approach when attempting to partly condition on survival, so that regression addresses surviving subjects only. 7. D ISCUSSION This paper describes modeling of longitudinal regression partly conditional on survival as a valid target of inference. A practical benefit of directly parameterized partly conditional regression is that longitudinal trends for a dynamic cohort can be described without having to condition on a specific survival time. For questions such as, “Are men or women more prone to ADL disability as they age?” the existence of a survival difference between these groups is acknowledged, but is not the focus for ADL analysis. By defining models in terms of the surviving cohort, RCA describes the measurable mean response as a function of time and covariates. The main focus of this paper is on the characterization of time trends for different groups of subjects. Group comparisons at specific follow-up times are made in light of survival differences. An alternative approach has been suggested by Rubin (personal communication) where “principal strata” are used to condition on potential survival under control treatment, Ai j (0) = 1, and potential survival under active treatment, Ai j (1) = 1 (Frangakis and Rubin, 2002). At each time point, only those subjects who would survive under both treatments are compared. Group comparisons of mean response are therefore balanced in terms of death. This causal model is partly conditional, since only current potential survival status and not the potential survival times are conditioned upon. Therefore, all the estimation issues discussed in Sections 2 and 3 remain important for valid characterization of multiple time points using regression methods. Although accommodation of dropout may be necessary, directly parameterized RCA estimation essentially treats observations as independent. Likelihood-based methods and GEE with non-independence working correlation will not directly provide regression estimates that condition on being alive, due to implicit imputation of postmortem responses. If dropout is handled by multiple imputation (Rubin, 1987) rather than IPC weights, the analysis may be conducted using linear, logistic or other straightforward regression techniques. Two-stage multiple imputation may also be considered for imputing missing response data and unknown survival times (Harel, 2003), but the resulting data sets would still be analyzed as independent responses to preserve RCA. Methods discussed here apply to cases where the response, survival and dropout processes are dependent only on observed quantities. When the dropout process depends on unobserved values of the response, sensitivity analysis in IPC weights can be applied to evaluate alternate parametric dropout models (Scharfstein et al., 1999). Survival status can generally be gleaned from public records, but censoring of survival information is an important consideration for RCA. Future work will address censored survival times in RCA analysis.

Directly parameterized regression conditioning on being alive

257

ACKNOWLEDGMENTS This work was partly supported by NIA AG-016976, NHLBI HL-072966 and a NHLBI Predoctoral Training Grant. We are grateful to Dr. Thomas M. Gill and the PEP, AG-017560, for providing data, and to Dr. Gill and Dr. Thomas Lumley for stimulating conversations about this research.

R EFERENCES B RAYNE , C., S PIEGELHALTER , D. J., D UFOUIL , C., C HI , L. Y., D ENING , T. R., PAYKEL , E. S., O’C ONNOR , D. W., A HMED , A., M C G EE , M. A. AND H UPPERT, F. A. (1999). Estimating the true extent of cognitive decline in the old old. Journal of the American Geriatrics Society 47, 1283–1288. C ROWDER , M. (1986). On consistency and inconsistency of estimating equations. Econometric Theory 2, 305–330. D IEHR , P., PATRICK , D. L., S PERTUS , J., K IEFE , C. I., M C D ONELL , M. AND F IHN , S. D. (2001). Transforming self-rated health and the SF-36 scales to include death and improve interpretability. Medical Care 39, 670–680. D IGGLE , P. AND K ENWARD , M. G. (1994). Informative drop-out in longitudinal data analysis (with discussion). Applied Statistics 43, 49–93. F ITZMAURICE , G. M. AND L AIRD , N. M. (2000). Generalized linear mixture models for handling nonignorable dropouts in longitudinal studies. Biostatistics 1, 141–156. F RANGAKIS , C. E.

AND

RUBIN , D. B. (2002). Principal stratification in causal inference. Biometrics 58, 21–29.

G ILL , T. M., D ESAI , M. M., G AHBAUER , E. A., H OLFORD , T. R. AND W ILLIAMS , C. S. (2001). Restricted activity among community-living older persons: incidence, precipitants, and health care utilization. Annals of Internal Medicine 135, 313–321. G ILL , T. M. AND K URLAND , B. F. (2003). The burden and patterns of disability in activities of daily living among community-living older persons. Journals of Gerontology Series A: Biological Sciences and Medical Sciences 58, 70–75. G ILL , T. M., W ILLIAMS , C. S. AND T INETTI , M. E. (1999). The combined effects of baseline vulnerability and acute hospital events on the development of functional dependence among community-living older persons. Journals of Gerontology. Series A, Biological Sciences and Medical Sciences 54, M377–M383. G RAY, S. M. AND B ROOKMEYER , R. (2000). Multidimensional longitudinal data: estimating a treatment effect from continuous, discrete, or time-to-event response variables. Journal of the American Statistical Association 95, 396–406. H AREL , O. (2003). Strategies for data analysis with two types of missing values, Ph.D. Thesis, The Pennsylvania State University Department of Statistics, University Park, PA. H EAGERTY, P. J. (2002). Marginalized transition models and likelihood inference for longitudinal categorical data. Biometrics 58, 342–351. H ORVITZ , D. G. AND T HOMPSON , D. J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47, 663–685. L AIRD , N. M. (1988). Missing data in longitudinal studies. Statistics in Medicine 7, 305–315. L IANG , K.-Y. 73, 13–22.

AND

Z EGER , S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika

L ITTLE , R. J. A. AND RUBIN , D. B. (1987). Statistical Analysis with Missing Data. New York: John Wiley & Sons. M ILLER , M. E., T EN H AVE , T. R., R EBOUSSIN , B. A., L OHMAN , K. K. AND R EJESKI , W. J. (2001). A marginal model for analyzing discrete outcomes from longitudinal surveys with outcomes subject to multiple-cause nonresponse. Journal of the American Statistical Association 96, 844–857.

258

B. F. K URLAND AND P. J. H EAGERTY

PAULER , D. K., M C C OY, S. AND M OINPOUR , C. (2003). Pattern mixture models for longitudinal quality of life studies in advanced stage disease. Statistics in Medicine 22, 795–809. P EPE , M. S., H EAGERTY, P. AND W HITAKER , R. (1999). Prediction using partly conditional time-varying coefficients regression models. Biometrics 55, 944–950. R EVICKI , D. A., G OLD , K., B UCKMAN , D., C HAN , K., K ALLICH , J. D. AND W OOLLEY, J. M. (2001). Imputing physical health status scores missing owing to mortality. Medical Care 39, 61–71. R IBAUDO , H. J., T HOMPSON , S. G. AND A LLEN -M ERSH , T. G. (2001). A joint analysis of quality of life and survival using a random effect selection model. Statistics in Medicine 20, 1173–1184. ROBINS , J. M., ROTNITZKY, A. AND Z HAO , L. P. (1995). Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association 90, 106–121. RUBIN , D. B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley & Sons. S CHARFSTEIN , D. O., ROTNITZKY, A. AND ROBINS , J. M. (1999). Adjusting for nonignorable drop-out using semiparametric nonresponse models (with discussion). Journal of the American Statistical Association 94, 1096– 1146. [Received May 11, 2004; revised November 9, 2004; accepted for publication November 9, 2004]