Optimal precision of coarse structural nested mean models to estimate the effect of initiating ART in early and acute HIV infection
-
Judith J. Lok
Abstract
Time-dependent coarse structural nested mean models (coarse SNMMs) were developed to estimate treatment effects from longitudinal observational data. Coarse SNMMs estimate the combined effect of multiple treatment dosages and are thus useful to estimate the effect of treatments that are initiated and then never stopped. Coarse SNMMs lead to a large class of estimators, with widely varying estimates and standard errors. To optimize precision, we derive an explicit solution for the optimal coarse SNMM estimator. We apply our methods by estimating how the effect on immune reconstitution of initiating 1 year of ART depends on the time between HIV infection and ART initiation, in the early stages of HIV infection. The CDC and the WHO are encouraging HIV testing, leading to earlier HIV diagnoses. Thus, more treatment decisions need to be made in early and acute infection. However, evidence is lacking about the clinical benefits of initiating ART in early and acute HIV infection, with guidelines developed mostly from analyzing patients with chronic infection. In the simulations and our motivating HIV application, naive coarse SNMM estimators render useless inference, whereas our new fitting methods render informative analyses. We thus hope that this article leads to broader applicability of SNMMs.
1 Introduction
Several causal inference approaches exist to estimate treatment effects from longitudinal observational data. Time-dependent coarse structural nested mean models (coarse SNMMs, [1]), for which we develop an optimal estimation method, describe the effect of time-dependent treatments, conditional on patient characteristics at the time of treatment initiation. Coarse SNMMs model the mean difference between the outcome with treatment initiated at time
In contrast, coarse SNMMs directly estimate the effect of multiple treatment dosages, and they are thus useful to estimate the effect of a treatment such as ART that once initiated is intended to not be stopped. This relates to target trial emulation [7,8] as follows. In the HIV application, coarse SNMMs are used to estimate the effect of initiating ART, regardless of whether patients choose to remain on ART. Thus, the effect estimated is an intention-to-treat effect, not a per-protocol effect. Alternatively, if reliable data on stopping ART and the reasons for stopping had been available, one could have estimated the effect of initiating ART and continuing on ART thereafter, related to a per-protocol effect. In this latter case, one could censor individuals when they stopped ART and apply inverse probability of censoring weighting (IPCW) for the resulting artificial censoring.
Marginal structural models (MSMs [9,10]) are another class of models to estimate the effect of time-dependent treatments from observational data. MSMs estimate how treatment effects depend on baseline, but not time-dependent, covariates. Robins [6] provides a detailed comparison of MSMs and SNMMs.
In contrast to most of the literature on efficient estimation of treatment effects from observational data (Imbens [11] provides an overview), this article focuses on optimal estimation of the effect of a time-dependent treatment. In addition, coarse SNMMs can estimate how the treatment effect depends on time-dependent pretreatment characteristics, which is not the focus of most of the literature on efficient estimation of treatment effects.
This article provides a description of the estimation methods and the assumptions needed to consistently estimate coarse SNMMs with an outcome measured over time. Detailed proofs of consistency and asymptotic normality can be found in Lok and DeGruttola [1] and Robins [4]. We focus mainly on doubly robust estimation with optimal precision, and on what the results imply for the treatment decisions of HIV-infected patients diagnosed early.
Robins [2] and Lok and DeGruttola [1] for time-dependent outcomes derived a large class of estimating equations for coarse SNMMs, all leading to consistent, asymptotically normal estimators for the treatment effect. It turns out that both estimates and standard errors may depend considerably on the choice of estimating equations within this large class, and a naive choice may render useless inference. Finding this in our HIV example motivated the current article. To optimize precision, this article develops an estimator that has, under an extra condition, optimal precision within a class that includes the estimators from Lok and DeGruttola [1]. This optimal estimator leads to the smallest possible asymptotic variance. The optimal estimator is also doubly robust: it is consistent and asymptotically normal not only if the model for treatment initiation is correctly specified but also if a certain outcome-regression model is correctly specified. The optimal estimator combines weighting and regression, and is therefore a mixed method (compared with mixed methods for point exposures described by Imbens [11]). Also without the extra conditions needed for optimality and in the presence of censoring, the optimal estimator is doubly robust; it just may not be optimal in such settings.
Optimal precision is simpler for coarse SNMMs than for more traditional SNMMs (see [4,12] for theory on traditional SNMMs), because coarse SNMMs avoid the need for accumulating effects over time. This article therefore includes an accessible illustration of the steps involved in calculating optimal estimators. The Web-Appendix has proofs for all theorems of Section 7 for time-dependent coarse SNMMs, resulting in a self-contained example.
We implement the optimal estimator and compare it to other estimators for coarse SNMMs, in a simulation study and in our motivating HIV application. ART is the standard of care for HIV-infected patients. The CDC and the WHO are encouraging HIV testing [13,14], leading to earlier HIV diagnoses. Thus, more treatment decisions need to be made in early and acute HIV infection. However, evidence is lacking about the clinical benefits of initiating ART in early and acute HIV infection, with most patients in the START trial [15] having chronic infection at baseline. Therefore, we estimate how the effect on immune reconstitution of initiating one year of ART depends on the time between HIV infection and ART initiation in the early stages of HIV infection. This HIV application includes correction for informative censoring, and bootstrap confidence intervals (CIs). Web-appendix A proves consistency of the bootstrap for all our estimators, under regularity conditions.
Simulation Section 9 and Application Section 10 show that estimators optimizing for precision perform substantially better in practice than arbitrarily choosing an estimator within the class of coarse SNMMs. This leads to a definitive answer to the question in the HIV application: whether the effect of 1 year of ART substantially increases when ART initiation is delayed in HIV-infected patients diagnosed in early and acute HIV infection.
The rest of this article is organized as follows. Section 2 introduces the motivating data. Section 3 introduces the setting and notation. Section 4 describes the treatment effect model. Section 5 introduces the assumptions necessary to estimate the treatment effect. Section 6 derives a large class of estimators. Section 7 derives the optimal estimator. Section 7.2 proposes doubly robust estimators and shows that they lead to a smaller asymptotic variance. Section 7.3 presents a theorem that guarantees optimality of estimators within their class. Section 7.4 gives an explicit expression for optimal estimators for the parameters of coarse SNMMs with an outcome measured over time. Section 8 describes implementation of the estimators. Section 9 compares the optimal estimator with other estimators in a simulation study. Section 10 compares the optimal estimator with other estimators in the motivating HIV application. Section 11 concludes this article with a Discussion.
2 Motivating data
ART is the standard of care for HIV-infected patients. Guidelines regarding ART initiation have been changing [14,16–18]. Historically, ART initiation was often delayed, depending on the state of the immune system as measured by the CD4 count. The delays were intended to avoid early viral resistance to ART, to thus conserve ART options when those were deemed to be most needed: when CD4 counts were low. Current treatment guidelines [14,18] recommend ART initiation at the time of HIV diagnosis. HIV testing and ART initiation at the time of HIV diagnosis are increasingly implemented, especially in the developed world but also often in resource-limited settings, in part because of the UNAIDS 90-90-90 initiative [19]. 90-90-90 aimed that by 2020, 90% of all people living with HIV would know their HIV status, 90% of all people with diagnosed HIV infection would receive sustained ART, and 90% of all people receiving ART would have viral suppression. While 90-90-90 did not completely reach its goal, the number of HIV-infected individuals who know about their HIV status increased substantially: 81% of HIV-infected individuals knew about their HIV status in 2020. Global access to ART also increased substantially, with 67% of patients diagnosed with HIV on ART in 2020, and the number of patients on ART in 2020 “more than tripled since 2010” [20]. PEPFAR, the U.S. President’s Emergency Plan for AIDS Relief [21], helped enable these increases.
Many patients are diagnosed with HIV when they are already chronically infected, and for those patients, the START trial [15] conclusively demonstrated the benefit of initiating ART immediately. However, with only a small number of patients in the START trial in acute or early infection, there is not a lot of evidence of clinical benefit for initiating ART immediately in patients diagnosed early. Finding such evidence is increasingly important, with the CDC and the WHO encouraging HIV testing [13,14] and the WHO also encouraging partner testing [14], leading to earlier HIV diagnoses. Thus, more treatment decisions need to be made during early and acute HIV infection. Our investigations shed light on the effect of efforts to diagnose HIV early, if early diagnosis is combined with immediate ART initiation as currently recommended.
The decisions to initiate ART are not only important for the HIV-infected patients themselves but also for their partners: ART may not only improve a patient’s own outcomes but has also been shown to reduce the risk of HIV transmission [22–24].
The observational AIEDRP (Acute Infection and Early Disease Research Program) Core01 data describe 1762 HIV-infected patients diagnosed during acute and early HIV infection [25] and thus provide a unique opportunity to estimate the effect of ART in early and acute HIV infection. Dates of HIV infection have been estimated using an algorithm that incorporates clinical and laboratory data [25,26].
By using the AIEDRP Core01 data, we will estimate how the effect on immune reconstitution of initiating 1 year of ART depends on the time between HIV infection and ART initiation. The effect on immune reconstitution of 1 year of ART initiated
Lok and DeGruttola [1] showed that in the AIEDRP data, ART use depends on covariates, such as the current CD4 count, that are prognostic for the outcome CD4 count; and that this leads to substantial confounding by indication. They showed that there is strong evidence that in these observational data, as expected, patients with a worse prognosis were treated earlier. A naive analysis therefore leads to underestimation of the effect of ART, or even to a reversal of the sign of this effect.
3 Setting and notation
Initially, all patients are assumed to be followed at the same times
4 Time-dependent coarse SNMMs
Our model for treatment effect is similar to [2], but differs in that as in Lok and DeGruttola [1], it allows for a time-dependent outcome:
Definition 4.1
(Time-dependent coarse SNMMS). For
We assume a parametric model for the treatment effect
(note that
Rank preservation holds if
Assumption 4.2
(Parameterizing coarse SNMMs).
The following example motivated this work on estimation with optimal precision:
Example 4.3
(Effect of ART depending on the time between estimated date of HIV infection and treatment initiation in HIV-infected patients). We initially assume that
where
The treatment effect
Following [3,4], we use the propensity score [31],
Assumption 4.4
(Parameterizing the propensity score).
Specifying a model
5 No unmeasured confounding and consistency
As shown in [1–3,5], to distinguish between the treatment effect and confounding by indication, we assume no unmeasured confounding: that information is available on all factors that both (1) influence treatment decisions and (2) possibly predict a patient’s prognosis with respect to the outcome of interest.
Assumption 5.1
(No unmeasured confounding - formalization).
The observed outcome
If a patient is not treated until time
Assumption 5.2
(Consistency). If
6 Estimation: unbiased estimating equations
This section describes the estimation methods from [1,2]. Proofs of Theorems 6.3 and 6.4 can be found in Lok and DeGruttola [1].
Definition 6.1
On
Example 6.2
(Effect of ART depending on the time between HIV infection and treatment initiation in HIV-infected patients). In the setting of Example 4.3, on
For the true
Theorem 6.3
(Mimicking counterfactual outcomes). Under consistency
Assumption 5.2, for
Thus, under no unmeasured confounding Assumption 5.1,
Under no unmeasured confounding Assumption 5.1, given the past treatment and covariate history,
Theorem 6.4
(Unbiased estimating equations). Under no unmeasured confounding
Assumption 5.1
and consistency
Assumption 5.2, consider any
If furthermore
stacked with the estimating equations for
For identifiability of the estimator, one needs as many estimating equations as parameters, in this case by choosing the dimension of
If
7 Estimating equations leading to optimal precision
7.1 Assumptions and restrictions on the estimating equations
From the vast literature on unbiased estimating equations, see for example Van der Vaart [35] Chapter 5, Theorem 6.4 implies that under regularity conditions,
In the following, we restrict to estimators satisfying the regularity conditions needed for (4) and thus (3) to hold. Among such estimators, we derive the optimal choice of
7.2 Doubly robust estimators lead to increased precision
Definition 7.1
Let
First, consider estimating equations for
Theorem 7.2
(Replacement of estimating equations by more efficient ones). Under no unmeasured confounding
Assumption 5.1, consistency
Assumption 5.2
and the usual regularity conditions for the sandwich estimator for the variance (equation (3)),
The equations based on
Usually,
Theorem 7.3
Replacement of
For the estimators that solve
Theorem 7.4
For
As for other SNMMs ([37], 3.10 page 23), estimators for coarse SNMMs resulting from
Theorem 7.5
(Double robustness). The estimator
It follows that both for robustness and for efficiency, the estimators with
7.3 A theorem that guarantees optimal precision
Theorem 7.6, a consequence of Theorem 5.3 from [38], provides a sufficient criterion for
Theorem 7.6
(Optimal precision criterion with
then no other
7.4 Explicit expression for estimating equations leading to optimal precision for coarse SNMMs with a time-varying outcome
This section finds the
Assumption 7.7
(Homoscedasticity). For
does not depend on
Without homoscedasticity Assumption 7.7, the optimal estimator derived below is still doubly robust, it just may not be optimal. Assumption 7.7 is a homoscedasticity assumption: a conditional covariance does not depend on
This is not far-fetched: under no unmeasured confounding Assumption 5.1, because of Theorem 6.3, the conditional expectation given
Rank preservation holds if
Theorem 7.8 describes the estimator with optimal precision in an example:
Theorem 7.8
(Estimator with optimal precision). Suppose that
(compare with
Example 4.3). Suppose that homoscedasticity
Assumption 7.7
holds. For
where
where
The class of estimators that solve estimating equations of the form
Theorem 7.9
(Estimator with optimal precision). Formulating
Theorem 7.8
for different treatment effect models
The term
Replacing the conditional variance of
The inverse of
The estimator with optimal precision requires estimating additional nuisance parameters. In small samples, this may be an issue, but as for many efficient estimators (see e.g. [39–42]), it does not lead to a larger asymptotic variance if all models are correctly specified:
Theorem 7.10
Suppose
results in the same asymptotic variance for
Simultaneously solving the estimating equations (9) leads to the same estimator
Instead of estimating
Under regularity conditions, we can use the bootstrap to create CIs for (functions of)
Theorem 7.11
(Consistency of the bootstrap). Under regularity conditions, the nonparametric bootstrap for all estimators in Section 7 is consistent under the conditions already adopted for consistency and asymptotic normality. That is, bootstrap CIs have asymptotically correct coverage.
8 Implementing these methods to estimate the effect of ART in early and acute HIV infection: estimation steps
8.1 Implementation: general remarks
Section 8 details the implementation of the estimators proposed in this article in our motivating HIV application. The model for
We adopt a pooled logistic regression model for the treatment prediction model
8.2 The preliminary estimator
ψ
˜
As the preliminary estimator
With the treatment effect model of Example 4.3 equation (2),
This procedure leads to estimating equations that are linear in
8.3 The optimal estimator
The mimicking outcome
Alternatively, also after plugging in
This procedure leads to estimating equations that are linear in
8.4 For comparison, a naive choice
For comparison, we implement two nondoubly-robust estimators, based on Theorem 6.4 and not using the optimal precision theory developed here. For these Theorem 6.4-based estimators, since in the HIV application, interest lies in the effect of 1 year of ART,
In the HIV application we choose
9 Simulations
We simulate data with monthly visits, and base simulation choices on the AIEDRP Core01 data on HIV-infected patients. We use an autoregressive model to simulate the course of the CD4 count, which may be more realistic in months 6–30 than before month 6, given the different behavior of CD4 counts in the first 6 months after HIV infection (Web-Appendix C). Therefore, we simulate data in months 6–30 and estimate the effect of treatment initiation in months 6–18. Web-Appendix C details the simulations.
We simulate two scenarios: 1: 1,000 datasets with 1,000 observations each, and 2: 500 datasets with 5,000 observations each. We fit model (2) with two parameters,
Table 1 shows results for the following estimators: 1a and 1b: Naive choices of estimators (Section 8.4), not doubly robust. 2:
Table 1 shows great improvements in precision from applying our theory in comparison with naive choices of estimating equations (Table 1 estimators 1a and 1b, described in Section 8.4). Choosing
Simulations: Comparison of rMSE and bias for the various estimators
| Model | Two parameters | Three parameters | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
||||||
| rMSE | (bias) | rMSE | (bias) | rMSE | (bias) | rMSE | (bias) | rMSE | (bias) | |
| 1,000 patients in 1,000 datasets | ||||||||||
| 1a. q as in (10), not DR | 3.5 | (0.07) | 0.29 |
|
3570 | (107) | 784 |
|
38 | (1) |
| 1b. q as in (11), not DR | 1019 |
|
94 | (2) | 736 |
|
163 | (2) | 8.0 |
|
| 2. restricted approx. optimal, not DR | 2.1 |
|
0.17 | (0.003) | 10.3 | (1.4) | 1.9 |
|
0.084 | (0.01) |
| 3. restricted approx. optimal, DR | 2.1 |
|
0.17 | (0.003) | 6.6 | (0.1) | 1.2 |
|
0.049 | (0.001) |
| 4. working identity covariances, DR | 1.6 |
|
0.10 | (0.003) | 3.9 | (0.04) | 0.53 |
|
0.017 | (0.0005) |
| 5. approximately optimal, DR | 1.3 |
|
0.079 | (0.002) | 3.1 |
|
0.41 | (0.004) | 0.013 |
|
| 5,000 patients in 500 datasets | ||||||||||
| 1a.
|
1.6 | (0.05) | 0.13 |
|
285 | (8) | 58 |
|
2.7 | (0.07) |
| 1b.
|
187 |
|
18 | (0.8) | 529 | (10) | 112 |
|
5.4 | (0.1) |
| 2. restricted approx. optimal, not DR | 0.94 | (0.003) | 0.073 |
|
4.2 | (0.2) | 0.78 |
|
0.034 | (0.002) |
| 3. restricted approx. optimal, DR | 0.94 | (0.005) | 0.073 |
|
2.9 | (0.04) | 0.51 |
|
0.022 | (0.0003) |
| 4. working identity covariances, DR | 0.75 |
|
0.047 | (0.0009) | 1.7 | (0.06) | 0.23 |
|
0.0075 | (0.0005) |
| 5. approximately optimal, DR | 0.59 |
|
0.035 | (0.0007) | 1.4 |
|
0.18 |
|
0.0056 | (0.00003) |
rMSE: root mean squared error. DR: doubly robust.
1a. and 1b. Naive choices of estimators within class (Section 7.4), not doubly robust.
2.
3.
4. Like the optimal estimator, but with working identity covariance matrices (Sections 6 and 7.3), doubly robust.
5. Optimal under correct specification of all models (Sections 6 and 7.3), doubly robust.
For all parameters, the true parameter values lie in between the 2.5 and 97.5% quantiles (over the simulated datasets) of the estimated parameters for estimators 2–5.
For n = 1,000, we also investigate choosing sparser models (Web-Appendix C.2) in the expressions for the prediction of treatment duration (for
Figure 1 shows the results for the datasets with 1,000 observations. Figure 1 compares the performance of estimators of the effect of ART on the 1-year increase in the CD4 count due to ART initiated at various months since HIV infection. This
How does the effect of 1 year of treatment depend on its initiation time? Comparison of root mean squared errors. rMSE of estimators 2–5 of the simulation study. For example, for month 11, the quantity in Figure 1 is the root MSE for the estimator of the expected difference in the CD4 count at month
10 The effect of ART in HIV-infected patients in acute and early HIV infection
We estimate how the effect on immune reconstitution of initiating 1 year of ART depends on the time between the estimated date of HIV infection and ART initiation, in the early stages of HIV infection. The effect on immune reconstitution of 1 year of ART initiated
We estimate the effect of ART
In this HIV application,
We assume:
Assumption 10.1
(Parameterization of coarse SNMM). For
with
Since interest in this HIV application lies in the effect of 1 year of ART, that is,
Because ART is assumed to only change at visit months
Table 2 provides estimates and bootstrap
The AIEDRP data: Various estimators and their bootstrap 95% CIs
| Model |
|
(width CI) |
|
(width CI) |
|
(width CI) |
|---|---|---|---|---|---|---|
| Two-parameter model | ||||||
| 1a. q as in (12), not DR | 22.4 (18.9,25.8) | (6.9) |
|
(1.58) | — | |
| 1b. q as in (13), not DR | 43
|
(338) |
|
(157) | — | |
| 2. restricted approx. optimal, not DR | 22.3 (19.1,25.3) | (6.1) | 0.18
|
(1.53) | — | |
| 3. restricted approx. optimal, DR | 24.1 (20.8,27.2) | (6.4) |
|
(1.37) | — | |
| 4. working identity covariances, DR | 25.3 (22.2,28.5) | (6.3) |
|
(0.99) | — | |
| 5. approximately optimal, DR | 24.8 (20.2,29.0) | (8.7) |
|
(3.44) | — | |
| 2b. as 2., sensitivity analysis | 24.0 (20.6,27.6) | (7.0) |
|
(1.71) | — | |
| 3b. as 3, sensitivity analysis | 26.0 (22.7,29.3) | (6.6) |
|
(1.45) | — | |
| 4b. as 4, sensitivity analysis | 25.7 (22.4,29.0) | (6.5) |
|
(1.01) | — | |
| 5b. as 5, sensitivity analysis | 25.4 (20.1,30.0) | (9.9) |
|
(3.48) | — | |
| Three-parameter model | ||||||
| 1a.
|
39
|
(203) |
|
(131) | 1.2
|
(16.2) |
| 1b.
|
38
|
(334) |
|
(239) | 1.6
|
(34) |
| 2. restricted approx. optimal, not DR | 19.5 (14.8, 24.1) | (9.3) | 2.0
|
(4.7) |
|
(0.45) |
| 3. restricted approx. optimal, DR | 23.4 (18.5,28.0) | (9.5) |
|
(4.9) |
|
(0.51) |
| 4. working identity covariances, DR | 25.6 (21.7, 29.5) | (7.8) |
|
(3.0) | 0.0095
|
(0.20) |
| 5. approximately optimal, DR | 25.9 (19.0, 31.8) | (12.8) |
|
(6.5) |
|
(0.44) |
| 2b. as 2., sensitivity analysis | 24.8 (19.7,31.0) | (11.3) |
|
(5.8) | 0.06
|
(0.74) |
| 3b. as 3., sensitivity analysis | 25.9 (21.2,30.6) | (9.5) |
|
(4.3) |
|
(0.43) |
| 4b. as 4, sensitivity analysis | 27.3 (23.2,31.4) | (8.2) |
|
(2.9) | 0.06
|
(0.19) |
| 5b. as 5, sensitivity analysis | 30.4 (20.7,33.6) | (12.8) |
|
(5.9) | 0.22
|
(0.43) |
DR: doubly robust.
95% CI: 95% CI based on bootstrap, Efron’s percentile method.
1a and 1b: Naive choices of estimators within class (Section 7.4), not doubly robust.
2:
3:
4: Like the optimal estimator, but with working identity covariance matrices (Sections 6 and 7.3), doubly robust.
5: Optimal under correct specification of all models (Sections 6 and 7.3), doubly robust.
Figure 2 compares the performance of our estimators of the effect of ART on the 1-year CD4 count increase due to ART initiated at different months after the estimated date of HIV infection. For example, for month 11, the quantity in Figure 2 is the estimated expected difference in the CD4 count at month
Table 2 also describes the results of a sensitivity analysis. In this sensitivity analysis, treatment initiation and dropout are modeled using model selection techniques (details in Web-Appendix D). While model selection in principle invalidates the CIs, this sensitivity analysis indicates that the results are somewhat sensitive to model specification, but the clinical implications are the same as those of the main analysis.
How does the effect of 1 year of ART depend on its initiation time? Estimates of the effect of 1 year of ART based on the AIEDRP data. Estimators 2–5 applied to the AIEDRP data (with 95% CIs). For example, for month 11, the quantity in the figure is the estimate the expected difference in the CD4 count at month
The estimated effect on the CD4 count of 1 year of ART initiated in acute and early HIV infection is substantial and significant. The effect decreases somewhat with increasing time between HIV infection and ART initiation, but this trend is not significant. The estimated effect of 1 year of ART initiated at the time of HIV infection is 304 (95% CI (266, 341)), and if initiated 12 months after HIV infection, the effect is 270 (95% CI (220,323)). The difference between the effect of 1 year of ART initiated 12 months after versus at infection is
The analysis in this HIV example has limitations. The results are obtained by analyzing the observational AIEDRP Core01 data. As in traditional SNMMs, we assume that all confounders are measured. This assumption cannot be tested using the available data. The author collaborated closely with Susan Little and Davey Smith, infectious diseases MDs at UCSD, and Victor DeGruttola, ScD, at Harvard, an expert in the statistics of HIV, to decide which covariates to include in the analyses: the variables that are predictive of both ART initiation and the outcome. Moreover, we assume that the treatment effect model of Assumption 10.1 is correctly specified. Yang and Lok [46] tested Assumption 10.1 and concluded that the AIEDRP data do not provide evidence that Assumption 10.1 is violated; however, this may be due to a limited sample size. In addition, one or more of the nuisance parameter models may be misspecified. That being said, only the treatment initiation model and the outcome regression model affect consistency and asymptotic normality, and double robustness implies that misspecification of only one of these two models preserves consistency and asymptotic normality.
Summarizing this HIV example: with the increased precision of our estimators, we conclude that ART is substantially effective when initiated in early and acute HIV infection. This contradict claims that ART might not be substantially effective in acute and early HIV infection. Current efforts to diagnose HIV early can be expected to be effective also in acute and early HIV infection, if diagnosis is combined with immediate ART initiation as currently recommended.
11 Discussion
Both in the simulation study and in the motivating HIV application, most of the naive coarse SNMM estimators lead to useless inference. This can realistically happen in practice if the nuisance function
Using optimal estimators substantially improves the precision of coarse SNMMs. In the simulation study, the optimal doubly robust estimator results in useful inference and performs best. In the HIV application, our methods also substantially improve precision. In the HIV application, the best performance is by a doubly robust estimator related to the optimal estimator, but using working identity covariance matrices, similar to the identity working covariance matrices approach in GEEs (e.g., [42] Section 4.6).
The suboptimal behavior of the optimal estimator as implemented in the HIV application may have several causes. It could be due to a combination of limited sample size and censoring. It could also be due to model (mis)specification of the nuisance parameter models. In particular, it is likely that in the HIV application,
| Parameter | Estimate | SE | 95% CI |
|
|
| CD4sqrtpred | 311 | 80 | (154,468) | 3.89 | 0.0001. |
These models for different
It is also possible that by only assuming that
One could hope that simplifying assumptions on the working covariance matrix might improve efficiency. Similar to model (2), one could consider models for this conditional variance like
on
Efficient estimation under heteroscedasticity is an interesting topic for future research.
In practice, if there is no reasonable subject-matter-informed way to specify a covariance matrix that depends on covariates, it may be more feasible to work with a working covariance matrix that does not depend on covariates. Given the theory on GEEs, our simulations, and the results in our HIV application, the working identity matrix could be used for the primary analysis, and the optimal estimator assuming that the matrix
An interesting topic for future research is investigating the properties of the proposed estimators when the treatment effect model is misspecified. Yang and Lok [46] developed goodness-of-fit tests for time-dependent coarse structural nested mean models and applied these to our HIV application by investigating violations of the model of Assumption 10.1; they found no significant deviations from this model. However, nonparametric estimation of the treatment effect
Yang and Lok [54] considered sensitivity analyses to violations to the assumption of no unmeasured confounding for time-dependent coarse structural nested mean models, and applied these to our HIV application.
Because the nuisance parameter models for treatment initiation, censoring,
In the HIV application, the improved precision of coarse SNMMs has implications for clinical practice. Currently, more HIV-infected patients are diagnosed in acute and early infection, due to increased HIV testing [13,14], including increased partner testing [14]. With increased HIV testing leading to earlier HIV diagnoses, more treatment decisions need to be made in early and acute infection. We estimate that the effect on the CD4 count of 1 year of ART initiated at the time of HIV infection is 304 (95% CI (266, 341)), and if initiated 12 months after HIV infection the effect is 270 (95% CI (220, 323)). Thus, our results indicate that also in acute and early HIV infection, ART initiation at the time of diagnosis substantially benefits patients. This result is also important for partners of HIV-infected patients because ART reduces the risk the risk of HIV transmission [22–24].
For a sufficiently smooth doubly robust estimator, if both models are correctly specified and each of these models is estimated at rate
The main text focuses on coarse SNMMs with a time-varying outcome; Web-appendix B shows how the calculations simplify with an outcome measured at the end of the study.
Currently, none of the theory presented in this article is routinely used when SNMMs are applied. That means, also the estimator with a working identity matrix instead of
Thus, we summarize that precision of estimators for coarse SNMMs depends substantially on the estimating equations chosen. The substantial improvement we find by choosing estimating equations leading to optimal precision creates a good starting point for a comparison of precision between MSMs and coarse SNMMs for settings where the treatment effect only depends on baseline covariates (in those settings, MSMs and SNMMs estimate the same quantities; it has been conjectured [6] that SNMMs have better precision). The substantial improvements in precision also suggest that the use of optimal estimators will encourage more widespread use of coarse SNMMs.
Acknowledgements
The author is grateful to the patients who volunteered for AIEDRP, to the AIEDRP study team, and to Susan Little, Davey Smith, and Christy Anderson for their help and advice in interpreting the AIEDRP data. The author thanks Ray Griner for extensive help with the SAS programming, and Shu Yang for programming the estimators in R. The author thanks James Robins and Victor DeGruttola for insightful and fruitful discussions.
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Funding information: This work was supported by the Milton Fund, the Career Incubator Fund from the Harvard School of Public Health, and the National Institutes of Health [NIAID R01 AI100762 to Lok, AI074621 and AI43638 (AIEDRP)]. The content is solely the author’s responsibility and does not represent the official views of the NIH.
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Author contributions: The author confirms the sole responsibility for the conception of the study, presented results and manuscript preparation.
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Conflict of interest: The author states no conflicts of interest.
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Data availability statement: The AIEDRP Core01 data and the SAS code are available upon request to the author.
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Articles in the same Issue
- Research Articles
- Decision making, symmetry and structure: Justifying causal interventions
- Targeted maximum likelihood based estimation for longitudinal mediation analysis
- Optimal precision of coarse structural nested mean models to estimate the effect of initiating ART in early and acute HIV infection
- Targeting mediating mechanisms of social disparities with an interventional effects framework, applied to the gender pay gap in Western Germany
- Role of placebo samples in observational studies
- Combining observational and experimental data for causal inference considering data privacy
- Recovery and inference of causal effects with sequential adjustment for confounding and attrition
- Conservative inference for counterfactuals
- Treatment effect estimation with observational network data using machine learning
- Causal structure learning in directed, possibly cyclic, graphical models
- Mediated probabilities of causation
- Beyond conditional averages: Estimating the individual causal effect distribution
- Matching estimators of causal effects in clustered observational studies
- Ancestor regression in structural vector autoregressive models
- Single proxy synthetic control
- Bounds on the fixed effects estimand in the presence of heterogeneous assignment propensities
- Minimax rates and adaptivity in combining experimental and observational data
- Highly adaptive Lasso for estimation of heterogeneous treatment effects and treatment recommendation
- A clarification on the links between potential outcomes and do-interventions
- Review Article
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