Abstract
The timing of an intervention such as surgery may happen sometime after treatment inception (e.g., diagnosis date) due to being put on a wait-list for treatment or being scheduled for treatment at a later time. If interest lies in the change in risk upon receiving the intervention, a Cox model with a time-dependent exposure may be employed. However, if there exists unmeasured confounding, maximum partial likelihood estimators of the hazard ratio are biased. An instrumental variable is a cause of the exposure of interest but not of the outcome except through the exposure. We propose a new framework involving potential outcomes for each possible waiting time, and derive an estimating equation for the hazard ratio of a treatment with a time-dependent start using an instrumental variable without making any assumptions about the form of involvement of unmeasured confounders. We conducted simulations to evaluate bias of our hazard ratio estimator. We illustrate the approach using two examples: First, we analyzed a procedural registry for which instrumental variable based estimators may be the only unbiased approach. Second, we demonstrate causal hazard ratio estimation for a randomized trial with time-dependent treatment initiation.
Keywords: Causal, Censoring, Endogenity, Estimating Equation, Marginal, Time-to-event, Unmeasured Confounder
Introduction
This paper addresses estimation of treatment effects when there is a waiting time between inception (e.g. diagnosis date, or date of index procedure) and start of treatment (e.g. transplant, reintervention). Procedural registries such as the registries of the Society of Vascular Surgery’s Vascular Quality Initiative (Martens, et al., 2006), contain information from the time of vascular procedures, such as a first and second intervention, but no other time-points. If interest lies in the effect of reintervention, covariates from the reintervention may be recorded but it may not be possible to adjust for these characteristics because patients without reinterventions before that time (e.g. the controls) will have no corresponding covariate information except at baseline. We are seeking methods for using information from a procedural registry to determine how reintervention effects subsequent risk relative to no reintervention.
For treatments that start sometime after inception, e.g. treatments with waiting times, a naive approach to the analysis is to group subjects by whether they ever get the treatment in question. However, this approach incurs immortal time bias (Platt et al. 2019), also known as guarantee-time bias. Cox’s model for time-dependent covariates (Therneau et al. 2017) can be used to avoid immortal time bias.
A treatment with a waiting-time is likely subject to confounding. That is, there may be variables that affect the risk of the primary endpoint but also impact the time to when the treatment is received. For instance, patients with severe disease may be prioritized for earlier treatment. This creates bias in typical estimators of the treatment effect. While adding covariates to Cox’s or other model, or using a propensity score based approach can account for confounding due to observed covariates, these approaches do not address confounding due to omitted covariates (unmeasured confounders). Instrumental variables can eliminate or ameliorate bias due to omitted confounders (Baiocchi et al. 2014; Boef et al. 2015).
Instrumental variable methodology includes two-stage techniques, such as two stage residual inclusion (2SRI), or two stage predictor substitution (2SPS) (Basu et al. 2018), as well as method-of-moments which has been adapted to binary regression (Foster 1997; Rassen et al. 2009) and Poisson regression (Johnston et al. 2008). Stukel et al. (2007) applied 2SPS to estimate a hazard ratio, by substituting the treatment variable with its predicted value given the first stage model. Wan et al. (2015) studied both 2SPS and 2SRI approaches to hazard ratio estimation. Our group observed that 2SRI introduces measurement error to the outcome model which can be accounted for using a frailty in Cox’s model to further reduce bias (Martínez-Camblor et al. 2019a, b). Wang et al. (2023) proposed an estimator of the hazard ratio using a binary instrument that allows specification of measured covariates in the model. Our group showed that, if there are no measured covariates, the Wang approach is equivalent to the hazard ratio estimator derived using principal stratification weights of the empirical distribution function (MacKenzie et al. 2016; O’Malley et al. 2023). Martinussen et al. (2019) derived an estimating equation for the hazard ratio of Cox’s model using othogonality of the IV and the survival curve at a fixed time and provided suggestions for choosing this fixed time. Instrumental variable based estimation for time-varying treatments have been addressed (O’Malley 2012; Cui et al. 2023) but we are not aware of any IV methodology for the effect of a time-varying treatment on a time-to-event outcome.
In this paper we propose an instrumental variable estimator of the hazard ratio for the effect of a treatment that has a waiting time. Our approach is based on generalized method-of-moments and removes confounding bias due to omitted covariates without explicitly modeling the relationship between those covariates and the treatment. Our contribution is a generalization of the approach we proposed for estimating the hazard ratio (MacKenzie et al. 2014) that we later adapted to the setting of a time-dependent hazard ratio (MacKenzie et al. 2021).
In the remainder of this paper, we extend our method to account for a time-dependent treatment allowing for the possible inclusion of multiple measured covariates in the model (Section 2), explore the frequentist operating characteristics of the extended estimator via simulation (Section 3), and apply it to both a comparative effectiveness study in vascular surgery and a heart transplant study (Section 4) before concluding the paper with a Discussion.
Methods
Assumptions concerning the instrument
Our approach to utilizing an instrument follows method-of-moments which is based on the independence between the instrument and the potential outcomes. Unlike two stage procedures for instrumental variables it does not utilize a model for the treatment. An example of an instrument could be the hospital or clinician preference measured by the rate of a surgery, but is known to have long waiting times. Another example would be study arm in a randomized trial in which the intervention starts some time after the decision to undergo the intervention.
We will denote the instrument by 
. To define the causal effect we introduce a set of potential outcomes. Let 
be the potential outcome (Rubin 1974) if treatment will ever be received when the instrument, 
. For each possible time at which treatment starts, i.e. waiting time, w, and each value of the instrument, i, there exists a potential time-to-event T(w, i) if treatment begins at time w. T(w, i) can be regarded as reception of treatment process 
where 
the standard indicator function (equal to 1 if A is true and 0 otherwise).
For each potental time-to-event, 
there is a corresponding potential censoring time, 
. We make the following assumptions, in addition to the model for the potential outcomes specified later in (6).
The instrument and the waiting time, W, are not independent;
.No direct effect: The instrument has no direct effect on the times-to-event; for any
and 
, the potential times-to-event satisfy 
. In other words, we can assume the existence of T(w) such that for all i, 
.- No confounding of instrument effect by omitted covariates: The instrument is independent of the potential time-to-event and censoring times conditional on covariates
, 
1 - The potential time-to-events and censoring times are independent conditional on covariates
, 
2
Remark
More work is needed to determine if our assumption concerning the joint independence of the censoring time, potential time-to-event and instrument can be further relaxed.
Causal estimand and model for potential outcomes
We are interested in the effect of a treatment received at some time after inception. The inception could be placement on a waiting list for an organ transplant, or the date of a diagnosis, and followed by a waiting time for a therapy used to treat that diagnosis. We refer to the time between inception and treatment reception as the waiting time, W. A complete list of relevant variables and their distributions is found in Table 1.
Table 1.
Simulation variables
| Random Variable | Symbol | Generation |
|---|---|---|
| Instrumental variable |  |
 (0,1) |
| Measured covariates |  |
independent 
|
| Quantile of time-to-event |  |
Uniform(0,1) |
| Time-to-event if not treated |  |
 |
Quantile of waiting time given 
|
 |
Uniform(0,1) |
| Waiting time | W |  |
| Time-to-event if treated at time w | T(w) |  |
| Observed time-to-event | T | T(W) |
| Censoring time | C | Uniform(0, ) |
The notation 
denotes the standard normal cumulative distribution function
We assume that the effect of the treatment is immediate and it has the same proportional multiplicative effect on the hazard of 
at all times subsequently. The causal estimand is 
. Among people who receive the treatment at time w, the hazard at time t is
 |
3 |
where 
is the baseline hazard. This is equivalent to the following survival curve formulation conditioning on survival up to the waiting time
 |
4 |
for any 
and any 
. This conditional probability formulation contrasts the decision to go through with the treatment, versus not, once the waiting time has elapsed.
If there exist covariates 
, possibly time-dependent, that are thought to affect the hazard, they can be accounted for using the following causal model,
 |
5 |
where 
and 
represent the k-dimensional random vector and observed values and 
.
Relation between potential outcomes under special case of equivalent quantiles
The joint distribution of potential outcomes are unidentifiable because only one is observed. Therefore, without loss of generality we can use deterministic relation between the potential outcomes that yields an elementary relation between them. We suppose that the potential outcomes have the same quantile of their respective distributions conditional on any covariates. This equi-percentile relationship between potential outcomes within a subject emulates the specification of additive linear constant-effect causal models (Small 2007; Hyunseung et al. 2016).
This yields the following framework for the potential outcomes based on Eq. (4)
 |
6 |
and
 |
7 |
where 
is the baseline cumulative hazard function of 
. For instance, if the hazard is constant (e.g., as is the case when the survival times follow an exponential distribution), 
and 
. For the conditional model (6), the corresponding potential outcomes framework is
 |
8 |
where 
is the baseline cumulative hazard function of T(w).
Estimating equations using instrumental variable
Let 
be the potential event indicator, and 
, the potential observed minimum of the time-to-event and censoring time. It follows from the independence of the instrument with 
that
 |
9 |
The difference 
(where 
) has an expected value of zero (Barlow and Prentice 1988) for any value of w, and is often referred to as the Martingale residual (of a stochastic process evaluated at the end of the study). Therefore, for any 
,
 |
10 |
Let 
be the observed time-to-event data (the actual realization among the potential outcomes). It follows that
 |
11 |
That is, our causal model (6) implies
 |
12 |
This suggests that, given an independent and identically distributed random sample of size n, the solution for 
in the estimating equation
 |
13 |
is a consistent estimator of the target parameter, 
. Notice that this equation can be written as
 |
14 |
where 
(
) is the counting process associated with the ith unit, and 
the maximum follow-up time.
The baseline cumulative hazard function, 
, is an infinite dimensional nuisance parameter. We propose to estimate it by the Breslow estimator,
 |
15 |
where 
, as is done when Cox’s partial likelihood is derived using nonparametric maximum likelihood (Zeng and Lin 2007). This approach of substitution of 
by the Breslow estimator was used by Martinussen et al. (2019) in the estimating equation they proposed.
When the Breslow estimator is substituted into the estimating equation above, we obtain
 |
16 |
This simplifies to the estimating equation we propose for 
,
 |
17 |
Notice that the explicit involvement in equation (17) of the instrument 
, the wait-time 
, observed covariates 
, and functions of the survival time (e.g., 
) in the above estimating equations for 
and 
emulates closed-form expressions for the method-of-moment IV estimator for a linear model (and generalized method-of-moments estimator for nonlinear models) derived by treating residual-orthogonality as a structural constraint such as in Foster (1997).
To estimate the parameters 
(the effects of the measured covariates) we use maximum partial likelihood estimates for a fixed value of 
and insert 
from equation (17). That is, to estimate 
, we solve the equation(s)
 |
18 |
for each 
.
Variance of the estimator
In this subsection we derive an asymptotic expression for the variance of our proposed estimator (17), for the setting without covariates; a variance formula for the setting of covariates is provided in the Appendix. We employ a sandwich variance: The variance of the solution of the estimating equation, 
, equals the sandwich expression 
where A is the matrix of partial derivatives
 |
19 |
Let 
be the time-dependent treatment indicator for the ith unit (
). In our waiting time scenario 
. The derivative, A, equals
 |
20 |
where 
for 
, and 
. Note 
.
Using properties of integrals of counting processes with respect to predictable processes, 
can be estimated by
 |
21 |
Remark 1
It is useful to graph the estimating function over some range of hazard ratios, e.g. 1/10 to 10, to ascertain if it is (i) monotonic and (ii) crosses zero within that range.
Remark 2
We require that the partial derivative, 
be nonzero. The expression for it shown above can be regarded as a weighted average over time of the covariance of the instrument and being exposed among those at risk. We require that it converges to a nonzero quantity as sample size increases.
Testing instrument strength
An instrument needs to be associated with the exposure (see the requirements reported in the previous section). If the association is weak then estimators based on this instrument perform worse. In the linear model case, it’s been recommended that the F-statistic for the association of the instrument and the exposure be at least 10 Staiger and Stock (1997). This uses the reasoning that asymptotic inference for Cox’s model does not involve the denominator degree of freedom, and that a chi-square with a single degree of freedom is equal to the numerator of an F-test with one numerator degree of freedom. We acknowledge that more work is needed to determine minimal lower bounds for instrument strength.
In our simulations, we adopt this criterion and restrict our results to datasets in which the likelihood 
-statistic of the coefficient of the IV in the Cox model that regresses the censored waiting time on the instrument and any measured covariates be at least 10.
Simulation
Simulation methods
Table 1 lists the random variables of the simulation and the symbol representing them. The quantile variables, drawn from an Uniform(0,1) distribution, are introduced for convenience. In this simulation, every subject has a non-zero chance of treatment and the waiting time to treatment follows an exponential distribution. Whether a subject does or does not receive treatment, depends on whether or not the waiting time is less than both the time-to-event and the censoring time.
Table 2 lists the parameters of interest: 1) The sample size, n; 2) Proportion with event observed (not censored), 
; 3) A measure of the association between the instrument and the potential time-to-treatment, 
, the Spearman correlation coefficient denoted 
; 4) A measure of the amount of unmeasured confounding, the rank correlation between the waiting time and 
denoted 
evaluated conditional on 
with zero indicating no unmeasured confounding; 5) the hazard of the event, 
, which we set equal to 1 (as for an exponential distribution); 6) the hazard for the waiting time distribution (also assumed to follow an exponential distribution), 
, whose value relative to 
controls the proportion of subjects who ultimately receive treatment; 7) the effect of the treatment 
, once received, on the hazard of the event measured in the time-dependent covariate model; 8) the effects of the three measured covariates 
, whose effects on the log hazard were 1, 0 and −1. Note: The single parameter specification represented by 
is simpler than needing to specify (i) a model for the dependence of the waiting time on the omitted covariates and (ii) a model for the dependence of the time-to-event on the omitted covariates.
Table 2.
Elements of the simulation: They are constant within a dataset but vary between datasets
| Element | Value Set |
|---|---|
| Sample size of each dataset | 500, 2000 |
| Expected censoring proportion | 0.50, 0.75 |
Rank correlation of instrument and waiting time, 
|
0.2, 0.4, 0.6 |
Rank correlation of  and W given Z, 
|
−0.3, 0, 0.3 |
Baseline hazard of the event if untreated, 
|
1 |
Log hazard ratio for covariates 
|
1, 0, −1 |
Median waiting time relative to median 
|
0.1, 0.2, 0.5 |
| Hazard ratio for treatment | 1/3, 1/2, 2/3, 1, 3/2, 2, 3 |
Before each dataset is constructed the parameters above are randomly drawn with equal probability from the value set
We obtain a time-to-event random variable whose dependence on the treatment and non-omitted covariates satisfies the desired model for the potential outcomes (6) while also being related to the waiting time (e.g. through omitted covariates) using a Gaussian copula. The Gaussian copula has a single parameter, which is a rank based correlation, 
. This is described in steps 4 and 5 below.
The steps of the random generation of each simulated dataset are:
Randomly generate the instrument
from a standard normal distributionRandomly generate 3 covariates from independent standard normal distributions
Randomly generate U from Uniform(0,1)
- Calculate
as 
, where 
is exponential with hazard 
Generate the waiting time, W, to have an exponential distribution with hazard
whose rank correlation with the instrument is 
and rank correlation with the untreated time-to-event is 
If
calculate 
Generate the censoring time, C, from the Uniform(0,
) distribution to obtain the desired proportion of events, e.g. 
for 
proportion of individuals
Simulation code can be found at the web site: https://github.com/toddamackenzie/Instrumental-Variable-Hazard-Ratio-for-Treatment-with-Waiting-Time.git.
Simulation results
We report the distribution of the estimate in relation to the true hazard ratio, depending on the strength of the instrument, the level of confounding by omitted covariates, as well as sample size and the censoring proportion. The distribution is conditional on the observed association of the instrumental variable (IV) with the waiting time being sufficiently strong, 
based on Cox’s model as explained in section 2.6.
Bias
Figure 1 shows the results for a sample size of 2000, and censoring of 50% (1000 events). For three levels of confounding by omitted covariates (top to bottom) and three levels of IV strength (left to right) we show the median and quartiles (bars) of the distribution of the estimator of the hazard ratio versus the true hazard ratio over 500 simulations of each combination. The level of confounding by the omitted covariate is determined by the rank correlation of the waiting time with the potential time-to-event (based on a Gaussian copula) if the subject was never to be treated (e.g. infinite wait) and was considered to be −0.3 or 0 (e.g. no omitted covariates) or +0.3. The strength of the IV is measured by its rank correlation with the waiting time (based on a Gaussian copula), which took one of 3 levels in each simulated dataset, 0.2, 0.4 or 0.6.
Fig. 1.
Sample size of 2000 and 50% censoring (1000 Events): Median estimate with interquartile range versus true value using our approach (Green) and Cox MPLE (Red) for 3 confounding scenarios and 3 values for the rank correlation of IV and waiting time. The line of identity is superimposed
In this setting of samples with 1000 events the median of our estimator appears to be equal to its target for a range of true values, strengths of instrument, and the three levels of confounding whereas the partial likelihood estimator is biased unless there is no confounding. As expected, the variability in our estimator decreases as the strength of the instrument increases which is true of all instrumental variable estimators (Martens et al. 2006).
Figure 2 shows the results of the simulations when the sample size is 500 with 75% censoring (125 events). When the rank correlation of the instrument and waiting time is high, 0.6, our estimator has good properties for all three settings of confounding. The median estimator based on our approach is close to the true value whereas the Cox maximum partial likelihood estimator has considerable bias for confounding in both positive and negative directions. When there is little correlation between the instrument and the waiting time, such as with a rank correlation of 0.2, our estimator is unreliable. It is just as biased as the Cox MPLE when the confounding is in the positive direction. This poor performance is not surprising if we consider the rule-of-thumb that the effective loss in sample size due to an instrumental variable analysis is proportional to the amount of variation in the exposure explained by the instrument (Martens et al. 2006). For instance, if the rank correlation of the instrument and exposure is 0.2, the proportion of variation explained is about 4%, so a sample with 125 events effectively has only 5 events. Not surprisingly, the root finding algorithm we utilize in the implementation of our estimating equation does not converge in over 25% of the datasets when there is only 125 events and the rank correlation of the instrument and exposure is 0.2.
Fig. 2.
Sample size of 500 and 75% censoring (125 events): See previous caption for description
Empirical coverage
We evaluated the empirical coverage of our interval estimator in our simulations. A 95% confidence interval for the log hazard ratio was constructed using the estimate plus and minus 1.96 times the standard error calculated using the variance formulas supplied in section 2.5. In a simulation to evaluate the coverage of a nominal 95% confidence interval estimator, we obtained an average coverage across all settings of 95.1%. Not surprisingly, the coverage of Cox’s maximum partial likelihood estimator is poor if there is confounding, which in our simulation corresponds to nonzero values, i.e., −0.3 or +0.3, of the rank correlation between the potential time-to-event 
and the waiting time, W. Table 3 presents these results. As expected, the length of our IV based intervals are wider than Cox based intervals, and the length of the interval is larger if the instrument is weaker.
Table 3.
Empirical coverage and median width of 95% confidence interval for our IV estimator and the MPLE, in several subsets of our simulations
| Empirical coverage | Median width | |||
|---|---|---|---|---|
| Subset | IV est. | MPLE | IV est. | MPLE |
| Overall | 95.1 | 42.6 | 5.94 | 2.00 |
| Censoring of 50% | 95.7 | 37.1 | 3.53 | 1.13 |
| Censoring of 75% | 94.4 | 48.1 | 6.54 | 2.10 |
| Sample size 500 | 95.2 | 51.6 | 6.48 | 2.10 |
No confounding;
|
94.6 | 95.1 | 6.23 | 1.99 |
+ Confounding;
|
95.0 | 12.4 | 5.96 | 1.87 |
− Confounding;
|
95.7 | 19.0 | 5.67 | 2.17 |
Weak instrument:
|
95.6 | 42.5 | 7.78 | 2.03 |
Strong instrument:
|
94.6 | 43.5 | 4.85 | 1.97 |
Hazard ratio
|
94.6 | 44.5 | 6.89 | 2.27 |
Hazard ratio
|
95.6 | 42.4 | 5.95 | 1.98 |
Hazard ratio
|
95.4 | 40.9 | 5.26 | 1.80 |
Examples
Using a procedural registry to estimate the effect of reintervention after endovascular abdominal aortic aneurysm repair
Abdominal aortic aneurysms (AAA) are aneurysms of the aortic wall that are weakened and are prone to rupture. Approximately 55% of patients who experience AAA rupture will die (Suckow et al. 2022). As such, repair is recommended when AAAs reach 5.5 cm in men, and 5.0 cm in women, and represent some of the most common vascular surgical procedures performed in the U.S (Rokosh et al. 2021; Suckow et al. 2018).
Nearly 75% of AAA repairs in the U.S. are performed using a minimally invasive technique, called endovascular AAA repair (EVAR), where stent-grafts are inserted through punctures in the femoral artery to exclude the AAA from the circulation, removing the risk of AAA rupture. EVAR has become the de facto first line treatment modality for AAA repair in the U.S. due to its low risk of perioperative complications and short hospital stay (often just 1 day) (Rokosh et al. 2021).
The Achilles heel of EVAR is reintervention, i.e., additional procedures performed after the index EVAR to treat the aneurysm. These procedures are most commonly performed to treat endoleaks, which are side branch arteries from the AAA that allow arterial blood to flow into the AAA despite the repair. Approximately one third of patients who have an EVAR will undergo a reintervention within 10 years (Columbo et al. 2021). These reinterventions are performed under the assumption that they will reduce future complications from the AAA, such as late AAA rupture (rupture of the AAA despite an EVAR). However, this hypothesis has never been tested in a randomized trial, and no trial of reintervention after EVAR is currently enrolling. As such, whether these procedures actually reduce long-term mortality remains in question, with wide variation in practice patterns across centers.
To address this question we used data from the Vascular Quality Initiative Registry (Martens, et al., 2006). This is a procedural registry; when a procedure is performed data on characteristics of the patient and procedure are entered into the registry. Data on any repeat procedures for patients is collected at their 1 year follow up visit, which is captured between 9 and 21 months. In addition to this follow up visit, patients registry file is linked to the Social Security Death Index for survival data. This creates a dataset where we have detailed patient information at baseline, an indicator of reintervention versus not at approximately one year, and long-term survival information. Patient characteristics are unknown except at times that a patients had a procedure. The decision to have a procedure is related to patient characteristics, including prior medical history, age, and operative variables from the index endovascular repair. However, we do not have these characteristics, except at baseline, for patients who decide, or are selected, not to have reintervention. For that reason we recommend and propose to employ estimation using an instrumental variable.
The dataset consists of follow-up of 56,497 patients from 319 hospitals in the USA (of whom 306 represented at least 30 patients). The follow-up is limited to 2 years. The number of individuals who died during the 2 year follow-up was 2,901. Of these patients, 2002 had reinterventions during the follow-up. Almost half, 954 had the reintervention at the time of endoleak diagnosis. Figure 3 panel A shows the Kaplan-Meier estimator for mortality and the competing risk cumulative incidence estimator of reintervention (Pintilie 2006).
Fig. 3.
A. Probability of mortality and reintervention over time since index operation. In almost 2% of patients the reintervention occurs during the index surgery. The rate visibly increases at 9 months which is common timepoint for a followup visit.B Mortality stratified by tertiles of the instrument which is the hazard of reintervention by hospital
Cox’s model for time-dependent covariates (e.g. reintervention surgery) was used and the estimator of the hazard ratio for reintervention was 3.46 (95% CI: 3.02 to 3.96). In other words, upon undergoing reintervention patients are at more than 3 times the instantaneous risk of death up to 2 years following Endoleak. After controlling for patient characteristics at the time of the index procedure, age, gender and smoking status, as well as hospital volume, the estimate was slightly smaller, 3.23 (95%CI: 2.91, 3.81).
We are concerned that this estimator is biased due to unmeasured confounding. For instance, the registry does not track covariate information on patients who do not receive reintervention. To overcome this we utilize an instrumental variable. We propose to use a measure of the preference for reintervention at a hospital as the instrumental variable. To measure the preference of reintervention by hospital we employed a Cox model for the hazard of reintervention with a frailty for hospital including covariates age, gender, smoking and log center volume.
The estimated frailties were extracted and used as the instrument. The 
association of this preference instrument with time-to-reintervention based on the coefficient from a Cox model controlling for age, sex and smoking status was 434.2, well above the threshold of 10 we suggested earlier. These reflect the hazard ratios of reintervention for each center shrunken toward the null. The median value of our proposed instrument was 0.96 with an interquartile range of 0.85 to 1.13. In other words the hospitals in the 75th percentile had a hazard (i.e. rate) of reintervention 1.34 times higher than those in the 25th percentile. The 10th and 90th percentiles were 0.75 and 1.32, respectively. Hospitals in the 90th percentile were at 1.75 times the rate of those in the 10th percentile.
The instrumental variable assumptions are, (i) it is associated with the exposure, (ii) it has no direct effect on the outcome, mortality, except through its association on use of reintervention, and (iii) it is not associated with any variables that affect the outcome. The third assumption would be violated if centers that reintervene at a higher rate use other survival impacting treatment and care practices at a higher (or lower) rate.
Figure 3 panel B shows Kaplan-Meier estimators of mortality stratified by tertiles of the instrument. There is a statistically significant association (P<0.0001) between our instrument and mortality. For an increase in the hospital’s rate of reintervention of 10%, the hazard of mortality increases 19.5%.
We estimated the effect of reintervention at the level of the individual using the estimating equations we proposed in (17) controlling for age, sex and smoking status. Our instrumental variable based estimator of the hazard ratio was 4.31 (95%CI: 1.68, 11.02). This estimator is even further from null than the maximum partial likelihood estimator. As typical with instrumental variable based estimators the confidence interval is considerably larger than the maximum likelihood approach. The instrumental variable based estimator with adjustment for patient characteristics and center volume (at time of index procedure) is 4.80 (95%CI: 2.03, 11.32). Overall, we conclude that reintervention occurring in the first 2 years following Endoleak increases mortality during that time period.
Our estimator is biased to the extent that the no confounders assumption is violated, e.g. centers with a proclivity for reintervention may have simultaneous practices that increase mortality, or centers with a proclivity for reintervention may have patients with higher risk of mortality not accounted for by age, sex, smoking as well as patient volume. For this reason we conducted a sensitivity analysis to determine how influential an omitted confounder between the instrument and mortality would be needed to eliminate the increased risk we estimated. We determined that a continuous omitted covariate that has a Spearman correlation of 0.2 and increases the risk of mortality by 50% for a change of one standard deviation in the instrument would nullify the association between the treatment and mortality we found using the instrument.
Note: For this example the root solving algorithm did not converge when zero is used as the starting value for the log hazard ratio for the treatment. We then plotted the estimating function and determined it did indeed cross zero, although it was somewhat flat for arguments close to zero, and then used a starting value close to the root observed on the plot.
Stanford heart transplant data using a fictional randomization arm
The Stanford Heart Transplant Data is a classic survival analysis dataset and was one of the first applications of Cox’s model with time-dependent covariates (Crowley and Hu 1977). It is readily accessible as part of the survival library in R (R Core Team 2021). This dataset captures follow-up of 103 individuals put on a waiting list for a heart transplant between 1967 and 1974, of whom 69 obtained transplants during follow-up. Cox’s PH model for time-dependent covariates estimates the hazard ratio comparing individuals who have had a transplant to those who have not. The estimate of the hazard ratio from this model without controlling for patient characteristics is 1.13 (95%CI: 0.63 to 2.04). When age and year are controlled for (both as continuous covariates) as well as prior heart surgery, the estimate of the hazard ratio is 0.99 (95%CI: 0.53 to 1.82). Of course, confounding by omitted covariates is a possibility. For instance, if the most critical patients were selected for transplants earlier, it would cause a bias in the direction of a deleterious effect of transplant.
To illustrate our method and how it could be applied to estimate treatment effects in randomized studies of treatments with waiting times, we have augmented the dataset with a fictional randomization arm. In this fictional randomized study, patients and their clinicians in one arm are encouraged to obtain a transplant as soon as possible (ASAP), while patients in the other arm are encouraged to Wait & See.
We emphasize this is a fictional augmentation. We found a 1:1 partition of the dataset into two fictional study arms that lead to a statistically significant finding that is easy for a user to code; it can be added to the existing jasa1 dataset in R using the single line of code
> jasa1$RandArm 
is.element(jasa1$id%% 12, c(0,1,4,9,10,11))
Figure 4-Panel A shows the estimated cumulative incidence of transplant for the ASAP and Wait & See arms accounting for death as a competing risk. As expected, the ASAP arm has a higher incidence, reaching 80% by 6 months. The subdistribution hazard ratio (Fine and Gray 1999) for undergoing transplant in ASAP vs. Wait & See is 2.12 (95%CI: 1.31 to 3.43). The corresponding Wald 
-statistic for the association of the instrument with the time to treatment is 9.3, which indicates that the fictional randomization is a weak instrument (e.g. < 10.0) and that leads to biased instrumental variable based estimates.
Fig. 4.
A The probability of receiving a transplant from time since randomization in both fictional arms.B Probability of death since randomization in each fictional arm
Figure 4-Panel B shows the Kaplan-Meier estimators of survival in both arms. Patients randomized to the ASAP arm have better survival than those randomized to Wait & See (log-rank P= 0.033). The intention-to-treat estimator of the hazard ratio is 0.61 (95%CI: 0.38 to 0.96). The interpretation in our fictional randomized trial is that patients encouraged to have a transplant as soon as possible have a 39% (95%CI: 4 to 62) lower hazard than those encouraged to Wait & See. The corresponding hazard ratio adjusted for age, year and prior surgery is very similar.
In general intention-to-treat estimands are attenuated toward the null relative to the effect of receiving treatment. The effect of the transplant itself (commonly referred to as the therapeutic effect of surgery), as opposed to the intention to transplant soon (the intent-to-treat effect), may be of more interest to patients and clinicians. This is the hazard ratio that contrasts the hazard of subjects if they have had the transplant versus the hazard of subjects who have not had a transplant after a given wait-time. To obtain this estimand, we can use our approach in which we employ randomization arm as an instrument.
Using our instrumental variable based estimator of the hazard ratio, we obtained the point estimate 0.12 (95%CI: 0.02 to 0.95), indicating an 8 fold reduction in the hazard of death. The corresponding p-value based on our method, obtained by referring the quotient of the estimate and its standard error to a normal distribution, is 0.036. Not surprisingly this is close to the p-value for the test of equivalence of survival across the two arms.
Discussion
We proposed an instrumental variable estimator of the hazard ratio for a treatment with a waiting time. Unlike two stage residual inclusion (control function) approaches to instrumental variable based estimation, our approach estimates a hazard ratio that is marginal with respect to the omitted covariates (the source of the confounding). In other words, this hazard ratio is not conditional with respect to the omitted covariates. For non-linear (or non-log-linear) models such as Cox’s model and logistic regression, conditional and marginal measures of effect (e.g. odds ratios) are not mathematically equivalent. Marginal hazard ratios are preferable if the application is to a population of subjects, as opposed to an individual.
The advantage of reporting hazard ratios conditional on omitted covariates is that they are more personalized to a patient than estimators that do not condition on omitted covariates. That is, they provide an estimate for people with similar values of the omitted covariate. However, omitted covariates are intangible. In observational studies omitted confounders are those that affect the outcome and also affect treatment selection. On the other hand, there will always be individual risk variables that affect the outcome but do not affect treatment selection, such as yet to be identified risk factors. Without knowing these risk factors and without them having an effect on selection, it is impossible to identify an effect that is truly personalized.
Our simulations demonstrated good properties of our estimator when either there are hundreds of events or the instrument is sufficiently strong. For a range of true hazard ratios, a range of strengths for the instrumental variable and for a range from positive to negative confounding, the median of our estimator has none or negligible bias if there are hundreds of events or the instrument has a strong association with the exposure. We do not recommend our approach if the number of events is 100 or less and the instrument explains a small proportion of variation in the exposure (e.g. less than 10%).
We illustrate our estimator using data from a procedural registry to evaluate the effect of a second procedure (reintervention). Multivariable modeling approaches such as Cox’s model for time-dependent covariates, or marginal structural models or sequential trial emulation cannot remove time-dependent confounding because procedural registries do not track covariates on the controls, e.g. information from those who do not have a reintervention. The approach of instrumental variables yields consistent estimators of the hazard ratio (see the model in formula (4) or (6)) if the instrumental variable assumptions hold. However, it is not possible to verify the instrumental variable assumptions that there is no direct effect of the instrument, and that there are no confounders of the instrument and the outcome.
Instrumental variables are relevant to randomized trials with imperfect compliance. The estimator we propose can be used to estimate the effect of a treatment in a randomized trial in which a treatment arm with a variable waiting time is compared to a control arm. This can be done by substituting the randomization indicator for the instrument and applying our estimator. We demonstrated this using a fictional randomization within the classic Stanford Heart Transplant dataset. In addition, our approach could be used to estimate the hazard ratio in a trial in which subjects are randomized to different starting times of treatment. It could be applied to a stepped wedge cluster randomized trial with longitudinal follow-up of patients across steps.
Another possible application of our approach is to Mendelian randomization studies in which the exposure of interest is a phenotype that develops at a variable age and the outcome is a time-to-event such as age at death. For instance, one could study the effect of diabetes on the incidence of a particular cancer using a cohort of all patients in which not all subjects have diabetes at the beginning of follow-up and using as an instrument, a genetic marker known to be linked to diabetes but not otherwise to the cancer of interest.
In this paper we discuss an approach for a treatment with a waiting time, i.e. a binary 0/1 time-dependent exposure which stays at the value 1 once it takes the value 1. Our estimating equation could be adapted to time-dependent covariates in general. For instance, it could be a binary covariate that moves back and forth from 0 to 1 corresponding to treatment being off and on. This is a more general setting with a larger space of potential outcomes. Whereas in our setting the set of potential outcomes is indexed by the real number line, 
, in a general time-dependent covariate setting the potential outcomes would be indexed by 
. More work is needed to adapt instruments to continuous and binary time-dependent exposures.
A limitation of our method is that it does not incorporate time-dependent covariates besides the time-dependent treatment indicator. For instance, our method does not reflect the treatment selection mechanism in which the decision to intervene at a particular time is dependent on patient characteristics that are varying over time. However, datasets such as procedural registries that we work with do not record information between procedures. This precludes the possibility of adjusting, or balancing with respect to, covariates between those receiving and not receiving a procedure because patient characteristics are not available for the latter. Such changes in baseline values of predictors are unmeasured confounders in the context of our real-world application, motivating further extensions of IV methods like that developed herein.
Appendix
Variance of the estimator in the setting of covariates
The variance of the solution of the estimating equation, 
equals the sandwich expression 
where 
is 
and 
is the covariance matrix,
 |
Let 
be the time-dependent treatment indicator for the ith unit (
). In our waiting time scenario 
. The partial derivatives of matrix A follow:
 |
22 |
where 
for 
, and 
and 
 |
 |
The bottom right component of matrix A, 
, is
 |
The components of the covariance matrix B follow: Using properties of integrals of counting processes with respect to predictable processes,
 |
 |
and
 |
Implementation
A function that implements the procedure above was coded using the R library Rcpp. The R function takes data in the counting process structure, i.e. a start time, stop time, and event indicator. A unit that receives treatment contributes two rows,
X=0, TimeStart=0, TimeStop= WaitTime, Z
X=1, TimeStart= WaitTime, TimeStop= End of follow-up, Z
A unit who does not receive treatment contributes a single row
X=0, TimeStart=0, TimeStop= End of follow-up, Z
Code can be found at the web site: https://github.com/toddamackenzie/Instrumental-Variable-Hazard-Ratio-for-Treatment-with-Waiting-Time.
Footnotes
Publisher's Note
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