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Statistical methods for elimination of guaranteetime bias in cohort studies: a simulation study
BMC Medical Research Methodologyvolume 17, Article number: 126 (2017)
Abstract
Background
Aspirin has been considered to be beneficial in preventing cardiovascular diseases and cancer. Several pharmacoepidemiology cohort studies have shown protective effects of aspirin on diseases using various statistical methods, with the Cox regression model being the most commonly used approach. However, there are some inherent limitations to the conventional Cox regression approach such as guaranteetime bias, resulting in an overestimation of the drug effect. To overcome such limitations, alternative approaches, such as the timedependent Cox model and landmark methods have been proposed. This study aimed to compare the performance of three methods: Cox regression, timedependent Cox model and landmark method with different landmark times in order to address the problem of guaranteetime bias.
Methods
Through statistical modeling and simulation studies, the performance of the above three methods were assessed in terms of type I error, bias, power, and mean squared error (MSE). In addition, the three statistical approaches were applied to a real data example from the Korean National Health Insurance Database. Effect of cumulative rosiglitazone dose on the risk of hepatocellular carcinoma was used as an example for illustration.
Results
In the simulated data, timedependent Cox regression outperformed the landmark method in terms of bias and mean squared error but the type I error rates were similar. The results from realdata example showed the same patterns as the simulation findings.
Conclusions
While both timedependent Cox regression model and landmark analysis are useful in resolving the problem of guaranteetime bias, timedependent Cox regression is the most appropriate method for analyzing cumulative dose effects in pharmacoepidemiological studies.
Background
Extensive studies have elaborated and documented protective effect of aspirin for the primary prevention of cardiovascular diseases, including myocardial infarction (MI), stroke, coronary artery disease (CAD) [1,2,3,4,5,6]. Distinct from other nonsteroidal antiinflammatory drugs (NSAIDs), aspirin has the capacity to irreversibly inhibit cyclooxygenase (COX) and suppress thromboxane production. Findings of experimental studies have suggested that COX2 suppression [7,8,9] or antiplatelet effect of aspirin [10, 11] could interact with cancer cells and play a significant role in blocking tumor angiogenesis, invasiveness and metastatic potential. Accordingly, several observational studies have reported an inverse association between aspirin use and cancer incidence or mortality. However, the results have been inconsistent and controversy still remains regarding its anticancer effects [12,13,14,15], in part due to variations in study designs and statistical methods for analyzing the effectiveness of aspirin. For example, Fraser et al. [16] conducted a populationbased study of breast cancer patients in Scotland, in which Cox’s proportional hazard models was used to assess the relationship between aspirin use and survival. The authors found significantly reduced allcause (HR = 0.53, 95% CI = 0.45 to 0.63) and breast cancerspecific mortality (HR = 0.42, 95% CI = 0.31 to 0.55) among participants who consumed aspirin following a diagnosis of breast cancer. Similarly, using Cox proportional hazards modeling, Jacobs et al. [17] reported that total cancer incidence is significantly lower among regular longterm aspirin users (RR = 0.84, 95% CI = 0.76 to 0.93). In contrast, a study by McMenamin and colleagues [18] found a nonsignificant decrease in the risk of lung cancerspecific mortality (p = 0.5975) with aspirin use. In this study, the authors used a timedependent Cox model whereby lowdose aspirin usage was treated as timevarying covariate, with users not considered to be exposed until after a lag of six months following initial prescription. Further, Cardwell et al. [19], in their conditional logistic regression analysis of nested casecontrol data, also found no evidence for protective effects of lowdose aspirin usage against colon cancerspecific mortality (OR = 1.06; 95% CI = 0.92 to 1.24).
The abovementioned studies of aspirin use show that when evaluating drug effects, results are often confounded by timerelated biases that tend to exaggerate the protective effects, even when the drug has no or little effect [20]. Timerelated biases can be further categorized into guaranteetime bias (also known as immortal time bias or timedependent bias), timewindow bias, or timelag bias. For the purpose of this paper, we focus on guaranteetime bias, which is frequently encountered in timetoevent analyses evaluating drug effects. Guaranteetime bias refers to bias arising from incorrect handling of the time from the beginning of followup to the first treatment exposure [21]. The bias occurs if treated patients are assumed to have already been treated from time of cohort entry. For example, in a study of patients with heart disease, heart transplant is a timevarying treatment. While some patients may receive a transplant at the start of followup, others may undergo transplant long after the beginning of followup or die before undergoing the transplant. Failure to account for timevarying feature of the treatment can result in biased estimation of the treatment effect, in this case, in favor of the treatment group [22].
This study aimed to evaluate the extent of guaranteetime bias in estimation of drug effects on disease outcomes. We conducted simulation studies in which timefixed, timedependent Cox model, and landmark analysis were compared for handling guaranteetime bias. For illustrative purposes, effect of cumulative rosiglitazone dose on the risk of hepatocellular carcinoma was investigated using the Korean National Health Insurance Data.
Method
GuaranteeTime bias
Guaranteetime bias in cohort studies can distort the results in favor of the treatment group, depending on the type of event (if it benefits the patient or not) [21, 23, 24]. In detail, consider this study as a cohort study and the situation where drug effect and event occurrence are independent. Event is defined as the disease of interest. Define the random variables W and T _{0} as the time to initiation of drug usage and time to occurrence of event. The binary random variable Z is defined to be Z = I[W ≤ T _{0}], which represents drug usage. The random variables T _{0} and T _{1} are defined as the time to event conditional on Z = 0 or 1 respectively, i.e. T = (1 − Z)T _{0} + ZT _{1}. As shown in Fig. 1, the person whose value of T _{0} is smaller than W is allocated to the nonuser group (person #1). Otherwise, individuals are allocated to the druguser group, according to cumulative drug exposure (persons #2 to 4). In other words, to have received the treatment implies that the subject survived or was eventfree, up to the initiation of drug use. As regards druguser groups, the longer the survival times, the higher the probability of belonging to the highdose group. This is the guaranteetime bias phenomenon, i.e. subjects must have lived long enough to receive the treatment and consume large amounts of drug.
Statistical modeling for GuaranteeTime bias
Our statistical modeling closely follows the paper by Nam and Zelen [25]. For convenience, we consider the situation where the drug usage is binary. The probability density functions of W , T _{0} , T _{1} will be denoted by g(w) , q _{0}(t) and q _{1}(t) respectively. The survival functions will be denoted by G(w) = Pr[W > w] , Q _{0}(t) = Pr[T _{0} > t] and Q _{1}(t) = Pr[T _{1} > t]. Note that by definition Z = 1 if the time for drug usage is observed. If the drug usage does not cause a change in the survival function, the conditional density function of nonuser group f(t z = 0) and druguser group f(t z = 1) would be the same.
Under the hypothesis that q _{0}(t) = q _{1}(t), if we simply compare the survival functions of the two groups according to the drug usage, the two groups of conditional density functions are not equal to f(t z = 0) ≠ f(t z = 1), as it has been proven in the paper by Nam and Zelen [25]. Therefore, it is not appropriate to simply compare the survival functions of two groups according to drug usage.
Landmark method
In the landmark method, selected landmark time τ _{0} is set, and binary random variable Z(τ _{0}) is defined as Z(τ _{0}) = I(W < τ _{0} T > τ _{0}). The conditional density functions for Z(τ _{0}) can be computed as follows [25]:
where A(t) = 1 − G(t).
Hence if q _{0}(t) = q _{1}(t), it can be seen that the two functions are the same as
As a result, it is possible to make a valid comparison of two survival functions between the druguser and nonuser groups using the landmark method. In this study, landmark analysis was conducted using timefixed Cox regression with drug exposure status defined prior to landmark time. One thing to mention in the landmark method is that it is very important to carefully select the landmark time.
Timedependent Cox regression model
In a timedependent Cox regression model, a timedependent covariate in the model tracks whether the classifying event has occurred during the estimation process [23]. This method eliminates guaranteetime bias by using drug usage as a timedependent covariate; subjects are classified as unexposed until the start of drug usage and exposed thereafter [26,27,28]. The timedependent Cox regression model has the advantage of using all study followup data since it starts analysis at the time of cohort entry. By including all data, this method has increased statistical power over the landmark method [29].
Simulation setting
We conducted extensive simulations to compare the performance of our three methods, Cox regression, timedependent Cox regression and landmark method, in terms of empirical type I error and power. We used a discretetime survival model to describe the effect of cumulative drug dose on the outcome using three indicator variables Z _{1}(t) , Z _{2}(t) , Z _{3}(t) which represent low, moderate and highdrug dose groups. We set N as total sample size and d as the number of drugusers. Among N subjects, d subjects consumed drug for different durations and the dosage was fixed at a constant dose of 0.5 per unit of time. We assumed that drug is taken continuously without interruption until the end of the study or event.
We divided the total observation period (10year) into ten intervals as needed for the discretetime survival model. The overall simulation setting is represented in Fig. 2. We assumed that the drug user rate is 5%. First, we randomly assigned time to initiation of drug use between 0 and 10 for each druguser. Using these initiation times, we calculated cumulative dose at each time t and categorized each subject into four groups: nonuser group and low(Z _{1}(t) = 1), moderate(Z _{2}(t) = 1), highdose(Z _{3}(t) = 1) groups. At each discrete time point, the probability of disease occurrence at time t is as follows:
If the event occurred within the observation period, survival time is the point of event occurrence. On the other hand, if the event did not occur, survival time is equal to 10 years and censored. For the landmark analysis, we considered subjects whose starting point of drug intake is before landmark time as drugusers, and otherwise nonuser. We used 5 and 7 as landmark times, respectively. Subjects who died or who were censored before the landmark time were excluded from the analysis.
Simulations were performed under two different scenarios. In the first scenario, type I error rates for each of the statistical methods were evaluated under the null hypothesis, whereby β _{1}, β _{2} and β _{3} values were set to zero to represent a case of no significant protective effect of drug on disease outcome. For the power comparisons in scenario 2, we simulated data from the alternative hypothesis (i.e. cumulative drug dose is associated with disease outcome) for each of the three β _{0} values denoting disease incidence: log(0.015), log(5^{∗}0.015) , and log(10^{∗}0.015). For each simulation setting, 1000 replication datasets were generated. All simulations were performed using the R statistical software [30].
Results
Simulation results
The empirical type I errors of three methods under the null hypothesis of no association between cumulative drug dose and disease are illustrated in Table 1. Each value represents the total number of cases out of 1000 replications for which the null hypothesis is incorrectly rejected. As expected, Cox regression model with fixed covariate values tended to produce inflated type I error rates, especially in the moderate and high dose groups. Timedependent Cox regression and landmark method, on the other hand, displayed wellcontrolled type I error rates; nearly all values remained close to the nominal level of significance of 0.05, regardless of the degree of disease incidence.
The power estimates for the alternative hypothesis are displayed in Table 2. Statistical power of the timedependent Cox regression approach was higher than that of landmark analyses in the highdose group. In the case of moderate and lowdose groups, the landmark method using landmark time 5 demonstrated higher statistical power compared to the timedependent Cox regression. This is attributed to the fact that there was extremely small number of cases for the highdose group, compared with the moderate group.
Table 3 summarizes the bias and mean squared errors (MSE) of hazard ratios for estimation of the association between cumulative drug dose and outcome. Timedependent Cox regression had much lower bias than the landmark method and generally showed smaller MSE values compared with the other methods. For instance, for the lowdose group, when the β _{0} value was log(0.015), the bias of the timedependent Cox regression model was 0.0009, while the bias of the landmark analyses were 0.0673 and 0.0416 for τ = 5 and τ = 7, respectively.
Real data example
We applied our simulation findings to a real data from the Korean National Health Insurance Database (NHID). The sample was followed for 12 years from 2002 to 2013. The event of interest was incidence of hepatocellular carcinoma (HCC). Among 47,738 patients with incident diabetes, 203 hepatocellular carcinoma cases were identified. Full details of the study design have been described elsewhere [31, 32]. In the current study, total prescribed doses of rosiglitazone were calculated at each year and summed to produce cumulative doses of followup years. Discretetime survival analysis was used to take into account of the intermittent administration of rosiglitazone and the varying dosage between patients. Based on cumulative dose of rosiglitazone use during the study period, drug users were categorized into low (<1350 mg), moderate (1350–4499 mg), high groups (≥4500 mg). In the landmark analysis, sixth year was chosen as a landmark time since the number of incident HCC was balanced at that time point. Data analysis was carried out using the SAS statistical software version 9.4 [33].
Results based on the NHID data are illustrated in Table 4. In the Cox regression, the risk of HCC incidence was lowest among subjects exposed to high cumulative doses of rosiglitazone (HR = 0.443, 95% CI = 0.218 to 0.899). However, the protective pattern of doseresponse relationship and effect sizes were considerably attenuated in the timedependent Cox regression and landmark analysis.
Discussion
In this study, we examined the phenomenon of guaranteetime bias through statistical modeling and simulation study. Specifically, our simulation study assessed the performance of three methods, namely Cox regression, timedependent Cox regression and landmark method based on timefixed Cox regression. These methods depend upon the proportional hazard assumption. According to our simulation results, timefixed Cox regression was shown to be vulnerable to guaranteetime bias [20, 22]. Pharmacoepidemiological studies typically involve the use of timevarying exposures, such as cumulative dose. Hence, under such situations, applying the timefixed Cox regression approach can induce bias due to model misspecification. Our results are in accordance with results previously reported by Mi et al. [29]. However, unlike previous studies, we performed a simulation by including cumulative dose groups as exposures.
Landmark analysis has been suggested in previous studies as a simplified alternative to timedependent Cox regression for elimination of guaranteetime bias in timetoevent data. In our simulation, landmark method was comparable to the timedependent Cox regression method in terms of type I error. But landmark analysis tended to slightly increase MSE. One possible explanation for these results could be the sample size. For landmark analysis to produce efficient estimates, it is imperative that optimal landmark time is specified. As illustrated by our results, if the landmark point is too early, there is a greater possibility of imbalanced observations among treatment groups. On the other hand, if the landmark point is too late, a significant proportion of events may be omitted, giving rise to insufficient number of cases to achieve adequate power. Thus, it is important that landmark studies are designed with sufficient number of participants to maintain adequate statistical power. Additionally, the landmark method will produce estimates with minimal bias conditional upon the treatment being evenly distributed across the followup [29]. Recently, the use of landmark super models, a pooled summary analysis of several landmarks, has been advocated to remedy the problem of low statistical power related to the landmark method [34, 35]. Further studies are warranted to compare the performance of landmark super models with the timedependent Cox model.
Nevertheless, the landmark approach has several advantages over the timedependent Cox regression model. Most notably, this method has the advantage of computational simplicity because drug use is defined as a timefixed covariate by using landmark time. Thus, one can visualize the survival curve of drug users using the KaplanMeier method. But, unconditional KaplanMeier estimates of timevarying status of drug users are unstable because the number of drug users can be quite small at early time point [36].
Conclusions
In conclusion, to avoid guaranteetime bias in observational studies of drug effects, we recommend incorporating timedependent exposure status in the analysis. While both timedependent Cox regression model and landmark analysis were found to be useful in resolving the problem of guaranteetime bias, timedependent Cox regression was the most appropriate method for analyzing cumulative and longterm drug exposure. We recommend the timedependent Cox regression for estimating hazard ratios of cumulative doses. Alternatively, due to its graphical capabilities, the landmark method may be a suitable alternative for visualizing survival curves for treatment groups.
Abbreviations
 HCC:

hepatocellular carcinoma
 MSE:

Mean squared error
 NHID:

National Health Insurance Database
 OR:

Odds ratio
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Acknowledgements
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Funding
This research received no specific grant from any funding agency in the public, commercial or notfor profit sectors.
Availability of data and materials
The data used in this study was simulated. The simulation program is in R and available from https://github.com/jihyeon1/yonsei_biostat.
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ISC, YRC, JHK, HRY, SYJ, and GRK contributed equally to the conception, analysis/interpretation of data, and drafting of the manuscript. CMN edited and revised the manuscript. All authors have read and approved the final manuscript.
Corresponding author
Correspondence to Chung Mo Nam.
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Keywords
 Cox regression
 Guaranteetime bias
 Landmark method
 Timedependent Cox regression