Bias due to coarsening of time intervals in the inference for the effectiveness of colorectal cancer screening

Abstract

Background

Observational studies are frequently used to estimate the comparative effectiveness of different colorectal cancer (CRC) screening methods due to the practical limitations and time needed to conduct large clinical trials. However, time-varying confounders, e.g. polyp detection in the last screening, can bias statistical results. Recently, generalized methods, or G-methods, have been used for the analysis of observational studies of CRC screening, given their ability to account for such time-varying confounders. Discretization, or the process of converting continuous functions into discrete counterparts, is required for G-methods when the treatment and outcomes are assessed at a continuous scale.

Development

This paper evaluates the interplay between time-varying confounding and discretization, which can induce bias in assessing screening effectiveness. We investigate this bias in evaluating the effect of different CRC screening methods that differ from each other in typical screening frequency.

Application

First, using theory, we establish the direction of the bias. Then, we use simulations of hypothetical settings to study the bias magnitude for varying levels of discretization, frequency of screening and length of the study period. We develop a method to assess possible bias due to coarsening in simulated situations.

Conclusions

The proposed method can inform future studies of screening effectiveness, especially for CRC, by determining the choice of interval lengths where data are discretized to minimize bias due to coarsening while balancing computational costs.

Key Messages.

• For observational studies of colorectal cancer (CRC) screening effectiveness, G-methods, or ‘generalized methods’, provide an appropriate method to estimate the effectiveness of a screening regimen by removing biases from confounding variables affected by screening.

• We develop a method to assess the bias in G-methods due to information loss when data are coarsened, a common practice for observational studies of screening effectiveness, using the example of colorectal cancer screening.

• We observe that the relative diagnosis rate under the faecal immunochemical tests (FIT) CRC screening regimen, compared with no screening, is artificially attenuated with a larger degree of coarsening, whereas the relative CRC mortality rate is amplified.

• Data coarsening can result in egregious bias of CRC mortality rates due to mislabelling diagnostic versus preventative screening exams; this should be considered when analysing longitudinal studies of CRC screening effectiveness.

Background

The utility of observational studies for evaluating disease screening, especially for cancer screening, is exemplified by studies showing that screening with colonoscopy or stool-based tests lowers the risk of colorectal cancer (CRC) incidence and mortality.^1–3 While practical for estimating population effectiveness, and when randomized trials are challenging, observational methods are subject to numerous potential biases. For example, when evaluating screening effectiveness using observational designs, each screen changes the future probability of disease outcomes; thus, the screening regimen is best viewed as a time-varying treatment given over a period of time.^4,5

While multistate models and time-varying Cox regression models have been used for the analysis of longitudinal data in the context of cancer screening,^6,7 G-methods, or ‘generalized methods’, have also provided appropriate ways to estimate and mitigate these biases.⁸ G-methods, originally derived for causal inference from complex longitudinal studies,^9–11 are a class of statistical methods that go beyond traditional linear and logistic regression by allowing for confounders varying over time.

CRC screening is an ideal complex model for studying observational study designs and models. CRC is the second leading cause of cancer mortality in the USA, with an estimated 52 550 deaths in 2023.¹² Colorectal neoplasms may be detected preclinically by multiple screening methods, including colonoscopy and faecal immunochemical tests (FIT).¹³ FIT detects human globin in the stool, is non-invasive, requires one stool sample, and unlike the invasive counterparts like colonoscopy, does not require any dietary restrictions or bowel preparation.¹⁴ The United States Preventive Services Task Force (USPSTF) recommends annual FIT screening for average-risk individuals aged 45–75 years; understanding its effectiveness has significant public health implications.^13,15 Finally, CRC screening is influenced by time-varying effects, including confounding by variables affected by screening; for example, polyps detected in the previous screening rounds would alter subsequent CRC screening decisions.

Motivated by unmet methodological needs to study the effects of screening exams on health outcomes, this paper examines biases from various ways in which coarsening (aggregating data on a discrete time scale, such as monthly to annually) can generate biased estimates, using FIT screening on CRC mortality as an example.

Development

Setting

The setting of cancer screening is unique with confounding by variables that are affected by treatment (i.e. screening). Unlike many other settings with repeated treatments, in each discrete application of cancer screening, the intervention may produce a nearly instantaneous effect on the key time-varying confounder diagnosis of a neoplasm (indicated by polyp detection), which is itself a discrete event.

However, observational data with a discretized time scale lack exact times when screening decisions are made and intermediate events occur. This may give biased inferences regarding screening efficacy. Additionally, even when the exact timings of various events are available, the computational burdens of using G-methods with precise data on event timing in large datasets can be quite substantial; the computation time increases exponentially as the number of time points increases.^16–18 Thus, coarsening is a necessary component of data analysis in many cases. In such circumstances, it is useful to understand the potential biases introduced by coarsening and the extent to which biases change with varying degrees of coarsening.

Explanation of coarsening bias

In observational data, the timing of the initial FIT is distributed over an interval rather than precisely on specific dates. For example, a guideline may recommend that screening start on a specific date, such as an age-recommended birthday, and then be repeated annually, presumably exactly on that same day. However, real-life variation for when screening starts and is repeated are common, resulting in variable time gaps between re-screens.

Coarsening of a few finer time periods counts the occurrence of an event as long as the event has occurred sometime during that time period. Coarsening the screening intervals leads to loss of information and potential biases. As an example, suppose two individuals are screened in two subsequent time periods; when these two time periods are coarsened, both are classified as FIT screened in this coarsened time period, but the information that the second individual was not screened in the first time period is lost.

Figure 1 is a directed acyclic graph (DAG) illustrating how coarsening results in biased estimates on FIT efficacy as a consequence of confounding by a variable (neoplasm diagnosis). In the DAG, diagnosis ($D_1$), which is an intermediate outcome, confounds the effect on CRC mortality ($Y$) of subsequent screening because:

Figure 1.

A directed acyclic graph for faecal immunochemical test (FIT) screening strategy for two time periods. Variables: $D$: diagnosis, $S$: Screening decision, $Y$: colorectal cancer (CRC) mortality and $U$: state of colorectal neoplasm. The colours and the numbers can be used interchangeably and are used in the text to describe confounding by the diagnosis variables

1. it is associated with CRC mortality ($Y$) by leading to treatment that reduces mortality (1A, Figure 1, $D_1\to Y\left)\mathrm{ }\mathrm{and}\mathrm{ }\mathrm{it}\mathrm{ }\mathrm{is}\mathrm{ }\mathrm{affected}\mathrm{ }\mathrm{by}\mathrm{ }\mathrm{the}\mathrm{ }\mathrm{state}\mathrm{ }\mathrm{of}\mathrm{ }\mathrm{colorectal}\mathrm{ }\mathrm{neoplasms}\mathrm{ }\right(U$) (1B, Figure 1, $D_1\leftarrow U\to Y$), which may be latent until diagnosis; and

2. it affects subsequent screening ($S_1$) as neoplasm precludes future screening and surveillance intervals (Figure 1, $D_1\to S_1)$.¹⁹

Collapsing the two time periods into a single time interval in analyses would hide the confounding effect of neoplasm diagnosis, leading to a biased inference for screening efficacy. This situation arises as individuals who develop neoplasms prior to their scheduled screening in the second period are misclassified as unscreened; bias occurs because these individuals will have a different mortality risk (see Condition i).

Application

To demonstrate these concepts, we evaluate three scenarios. Scenario 1 is a hypothetical randomized control trial (RCT) with two arms: no screening versus FIT screening, and two time periods for screening. Scenarios 2 and 3 use the same data-generating process as Scenario 1. However, similar to observational studies and unlike in RCTs, they do not have access to the randomization information.

In Scenario 1, a total of 20 000 individuals are randomized to no screening versus annual FIT screening and followed for 1 year. Those in the FIT screening regimen are randomly split into either receiving FIT exactly at the beginning of the calendar year or receiving it exactly 6 months later in the same calendar year. We refer to these two groups as immediate FIT and delayed FIT groups. Each of these 6 months constitutes a time period, i.e. we have a study with two time periods. Here, and in more complicated scenarios considered later, the initial randomization or unconditional unconfoundedness allows for more transparent calculations; adjustment for baseline confounders is a straightforward extension of the approaches presented here. The structured tree graph in Figure 2 illustrates this scenario.³

Figure 2.

A structured tree graph for faecal immunochemical test (FIT) screening over two time periods using hypothetical data. The leftmost circle denotes the initial randomization of 10 000 out of the 20 000 to be screened in one of two periods. The remaining circles illustrate the number of individuals with and without neoplasm diagnosis in the top and bottom paths, respectively. The numbers at the end of each path denote colorectal cancer (CRC) mortalities

Figure 2 encodes that FIT immediately increases the detection of neoplasms, but subsequently, after a negative screening result, the probability of diagnosis (i.e. disease incidence) declines to a level below that of the never-screened group. Similarly, the average CRC mortality risk among all screened participants is lower in the same period than among unscreened individuals, given the benefits of early detection and treatment among those whose neoplasms are detected or whose pre-cancerous polyps are removed.

In Scenario 2, the corresponding observational study to Scenario 1 is considered, but the information on the initial randomization, i.e. who is randomized to be screened, is unknown. The structured tree graph for Scenario 2 is shown in Figure 3, in which only the time when an individual receives treatment is observed, not the exact date when they were scheduled to be treated as in the randomization design. Thus, the 200 individuals diagnosed with cancer in the first 6 months of the study who were randomized to be screened in the second 6 months in Scenario 1 are inappropriately recorded as ‘never screened’ in this observation study analogue (Scenario 2). This is because they were diagnosed with a neoplasm before they started scheduled screening, which precluded them from a CRC screening.

Figure 3.

Panel (a) shows the corresponding structured tree graph for faecal immunochemical test (FIT) screening for the observational data based on Figure 2. Panel (b) highlights certain pathways and converts frequencies to probabilities to show how to apply the g-formula to calculate colorectal cancer (CRC) mortality risk under the no-screening regimen. The colours trace the three paths along which we calculate the CRC mortality risks and then add in the g-formula

Scenario 3 exemplifies a different observational data analysis, where information on screening is obtained only in every other period. It assumes first that we classify individuals only by immediate FIT and ignore the delayed scenario under the FIT regimen. In other words, the study classifies all who did not receive a FIT screening at the beginning of the study as ‘never screened’. Table 1 depicts our observational data and calculations.

Table 1.

Risk calculations with data from Figure 2 when information on screening is obtained every other period

Baseline FIT	Total	Neoplasms	Risk	CRC deaths	Risk
No	15 000	2000	0.133	245	0.016
Yes	5000	1200	0.240	60	0.012
Risk difference			0.107		−0.004
Risk ratio			1.800		0.735

A randomization procedure assigns 10 000 out of 20 000 to be screened, of which 5000 each are randomized for screening in the first and second periods, respectively, and the remaining 10 000 are not screened. Due to coarsening, only the 5000 are counted to receive baseline faecal immunochemical test (FIT).

Application to more complex scenarios with finer time scales and multiple screenings over longer time periods

In real-world settings, FIT screening is recommended annually or, in some settings, biannually and data are accumulated on a relatively fine time scale. It is thus of interest to understand the relation of the degree of coarsening, i.e. the length of the discretized time interval, to the bias; we study this below in complex settings. We describe our settings below to evaluate coarsening bias in these real-world settings. Rate calculations for such scenarios are based on simulated data, in which specified parameter values for the data generating models are partially based on existing literature on diagnosis rates without screening,^13,20 after a FIT-negative result of interval cancer^2,21 and in a subsequent screening^22,23 CRC mortality rates with different instances of diagnosis.^24–26 Details of these parameters are provided in Supplementary File 1 (available as Supplementary data at IJE online) along with the R statistical software computer code in Supplementary File 2 (available as Supplementary data at IJE online)²⁷ where these rates can be modified to calculate risk, specifying the degree of coarsening (see Supplementary Section S2, available as Supplementary data at IJE online).

Application to finer time-scaled data for a short period

We considered a ‘study’ where each individual is recorded each month for 12 months and: (a) there are no baseline covariates, (b) in each month, receipt of screening FIT within that month is first determined and then whether a neoplasm was detected (there is no censoring) and (c) any death is because of CRC.

Note that (a) relates to the study design and is made to simplify the illustration. Assumptions (b) and (c) simplify calculations. The assumptions are not likely to affect the effect the level of bias from the degree of coarsening.

The ‘randomization’ in this setting is as follows. At the start of the study, each individual is randomly assigned with 50% probability to screening FIT protocol versus no screening. For individuals assigned to FIT, the assignment probability to a specific time (month) for screening is 1/12 for each of the 12 months of the study period. As in the previous section, this randomization information to different months throughout the year is hidden in an observational study. Thus, g-formula does not assume this equal probability randomization.

Our interest is in the risks of neoplasm and cumulative CRC mortality under FIT screening compared with no screening. In this illustration, these parameters are assumed to be known (see Supplementary Section S3, available as Supplementary data at IJE online). Thus, the calculations below are not affected by any sampling variabilities. In Supplementary Section S1 (available as Supplementary data at IJE online), we provide these formulas for a general coarsening for the FIT and no screening regimen.

Application to finer time-scaled data for longer study period with multiple annual FIT screenings

We consider a 10-year-long study period with the same degrees of coarsening. Half of the population is randomized to be not screened. In the other half, an individual is randomly assigned to be screened at one of the first 12 months with equal probability, and, while eligible, is screened at the same month each subsequent year. Individuals in the FIT screening regimen remain eligible for screening so long as they have not had a CRC diagnosis. Rate calculations use simulated data. As noted before, these calculations require specifications of different rates, e.g. diagnosis rate at screening after two consecutive FIT screenings or between the second and third screenings. We specified these parameters guided by a review of existing literature.^2,13,20–26Supplementary Section S3 (available as Supplementary data at IJE online) details the data generating model.

Results

In Scenario 1, 24% of individuals randomized to FIT are diagnosed with a neoplasm, but only 8% not randomized to screen are diagnosed, yielding a risk difference of 16% and a risk ratio of 3 for neoplasm diagnosis (Table 2). The 1-year neoplasm risk is the same in both the immediate and delayed FIT groups, consistent with the randomization strategy. The 1-year CRC mortality risk in the FIT screening regimen is 1.35%; 60/5000 = 1.2% in the FIT immediate group and 75/5000 = 1.5% in the delayed FIT group. The mortality in the no-screening group is 170/10 000 = 1.7%. The mortality risk difference is −0.35% and the risk ratio is 0.794.

Table 2.

Risks for different screening groups using the hypothetical RCT in Figure 2

	Neoplasm incidence			CRC mortality
FIT regimena	Risk	Risk difference	Risk ratio	Risk	Risk difference	Risk ratio
No	0.08			0.017
Initial	0.24	0.16	3	0.012	−0.005	0.706
Delayed	0.24	0.16	3	0.015	−0.002	0.882
Overall	0.24	0.16	3	0.014	−0.004	0.794

The randomization procedure assigns 10 000 out of 20 000 to be screened, of which 5000 each are randomized for screening in the first and second period, respectively, and the remaining 10 000 are not screened. While faecal immunochemical test (FIT) detects more neoplasms, colorectal cancer (CRC) mortality risk is lower under FIT since early detection of neoplasms reduces mortality risk.

^aNo, no screening regimen; Initial, faecal immunochemical test (FIT) screening only in the first period; Delayed, faecal immunochemical test (FIT) screening only in the second period; Overall, faecal immunochemical test (FIT) screening either in the first or second period.

Because there are still no unmeasured confounders in Scenario 2, appropriate adjustments give consistent effect estimates. Applying the g-formula, we obtain the risk of neoplasm diagnosis under the regimen: never screen with FIT as 0.08. We look at the risk of CRC mortality under the same regimen: {1 × 0.4} × 0.3 + {1 × 0.96 × 1 × (1/24)} × (1/8) + {1 × 0.96 × 1 × (23/24)} × 0 = 0.017. These numbers are identical to the previous calculation in Table 2, where randomization was performed.

The structured tree graph in Figure 4 highlights a screening regimen in which FIT is delivered each year, half at the beginning, half at 6 months. Applying the g-formula, the risk of neoplasm is 0.24. The risk of CRC mortality is 0.0135. Thus, again, the g-formula recovers the same risks as the randomized trial in Table 2.

Figure 4.

Corresponding structured tree graph for faecal immunochemical test (FIT) screening for the observational data based on Figure 2. Paths are highlighted for the regimen: provide FIT each year, half at the beginning, half at 6 months, and we show the probabilities along the paths for the corresponding g formula. The colours trace the six paths from the root of the tree to the terminal events along which we calculate the colorectal cancer (CRC) mortality risks and then add in the g-formula under the FIT screening regimen

In scenario 3, the measures of effect for neoplasm diagnosis are attenuated relative to the true measures of effect, e.g. a risk difference 0.107 compared with the true measure of 0.16. This is expected, as the comparison is made after moving half of the high-risk group assigned screening into the lower-risk unscreened group. However, the measures of association for mortality are somewhat larger than the true effects, e.g. risk difference of −0.00433 compared with the true measure of −0.0035. This can be explained by noting that, as seen in Table 2, the mortality risk among individuals assigned to delayed screening is more akin to the mortality risk among individuals not assigned to screening than among individuals assigned to immediate screening with the lowest mortality risk. Thus, when the comparison combines the first group with the second group, the relative mortality risk of the combined ‘no FIT’ group is larger than the true risk of no screening versus FIT screening.

Applications to more complex scenarios

Finer time-scaled data for a short period

Figure 5 plots the risks, risk differences and risk ratios for different levels of coarsening of 12 months into equal-length intervals for application to shorter time period. A snapshot of the numbers is given in Table 3. The relative diagnosis rate under the FIT regimen over the no-screening regimen is reduced with a larger degree of coarsening. The relative CRC mortality risk, on the other hand, is amplified. On the log risk ratio scale, this bias is pronounced. This bias occurs because higher-risk individuals may be classified into the no-screening regimen, despite plans to be screened later, because they were diagnosed before the scheduled screening. The numbers that relate to this figure are given in a table in the Supplementary Section S4 (available as Supplementary data at IJE online).

Figure 5.

Patterns of cumulative neoplasm diagnosis and colorectal cancer (CRC) mortality rates over 1 year for different degrees of coarsening

Table 3.

Cumulative neoplasm diagnosis rates in years 3, 5 and 10 for a range of degrees of coarsening under faecal immunochemical test (FIT) screening regimen or no screening regimen based on a synthetic longitudinal model of colorectal cancer (CRC) screening over a decade

		Length of a coarsened interval
Year		1	2	3	4	6
3	FIT regimen	0.21064	0.20979	0.20896	0.20817	0.20673
	No screening regimen	0.16604	0.16697	0.16781	0.16855	0.16970
	Log risk ratio	0.23793	0.22828	0.21931	0.21113	0.19740
5	FIT regimen	0.37614	0.37508	0.37407	0.37311	0.37135
	No screening regimen	0.31291	0.31340	0.31379	0.31409	0.31435
	Log risk ratio	0.18403	0.17968	0.17572	0.17221	0.16664
10	FIT regimen	0.65688	0.65577	0.65471	0.65371	0.65188
	No screening regimen	0.68711	0.68642	0.68566	0.68479	0.68275
	Log risk ratio	−0.04499	−0.04569	−0.04618	−0.04645	−0.04627

The differences in the numbers across columns show the implication of coarsening.

Larger study period with multiple annual FIT screenings

Under different degrees of coarsening, we compare a FIT screening and a no-screening regimen. For a general coarsening structure, Supplementary Section S2 (available as Supplementary data at IJE online) provides the formula for the calculations. Similar to the earlier sub-section, these calculations require us to specify different rates, e.g. diagnosis rate at screening after two consecutive FIT screenings or in between the second and third screening. These parameters are specified and guided through a literature review.

Conclusions

This paper evaluated how bias may occur and could be mitigated in observational studies of cancer screening with time-varying confounding using worked examples from CRC screening. Here, we showed how G-computation analysis recovers the correct effect estimates when the variable is recorded in original forms even with observational data, but biased estimates result when the time-related variable is discretized or coarsened. We provide scenarios as well as examples that mimic real-world repeated annual FIT screenings events that may occur over 10 years to match the recommended interval for screening colonoscopy. A simulated randomized study is used as the basis for assessing bias from coarsening.

In observation studies, coarsening bias may occur because of classifying higher-risk individuals into no-screening regimens. Using theoretical arguments and detailed numerical studies, we demonstrated that statistical inferences for CRC screening efficacy based on longitudinal studies could be biased if time-coarsened data are used. This is related to two unique features of cancer screening: (i) screening is an intermittent series of discrete actions with significant gaps in between; (ii) in each discrete application, the event of screening itself may produce a nearly instantaneous effect on the diagnosis of a neoplasm, the key time-varying confounder, which is itself a discrete event. This coarsening bias occurs separately from bias due to unmeasured confounding. Based on our analysis, we recommend using shorter time intervals that are still computationally feasible and evaluating the sensitivity of the analytical results to this interval size. Our study also suggests that appropriate recording of the times of screening, possible diagnosis and subsequent events is important for accurate analysis of screening efficacy based on such data. Specifically, mislabelling the event to a wrong interval will increase the bias of the inference with the wider, coarsened intervals.

Regarding screening for CRC, it may be of interest to compare different screening methods. A longer study period is required for screening colonoscopy as it is recommended every 10 years for average-risk people, aged 45–75 years. While our formulas can still be used, we did not analyse this context as the existing literature did not provide sufficient details to specify diagnosis rates across multiple screening regimens and over a long study period. However, as results for comparative effectiveness studies with long study periods become available, our methods can guide additional assessments of the biases in coarsened comparisons across multiple screening methods.

Lung and breast cancer screening are parallel examples where our investigations could guide epidemiologists in the evaluation of the comparative effectiveness of different types of screening regimens. Efficacy estimation for vaccines with multiple doses over time is another application of the approaches described. As our calculations easily adapt to many different scenarios—using the general formulas given in the Supplementary Material (available as Supplementary data at IJE online)—we hope to have simplified the process of evaluating the potential influence of coarsening for researchers and suggest methods for selecting more optimal intervals.

Ethics approval

Approval is not needed since the study did not use any real individuals.

Data availability

The data underlying this article are available in the article and in its Supplementary data at IJE online.

Supplementary data

Supplementary data are available at IJE online.

Author contributions

B.K. co-designed and conceptualized the study, led the statistical analysis, wrote the original draft of the paper and assisted in subsequential rounds of writing and editing. M.M.J. co-designed and conceptualized the study, supervised the analytic strategy, provided substantial edits to the original draft and assisted in subsequential rounds of writing and editing. A.I.H. acquired data from an extensive literature search and assisted in the writing and editing of the manuscript. Y.K.L. assisted in the formal analysis as well as the writing and editing of the paper. A.G.Z. provided important intellectual content to the design of the study and assisted in the writing and editing of the paper. C.A.D. obtained funding for this project, provided important intellectual content and oversight to the manuscript and assisted in the writing and editing of the manuscript.

Funding

This research was supported by the National Cancer Institute of the National Institutes of Health under Award Number R01CA213645. Additional funding was obtained through National Institutes of Health (NIH)/NCI Cancer Center Support (grant number P30 CA008748) (AGZ and AIH). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

Conflict of interest

None declared.