We now demonstrate how the foregoing development is specialized to a longitudinal study with dropout by application to ACTG 175. Recall that interest focuses on *β* = *E*(*Y* ), where *Y* = *Y*_{5} = CD4 count at 96±5 weeks, the mean CD4 count for the HIV-infected population if assigned to regimens ZDV+ddI, ZDV+ddC, or ddI; *M* = 4; and *m*(*Z*, *β*) = *Y* − *β*. The baseline covariate vector *X* includes age (years); weight (kg); Karnofsky score (karnof), an index reflecting ability to perform activities of daily living (0 to 100); days of prior antiretroviral therapy (antidays); and binary indicator variables for hemophilia (hemo), homosexual activity (homo), history of intravenous drug use (drug), ZDV within 30 days of the trial, race (0 = white), gender (0 = female), antiretroviral history (hist; 0 = naive, 1 = experienced), and symptomatic status (symp; 0 = asymptomatic).

We consider estimation of

*β* by the simple IPW estimator, which corresponds to solving (

8) with all of the

*h*_{r}{

*G*_{r}(

*Z*),

*ξ*} set equal to zero,

_{ipw}; two versions of the estimator

_{br}_{*} of

Bang and Robins (2005); and two versions of the proposed estimator

_{opt}_{*}. The CAR assumption (

1) is not unreasonable; it is widely acknowledged in longitudinal HIV studies that subjects with baseline characteristics such as intravenous drug use and/or lower evolving CD4 counts prior to dropout, reflecting compromised immunologic status, may be more likely to drop out. Under CAR/MAR, the naive estimator, the sample mean of CD4 counts for the complete cases at 96±5 weeks, equal to 348.7 cells/mm

^{3} with standard error (SE) 5.76, thus may be an overestimate if subjects with poorer immunologic status are more likely to drop out.

Using the notation at the end of Section 2, we represent the models we now present by replacing

*C* by

*R* and

*G*_{r}(

*Z*) by

_{j} and indexing visits by

*j* in obvious fashion. For use with all estimators, logistic regression models for the discrete hazards at each

*j* were developed with main effects in elements of

_{j} identified via separate ML fits at each

*j* to the data on all subjects with

*R* ≥

*j* using forward selection with entry level of significance 0.15; we also considered other levels, with no qualitative differences. This yielded models

,

*j* = 1, …, 4, where expit(

*u*) =

*e*^{u}/(1 +

*e*^{u}),

, and

is the subset of

_{j} selected;

, and

. Finding the MLE

then reduced to carrying out individual ML fits of these models for each

*j*.

Noting that

*E*{

*m*(

*Z*)|

_{j}} =

*E*(

*Y* |

_{j}) −

*β* for each

*j*, developing models

*h*_{j}(

_{j},

*ξ*) and

,

*j* = 1, …, 4, for

_{opt}_{*} and

_{br}_{*}, respectively, corresponds to developing models for the regression of 96±5 week CD4 count on

_{j}; i.e., for

*E*(

*Y*|

*Y*_{1}, …,

*Y*_{j},

*X*). To develop models

*h*_{j}(

_{j},

*ξ*), we assumed that the longitudinal data follow the linear mixed model

where

*α*_{i} = (

*α*_{0}_{i},

*α*_{1}_{i})

^{T} ~

*N*{(

*μ*_{α}_{0},

*μ*_{α}_{1})

^{T},Σ

_{α}};

are iid for all

*i*,

*j*; the

*α*_{i},

*i* = 1, …,

*n*, are independent of each other and all

*e*_{ij}; and

= (weight,karnof,hist,symp) was identified by fitting (

14) by ML with all of

*X* included and retaining only those elements for which the usual

*t*-test of whether or not the associated coefficient is equal to zero had p-value less than 0.05. Under (

14), standard results for the multivariate normal distribution yield the required conditional expectations

*E*(

*Y*|

*Y*_{1}, …,

*Y*_{j},

*X*) =

*E*(

*Y*|

*Y*_{1}, …,

*Y*_{j},

), all of which depend on the common

; see Web Appendix E. To obtain the first version of

_{opt}_{*},

, say, we estimated

*ξ* in the implied models

*h*_{j}(

_{j},

*ξ*) using (

12). For direct comparison of the Bang-Robins approach to the proposed method using the same covariate information, we let

for each

*j* be linear regression models including main effects in all CD4 counts up through

*j* and

and estimated the

*ξ*_{j} by separate ordinary least squares (OLS) regressions for each

*j* based on the observed data at

*j*; denote the resulting estimator by

. For a second version of

_{br}_{*}, denoted

, we instead considered for each

*j* all of

*Y*_{1}, …,

*Y*_{j},

*X* as potential main effects in linear models, and developed and fit these separately by OLS with forward selection on the elements of

*X*. The resulting

contained (age,karnof,race,gender,hist), (age,hemo,drug,karnof,antidays,gender,symp), (age,hemo,karnof,gender), and (age,hemo,karnof) for

*j* = 1, 2, 3, 4, respectively, along with (

*Y*_{1}, …,

*Y*_{j}). We implemented both

and

as described by

Bang and Robins (2005, Section 3). A second version of the proposed estimator,

, was derived by, rather than taking

*ξ* common across

*j*, letting the models implied by (

14) for each

*j* have

*j*-specific parameters

*ξ*_{j}. We then let

, and estimated

*ξ* using (

12). For all estimators, we obtained SEs via the sandwich technique.

Estimation of

*ξ* by solution of the estimating equations based on (

12) may be carried out via standard techniques, such as a Newton-Raphson updating scheme. Thus, in principle, implementation is no more complex than for the Bang-Robins approach, where the

*ξ*_{r} are estimated by separate solutions to

*M* sets of estimating equations. Computation of

_{opt}_{*} is likely a higher-dimensional problem than the separate ones; however, here and in Section 6, we encountered no numerical difficulties with either method.

For comparison, we also fit the mixed model (

14) directly by normal ML using SAS proc mixed (

SAS Institute, 2009) and estimated

*β* by the marginal predicted value

_{mixed} at 96±5 weeks obtained by setting

equal to its sample mean with SE from the associated estimate statement, which treats the sample mean of

as fixed.

The resulting

_{ipw} = 332.96, (SE 5.10),

. Recognizing that this is a single data set, it is encouraging to note that the estimates are virtually identical, and, consistent with the theory, the IPW estimator is inefficient relative to the AIPW competitors on the basis of estimated SE. Moreover, both versions of the proposed estimator achieve or surpass the performance of the Bang and Robins estimators, although not dramatically, and all estimates are indeed smaller than the naive estimate, as expected. We also obtained

_{mixed} = 346.20 (4.92); in contrast to the AIPW estimates, this estimate is not appreciably different from the naive.

We deliberately chose the ACTG 175 study to demonstrate the methods because of a unique feature that highlights the advantage of consideration of the general setting of monotone coarsening. Although subjects in the study ceased to attend clinic visits and provide CD4 counts after some time point, so effectively did “drop out” of the study with respect to the response of interest, follow-up of all subjects continued. Thus, additional information on each subject throughout the entire 96-week period, regardless of whether or not s/he ceased to attend clinic visits, is available, which we summarize in four time-dependent covariates dis_{ij} = *I*{subject *i* discontinued study treatment during (*t*_{j}, *t*_{j}_{+1}]}, *j* = 1, …, 4; we did not include dis_{j} in the definitions of *L*_{j} in the foregoing analysis for illustrative simplicity, although we could have done so. Acknowledging these data takes this situation out of the realm of the standard longitudinal dropout setting and notation at the end of Section 2, which assumes that no data are available beyond visit *j* if the subject was last seen at *j*. However, the present setting may still be cast as a case of monotone coarsening and these additional data incorporated in the analysis, as we now demonstrate.

Reverting to the general notation, *Z* = (*X*, *Y*_{1}, *Y*_{2}, *Y*_{3}, *Y*_{4}, *Y*,dis_{1},dis_{2},dis_{3},dis_{4}); and, with *C* = *r* indicating that the subject last provided a CD4 count at visit *r*, we observe *G*_{r}(*Z*) = (*X*, *Y*_{1}, …, *Y*_{r},dis_{1},dis_{2},dis_{3},dis_{4}), *r* = 1, …, 4, and *G*_{∞}(*Z*) = *Z* for *r* = ∞. Clearly, the coarsened data satisfy the monotonicity requirement. This demonstrates that one need not think strictly temporally in characterizing monotone coarsening in longitudinal data.

Recall that the goal is to estimate

*β* = mean CD4 count at 96±5 weeks for the population

*assigned* to ZDV+ddI, ZDV+ddC, or ddI, so regardless of whether or not subjects stayed on these regimens for the entire 96 weeks. We illustrate by calculating

_{opt}_{*} and

_{br}_{*} as follows. For both estimators, we derived the discrete hazard models by the same strategy as in the previous analysis, considering all elements of

*G*_{r}(

*Z*) as possible main effects in the linear predictor of a logistic regression model for each

*r* and retaining a subset of these terms by forward selection. This yielded logistic regression models

*λ*_{r}{

*G*_{r}(

*Z*),

*ψ*_{r}) that included main effects for (

*Y*_{1},age,drug,karnof,antidays,race,hist,symp), (

*Y*_{2},age,homo,drug,antidays,karnof,dis

_{1},dis

_{2}), (

*Y*_{3},dis

_{1},dis

_{2}), and (

*Y*_{1},

*Y*_{3},hemo,karnof,race,dis

_{2},dis

_{4}) for

*r* = 1, 2, 3, 4, respectively. To derive models

*h*_{r}{

*G*_{r}(

*Z*),

*ξ*) for

_{opt}_{*}, we used the form of

*E*(

*Y*|

*X*,

*Y*_{1}, …,

*Y*_{r},dis

_{1},dis

_{2},dis

_{3},dis

_{4}) implied by the linear mixed model

*Y*_{ir} =

*α*_{0}_{i} +

*α*_{1}_{i}t_{ir} +

*γ*^{T}_{i} +

_{1}*I*(

*r* ≥ 3)dis

_{i}_{2} +

_{2}*I*(

*r* = 5)dis

_{i}_{4} +

*e*_{ir}, where the random effects and within-subject deviations are normal as above, and now

= (weight,karnof,symp); see Web Appendix E. The common

was then estimated via (

12). For

_{br}_{*}, we took

, so

, which was estimated by OLS for each

*r*. Using these estimated discrete hazards to also calculate

_{ipw},

_{ipw} = 325.32 (5.80),

_{opt}_{*} = 328.10 (5.05), and

_{br}_{*} = 327.46 (5.49). As before, performance of the estimators based on estimated SEs is consistent with the theory.