Comparison of Approaches to Weight Truncation for Marginal Structural Cox Models

2.1 Notation and definition

We consider a longitudinal study in which N subjects are followed at regular time intervals (e.g., every 6 months) from entry into the cohort (time zero), and at each follow-up visit k, k = 1,…, m, subjects’ covariates are measured and their current treatment assignments are then decided. We denote by T_i, the observed follow-up time of subject i, and V_i, the vector of baseline covariates, measured at time zero. For subject i at time t, let A_i(t) be a binary indicator of the received treatment and L_i(t) be the current value of the time-dependent covariate L. Since subjects are assessed only at discrete time intervals, we assume that the treatment A_i(t) and covariate L_i(t) remain the same within the interval between any two adjacent assessments . Let Y_i(t) = 1 if subject i had an event at time t, and 0 otherwise. Those who are either lost to follow-up or do not have an event until the end of study are rightly censored at the respective time. We use an overbar to represent a covariate history, thus, Ā(t) indicates the treatment history up to t, and indicates the history of the time-dependent confounder until that time.

Let T_ā denote a random variable representing a subject’s counterfactual survival time, followed a given treatment history ā(t) = {a(u);0 ≤ u ≤ t}. For each ā, the COX MSM is defined as (Hernán et al., 2000) follows:

[1]

where λ_Tā(t|V) is the hazard at time t for a subject with the baseline covariate vector V had, contrary to fact, the subject followed the treatment history ā; β₁ and β₂ are unknown causal parameters; λ₀(t) is the unspecified baseline hazard at time t for a subject who is never treated ā(t) = , with V = 0; and f (·) is an analyst-defined function of the treatment history. The choice of baseline covariates to include in an MSM falls to the analyst. Of course, because the true form of the model is not known, as the number of included covariates increases, the risk that some of their effects are misspecified increases as well.

As noted above, the causal parameters β of a MSM can be estimated using IPTW to account for the time-dependent confounding effect of (Robins, 1997, 1999a). A commonly used weight is the stabilized weight, , defined by Robins et al. (2000), as follows:

[2]

where m(t) is the total number of visits up to t, including the initial visit (k = 1, t = 0). The denominator of the stabilized weight is the probability that a subject received his own observed treatment at time t, A(t), given his own past treatment history, confounder history, and baseline covariates. The numerator, the probability that a subject received his observed treatment at time t conditional on only his past treatment history and baseline covariates, but not on time-dependent confounder , is used to reduce the variability of the original “unstabilized”weights (Hernán et al., 2000). By accounting the treatment history and baseline covariates in both the numerator and the denominator, the stabilized weight reflects an incremental effect of the time-varying confounder on the current treatment choice, over and above the other determinants of treatment.

Four assumptions are needed to consistently estimate the causal parameters with the IPTW estimator: consistency, exchangeability, no-model misspecification, and positivity (Cole and Hernán, 2008). The consistency assumption is the fundamental assumption that links the counterfactual data T_ā to the observed data T_Ā; it states that the observed outcome T_Ā is equal to the counterfactual outcome T_ā under the observed treatment history, that is, ā = Ā. Exchangeability states that given the recorded covariates in V and L(t), there are no other unmeasured confounders. The no-model misspecification assumption postulates that the unknown probabilities are modeled through a correctly specified model, and the marginal structural model is correctly specified. However, in practice, any unsaturated models may be misspecified, and thus may lead to biased causal effect estimates (Robins and Hernán, 2008). Neugebauer and van der Laan (2007) developed a nonparametric MSM (NPMSM) approach, which does not require correct specification of a parametric model but relies on a (potentially misspecified) working model instead.

2.2 Impact of the ETA violation on the IPTW estimates

A practical ETA violation can lead to some observations being assigned very large weights. To illustrate how observations with the most extreme weights (i.e., the most influential observations) can affect the IPTW estimates, we purposely selected four simulated samples that yielded IPTW estimates for the current treatment effect A(j) that were exhibited a relative difference of more than 50% from the true parameter value. The highest weight in each selected sample was at least 40% higher than the second highest weight in the same sample.

For each selected sample, we generated B = 100 resamples, and for each resample, the untruncated IPTW estimate was calculated. The impact of the most highly weighted observation on the IPTW estimate can then be observed from the distribution of the resample-specific estimates by tracing whether a resample included this observation. In both top panels of Figure 1, it is evident that the resample-specific estimates have a strictly bimodal distribution: in all those resamples in which the observation with the extreme weight was included (dark dots) the treatment effect is highly overestimated, while all other resamples (light dots) yield estimates reasonably close to the true effect (indicated by the dashed line). In both samples, the event associated with the highest weight occurs for subjects who are treated and have high values of the confounder, which could suggest that treatment is associated with a higher hazard of the event. Thus, because of the extremely high weight assigned to these unusual observations, the weighted maximum partial likelihood estimator (MPLE) of the treatment effect yields a counter-intuitive log(HR) > 0. The impact of those observations on the IPTW estimates can also be illustrated by the dramatic change in the estimates that was observed when we just artificially reversed the survival status of the observation associated with highest weight from “event” to “no event”: the estimated effects of current treatment A(j) changed from 0.444 to −0.654 and from 0.665 to −1.179 for samples (a) and (b), respectively.

In contrast, the two bottom panels of Figure 1 show results for the two samples where those observations that are assigned extreme weights are not associated with the event. In those two samples, the inclusion or the exclusion of the observation with the highest weight has only a minor impact on the untruncated IPTW estimates. This confirms that censored individuals are less influential in the survival analysis.

The examples presented here indicate that a single observation may determine and indeed bias the results for the entire sample. This can be avoided by truncation of the weights, which reduces the influence of some observations, e.g., with fixed 99.5th percentile weight truncation, the estimated effects of current treatment A(j) changed from 0.444 to −0.569 and 0.665 to −0.578 for samples (a) and (b), respectively. We are thus motivated to find a principled and optimal means of performing the truncation.

2.3 Alternative truncated IPTW estimators of the Cox MSM

2.3.1 The problem

Since the variability of the IPTW estimators is typically due to the impact of a few observations with above-illustrated extreme weights, weight truncation has been proposed to limit the maximum contribution that any observation in the data can have on the fitted Cox MSM at any time t (Cole and Hernán, 2008). Specifically, the truncated IPTW estimator relies on weights that are truncated at a prespecified constant M: , i.e., any weight greater than Mis replaced by the value (truncated weight) M (Cole and Hernán, 2008).

Figure 1

Investigation of the impact of the extreme weights on the IPTW estimates of A(j) using resampling. Dark dots indicate that the subject with the highest weight in the original sample was included in a resample. The solid line represents the estimated effect of A(j) in the original sample and the dashed line represents the true effect. The treatment (A), the event (Y), and the weight for the observation with the highest weight are indicated in the title of each panel.

The direct effect of weight truncation is the reduction in the instability of the IPTW estimator (Moore et al., 2009). However, a consequence of weight truncation is the introduction of bias in the estimated weights, which in turn leads to bias of the IPTW estimator of the treatment effect. Thus, weight truncation requires a bias-variance trade-off (Cole and Hernán, 2008). Ideally, an optimal truncation level should be selected so that the corresponding truncated IPTW estimator yields the minimum mean-squared error (MSE) (Bembom and van der Laan, 2008b). However, it is difficult, in practical applications, to assess the MSE of an IPTW estimator. Bembom and van der Laan (2008b) proposed a closed-form estimate for the expected MSE of a truncated IPTW estimator in order to select an appropriate truncation level; however, the approach is limited to cross-sectional data with either continuous or binary outcomes. Although it is possible to extend the approach for longitudinal data, the computation can become burdensome as the number of time intervals increases and the development of a similar closed-form equation for longitudinal, right censored time-to-event data is not straightforward.

2.3.2 Proposed empirical criteria to select the truncation level

Based on the principle of MSE minimization considered by Bembom and van der Laan (2008b), we propose two easily implementable approaches to approximate the expected MSE, based on which the optimal truncation level can be selected. Specifically, we estimate the expected MSE of the truncated IPTW estimator for each of several truncation levels considered, then select the truncation level that corresponds to the smallest MSE. The MSE of an estimator is defined as follows:

[3]

where M denotes the truncation level, β denotes the true causal treatment effect, denotes the truncated IPTW estimate, SD denotes the standard deviation of the estimator, and MSE denotes the mean-squared error.

In practice, it is impossible to calculate the true MSE because β is unknown. However, if one were able to find an estimable quantity which is monotonically associated with the true MSE, it would still be possible to identify the optimal truncation level based on this approximating quantity. We consider two approaches to select the optimal truncation level that approximate the true MSE (specifically, the bias component) in different ways. In both approaches, the variance component of MSE in eq. [3] is estimated by the square of the estimated standard error (SE) of the corresponding truncated IPTW estimator.

The first approach to approximate the bias component in eq. [3] uses an estimable “proxy” for the unknown β. The expected MSE is then calculated as follows:

[4]

where SE represents the standard error of the truncated ITPW estimator , and denotes the proxy, or substitute, for the true causal parameter β. Three candidates are considered for : (i) the untruncated IPTW estimate; (ii) the truncated IPTW estimate with weights truncated at the 99.5th percentile of sample weights; and (iii) the truncated IPTW estimate based on the 99th percentile. The percentiles considered (99.5th and 99th) are consistent with Cole and Hernán (2008), who suggested truncating weights at the first and 99th percentiles. In preliminary simulations, no benefit was observed with truncation of weights at the left tail of the weight distribution. Thus in all the percentile-related approaches considered in this article, weights are truncated only in the right tail (that is, restricting only the maximum weights).

We also consider a second approach, in which the data are divided into halves, and the untruncated IPTW estimate from one subset is used as the proxy for the other half, and vice versa. That is, the data are first randomly divided into two equal-sized subsets. The MSE is then estimated using eq. [4] separately for each subset by using the untruncated IPTW estimate from the other subset as the proxy . The average of the two estimated MSEs from the two subsets is used to approximate the expected MSE of the entire dataset. Since we require a large sample size to ensure a better estimate of the “truth” from one subset, increasing the number of folds may be risky as it may result in small sample size in each subset. Therefore, in order to improve the estimation, we considered repeating this two-fold procedure r times, taking the mean of the r MSEs as the final estimate of MSE. The rationale behind the repetition of the cross-validation (CV) r ≥ 2 times is to reduce the possible impact of sampling error of the dissimilarity of the two halves of the datasets. We refer to this cross-validation-like method as the CV approach.

3.1 The data

We simulated a prospective study of a hypothetical cohort of N HIV-positive patients. Using the notation of Section 2.1, for the ith (i = 1,…,N) patient, the event time, from the start of the follow-up to an AIDS-defining event, is denoted by T_i, measured in years. Patients were evaluated every 6 months, and at each visit, the decision to initiate, continue, or interrupt treatment was made. At each visit j, (j = 1,…,m), the time-varying binary treatment, highly active antiretroviral therapy (HAART), is denoted by A_i(j) where A_i(j) = 1 if treatment was received and A_i(j)=0 otherwise, and a time-dependent continuous confounder, CD4 cell count, is denoted by L_i(j). Whether a subject had an event in the interval between visits [j, j + 1) is denoted by Y_i(j) with 1 indicating that the event occurred and 0 otherwise. The only baseline covariate was the pretreatment value of the confounder: L_i(1). The time-dependent covariate L_i(j) was assumed to act as both a confounder and a mediator for the treatment effect on survival throughout. The causal diagram is shown in Figure 2 with the subscript for omitted subject.

The data were generated using the same methods and data-generating distributions as used in Xiao et al. (2010). The data generation algorithm was briefly described in Appendix A.1, and the details are described in section 3.2.1 of Xiao et al. (2010). R code is available on request from the corresponding author. In brief, we generated baseline covariates from a uniform distribution, and then alternated between (i) generating visit-specific treatment decisions (on or off treatment) using a logistic model, which depended on the most recent confounder and treatment values and (ii) generating the time-varying confounder whose value depended on its most recent values and on the most recent treatment assignment. The expected time-to-event was generated using a conditional Cox PH model in which a subject’s hazard of having an event at a given time depended only on his current values of confounders and treatment. The noninformative censoring time was generated from a uniform distribution. The observed time-to-event and the corresponding censoring status were then determined by comparing the expected time-to-event, the generated censoring time, and the maximum follow-up time. Note that, unlike the algorithm of Young et al. (2009), this data generation approach does not use the exact marginal causal parameter β. However, in situations such as the simulations we performed, where the event rate in any interval is small (on an average, about 4% in each of the ten between-visits intervals), the noncollapsibility of the hazard ratios is negligible and hence so is the difference between the marginal and conditional parameters.

Figure 2

Causal diagram for the data-generating mechanism of the hypothetical cohort of HIV-positive patients used in the simulation study.

Two thousand independent random samples were generated, and sample sizes of N = 400, 2,000, and 10,000 were considered. The maximum follow-up time was 5 years (m = 10 visits). The average censoring rate for the main scenario, with N = 2,000, was approximately 60%. Appendix A provides information on the true models used to generate the data, the underlying assumptions, as well as the basic descriptive statistics on the distribution of the estimated inverse-probability-of-treatment (IPT) weights at selected time points.

3.2 Analysis of the simulated data

To estimate the total causal effect of current treatment A(j) (which acts directly on the outcome) and past treatment A(j − 1) (which acts indirectly via L(j)), on the time-to-event, we fitted the following marginal structural Cox model for each sample:

[5]

with weights truncated according to the different approaches described in Section 2.3.2.

We analyzed each simulated data using several different IPTW estimators of the Cox MSM, as given in Table 1. The untruncated IPTW estimator of the Cox MSM (Estimator 1) was compared with different weight truncation approaches, including weights truncated at (i) fixed values, (ii) fixed percentiles of the weight distribution in a given sample, and (iii) truncation levels selected using the proposed approaches described in Section 2.3.2. For fixed values of truncation, we considered a total of 23 values (spread between 3 and 100: 3–15 incremented by 1, 20–50 incremented by 5, 75, and 100); for fixed percentiles, we used the 99.5th (Estimator 2) and 99th (Estimator 3) percentiles of the sample distribution of stabilized weights. Estimator 4 used the untruncated IPTW estimate as the proxy , while Estimators 5 and 6 used the 99.5th and 99th percentile truncated IPTW estimates as the proxy, respectively. Estimator 7 is based on the two-fold cross-validation approach with r = 1 or 5, and Estimator 8 corresponds to the “oracle scenario” that uses the true causal parameters in the calculation of the MSE. For each estimator, the mean of the MSEs for A(j) and A(j − 1) was calculated to estimate the overall MSE for a given truncated estimator, and the optimal truncation level for a given sample was selected to correspond to the minimum estimated MSE.

Stabilized weights were used throughout and calculated for each person-visit, using eq. [2]. The denominator (i.e., the treatment assignment probability) was estimated with the correctly specified logistic regression model. Specifically, it was calculated as the estimated probability of receiving the observed treatment, A(j), for subject i at visit j, conditional on the current value of the confounder, L_i(j), and the indicator of the previous treatment, A_i(j − 1). The numerator was calculated as the probability of receiving the observed treatment estimated as a function of the baseline value, L_i(1), and treatment history, A_i(j − 1), only. The marginal structural Cox models were then estimated via a time-dependent weighted Cox regression model adapted for the person-visits data (Xiao et al., 2010). The robust variance estimators were used to account for within-subject correlation induced by the use of time-dependent weights. All analyses were carried out using R (R Development Core Team, 2011).

The bias of the IPTW estimators was estimated as the mean difference between the 2,000 estimates and the corresponding true value of the respective causal parameter. The ratio of the SD of the 2,000 estimates to the mean SE was calculated to evaluate the accuracy of the SE estimators. To quantify the bias-variance trade-off, the root mean-squared error (RMSE) was calculated as the square root of the sum of squared bias and variance. The 95% coverage rate was calculated as the proportion of samples in which the nominal 95% confidence intervals (CIs) included the corresponding true parameter.

Table 1

Description of different estimators of the marginal structural Cox model.

Description	Specification	Estimator	Label
Untruncated weights		1
Weight truncated at a fixed	99.5th	2
percentile of weights distribution	99.0th	3
Weight truncated at the selected	Estimator 1	4	Untruncated
level giving minimum expected MSE,	Estimator 2	5	Perc995
which was calculated with =	Estimator 3	6	Perc99
	2-fold cross-validation	7	CV
	True causal parameters	8	True

We assessed the performance of the proposed weight truncation schemes across different sample sizes (N = 400, 2,000, and 10,000). We also evaluated the performance of the proposed methods under different degrees of violation of the ETA assumption, which was controlled by modifying the association between the treatment and the confounder. A weaker treatment-confounder association induces a lesser ETA assumption violation, while a stronger one’s association induces a greater violation.

4.1 Main results

Figure 3 displays simulation results for Cox MSM parameters estimated with the stabilized weights truncated at alternative fixed values. The bias and the empirical SD over 2,000 estimates are reported. As expected, the estimators of the Cox MSM that uses untruncated IPTW are unbiased; this corresponds to truncation at infinity (denoted by “inf”) at the right end of the graphs. However, the untruncated IPTW estimators have large variability due to a small number of observations with extremely large weights. As the weights are truncated at progressively smaller values, the variance of IPTW estimators decreases while bias increases. The impact of truncation on bias is more marked when the truncation level decreases below 20 (Figure 3, top panel).

Figure 3

Properties of IPTW estimators of Cox MSM with the stabilized weight truncated at fixed values or not at all (“inf”).

Each simulated sample was analyzed using eight different estimators specified in Table 1 and the results are compared in Table 2. As noted in Figure 3, the untruncated estimator (Estimator 1) yields unbiased but highly variable estimates for both direct and indirect treatment effects, resulting in large RMSEs. Furthermore, the SD-to-SE ratios are much greater than 1, indicating that the SEs are seriously underestimated in the untruncated Cox MSM. Accordingly, the 95% coverage rates are much lower than the nominal 95% (first row of Table 2). Among the weight truncation approaches (Estimators 2–7), all the other approaches show a significant improvement in the SDs, SD/SE ratios, RMSEs, and 95% coverage rates over the untruncated estimators except for the approach using the untruncated estimates as the proxy (Estimator 4). This comes at the expense of a small degree of bias that increases with decreasing truncation threshold.

The methods using fixed percentile weight truncation (Estimators 2–3) and the proposed MSE-based approaches using estimates with fixed percentiles as the proxy (Estimators 5–6) yielded similar results and, on average, performed the best. Estimator 2 is the best in terms of the 95% coverage rates and Estimator 6 yields the smallest RMSE. In addition, all four percentile-based estimators (Estimators 2, 3, 5, and 6) produced more accurate estimates of SE, with the SD/SE ratios between 1.02 and 1.14. Although the results of the two-fold cross-validation approach (Estimator 7 with r = 1) are slightly worse than the methods based on percentiles; this approach is more objective, since it does not require the user to arbitrarily specify a fixed percentile or a fixed proxy for weight truncation, the optimal values of which may vary across different data structures. The performance of the CV method was improved significantly when we repeated this procedure five times, the SD/SE ratios, RMSE, and 95% coverage rates all approaching the results of percentile-based methods. Although the untruncated IPTW estimator is consistent for the true causal parameter, using the untruncated IPTW estimate as the proxy produced suboptimal results (Estimator 4). Because of the large variability of untruncated IPTW estimates, the estimates from a single dataset may deviate considerably from the true parameter and thus the estimated MSE cannot be well estimated by this approach.

To further evaluate the performance of the proposed MSE-based methods (Estimators 4–8), in Figure 4 we compare the frequency distribution of the selected optimal truncation levels using different methods. The different methods yield different distributions of the selected optimal truncation levels. Except for the method with untruncated IPTW estimates as the proxy (Estimator 4), all approaches selected relatively low truncation thresholds (3–5) for the majority of the samples. For the two percentile-based proxies (Estimators 5 and 6, or “Perc99” and “Perc995” in the figure respectively), the optimal truncation levels did not exceed 15 for any sample. In contrast, the optimal truncation levels selected using two-fold cross-validation approach varied across the range of thresholds, and exhibited a pattern similar to those selected using the method based on the true parameters (Estimator 8). Using the untruncated IPTW estimates as the proxy yields, a distribution of optimal truncation levels that differs considerably from the other methods, with the highest truncation level selected in about 30% of samples.

The performance of the different methods was further assessed by calculating the probabilities of selecting the same truncation level as selected by the “gold standard” method, i.e., the MSE-based method with the true parameters (Estimator 8). Among the 2,000 samples, the probabilities for each method of selecting the same truncation level as the method using true parameters are as follows: 57.3% for “Perc99”, 44.0% for “Perc995”, 67.5% for “CV”, and 24% for the “untruncated”, indicating that the cross-validation method approximates the “gold standard” distribution slightly better than the other methods.

The performance of the proposed methods across the different sample sizes is compared in Figure 5. As the sample size increases, the RMSEs of all the models decrease; however, increasing sample size does not improve the estimation of SEs with slightly increasing SD/SE ratios. The results of the different methods are similar across different sample sizes. The four percentile-based methods (Estimators 2, 3, 5, and 6) yield smaller RMSEs and more accurate variance estimates with the SD/SE ratios close to 1. The two-fold cross-validation method (Estimator 7) yields similar results to the percentile-based methods for small and medium sample size, but its performance deteriorates at the largest sample size, especially in terms of underestimation of SE.

Finally, we investigated the performance of different methods under different degrees of violation of the ETA assumption. We found that in the scenario with only minor violation, the estimator without truncation yielded a similar RMSE to our proposed optimal methods, and the coverage rates for all the estimators was close to the nominal 95% (data not shown). In the scenario with the most serious violation, the proposed optimal methods (except for the method using the untruncated IPTW estimates as the proxies) exhibited the same performance as in the main scenario (Table 2). The method using fixed 99th percentile weight truncation yielded the best 95% coverage rate, with SD/SE ratio close to 1, while the MSE-based method using 99th percentile weight truncation IPTW estimate as the proxy gave the smallest RMSE.

4.2 Bootstrap-based SE versus Robust SE

We compared the robust sandwich SE estimator with the bootstrap-based SE estimator of the estimated treatment effects. Specifically, we randomly selected 1,000 out of 2,000 samples, and, for each sample, we generated B = 100 resamples by first randomly resampling subjects with replacement and then including all the observations of those selected subjects. The resamples were then analyzed using the untruncated and truncated IPTW estimators. The bootstrap-based SE was estimated as the SD of the 100 resample-specific IPTW estimates. On average, the bootstrap SEs were only slightly higher than the robust SEs, with the ratio of mean SE values equal to 1.08 for untruncated IPTW estimates (data not shown). Furthermore, for truncated estimates, the two types of SE estimators yielded very similar results (e.g., with weights truncated at 100, the ratio of the mean SE values is 1.00). However, in a small fraction of simulated samples, especially for those where the estimated treatment effect was largely affected by a single observation with an extreme weight, the bootstrap SE was occasionally much larger than the robust SE. For example, for the two samples a and b in the top panels of Figure 1, where the resample-specific estimates had a strictly bimodal distribution, the bootstrap-based SE was several times higher than the robust SE (0.524 versus 0.115, and 0.806 versus 0.211, respectively). This indicates that, in some samples, the bootstrap SE estimator is able to capture the numerical instability of the estimates that arises due to a single strongly influential outlier, whereas the robust SE estimator is not.

Figure 4

Distribution of the selected optimal truncation levels using different optimal truncation methods: “Untruncated”, “Perc995”,“Perc99”, “CV”, and “True” corresponds to the Estimators 4–8, respectively, in Table 1.

Figure 5

Comparison of the different weight truncation approaches for various sample sizes (n = 400, 2,000, 10,000). The numbers 1–8 refer to the Estimators 1–8 specified in Table 1.