What is immortal time bias?
What immortal time bias is, how it gets into an analysis, classic examples, and how to avoid it: align time zero, use time-varying exposure or a landmark.
Immortal time bias can occur in observational studies when, because of how exposure is defined, people must stay event-free for a period of follow-up in order to be counted as exposed. During that period, often called “immortal time”, the outcome cannot occur in the exposed group, and the bias arises when the period is counted as exposed time or left out of the analysis.
Put simply, immortal time bias occurs when people are placed in a group at the start of follow-up using something that only happens later, so everyone in that group is guaranteed to be event-free until it happens.
For example, consider a study examining the effect of a new drug on mortality in patients with a certain disease. If the new drug is started some time after diagnosis, for example 6 months later, but patients who receive it are counted as treated from diagnosis, they are immortal during those first 6 months: they had to survive them to be treated at all. This makes the drug look more beneficial than it is, as those who did not receive the drug will include those who died before having a chance to get treated.
Two ways it gets into an analysis
- Misclassified immortal time. Follow-up starts at cohort entry for everyone, but people who start treatment later are counted as treated from entry. The time before treatment, which they had to survive, is credited to the treated group.
- Excluded immortal time. Follow-up starts at cohort entry for the untreated but at the start of treatment for the treated, so the time the treated survived before starting is dropped. That time was event-free and should have counted as untreated, so leaving it out makes the event rate in the untreated look higher than it is.
Both make the treated group look better than the untreated. The first moves event-free time from the untreated to the treated, which lowers the treated group’s event rate and raises the untreated group’s; the second throws that time away, which raises the event rate in the untreated. Suissa (2008) describes both in pharmacoepidemiology.
A small example shows the size of the bias. Suppose a drug has no effect on death. People who eventually take it spend 50 person-years in total between cohort entry and starting it, with no deaths, because they had to survive to start; they then spend 150 person-years on the drug, with 15 deaths. People who never take it contribute 150 person-years, with 20 deaths. Some of those 20 deaths are in people who would have started the drug had they lived, which is why the 50 person-years before starting contain no deaths.
- Exposure treated as time-varying (correct). Treated: 15 deaths in 150 person-years, 10 per 100 person-years. Untreated: 20 deaths in 150 + 50 = 200 person-years, also 10 per 100. The rate ratio is 1.0.
- Misclassified immortal time. Treated: 15 deaths in 200 person-years, 7.5 per 100. Untreated: 20 deaths in 150 person-years, 13.3 per 100. The rate ratio is 0.56.
- Excluded immortal time. Treated: 15 deaths in 150 person-years, 10 per 100. Untreated: 20 deaths in 150 person-years, 13.3 per 100. The rate ratio is 0.75.
A drug that does nothing appears to cut the death rate by 44% when the immortal time is misclassified, and by 25% when it is excluded. The bias grows with the amount of immortal time (Suissa, 2008).
Examples
- Oscar winners. A 2001 study reported that Academy Award winners lived almost four years longer than their less successful peers, but its headline estimate counted the years before a win towards the winners’ survival (Redelmeier and Singh, 2001). Reanalysed with winning treated as time-varying, the advantage was closer to one year and not statistically significant (Sylvestre, Huszti and Hanley, 2006).
- Statins and diabetes. A cohort study reported that statins delayed the need for insulin in type 2 diabetes (hazard ratio 0.74), but it counted patients as statin users from cohort entry (Yee et al., 2004). When the study was replicated with statin use treated as time-varying, the hazard ratio was 1.97, so the apparent protection came from the analysis (Lévesque et al., 2010). That reversal is not, on its own, evidence that statins cause harm.
- Heart transplantation. Early analyses compared recipients with non-recipients from acceptance into the programme. Recipients had to survive the wait for a donor heart, so transplantation looked more beneficial than it was until the waiting time was handled properly (Gail, 1972; Mantel and Byar, 1974).
Where it occurs
A randomised comparison analysed as randomised is protected, because each patient’s group is fixed at randomisation, which is also the start of follow-up. Comparisons within a trial that group patients by something that happens later, such as response to treatment or adherence, are observational, and can suffer from it. It can also occur in many types of observational study, including cohort and case–control studies.
How to spot it
Ask of any comparison: when does follow-up start for each group, and is anyone’s group decided by something that happens after that? If people must survive, or stay event-free, for a while in order to join a group, that group has immortal time. Exposure definitions that need a minimum amount of use, such as a year of treatment or two prescriptions, are a common source.
How to avoid it
- Line up time zero. Start follow-up, check eligibility and assign groups at the same moment, as a trial would. Specifying the target trial that an observational analysis emulates makes this explicit (Hernán et al., 2016).
- Treat exposure as time-varying. Count each person as unexposed until they start treatment and exposed from then on, for example with a time-varying covariate in a Cox model. The time before treatment then belongs to the period in which it happened. This removes the immortal time but not confounding. The reasons people start treatment often change over time and must be adjusted for as well, and when earlier treatment affects them, a standard Cox model cannot do this and methods such as inverse probability weighting are needed. A Kaplan–Meier plot by eventual treatment group brings the bias back, even when the model handles it correctly.
- Use a landmark. Classify exposure at a fixed time after entry, and start follow-up there among those still at risk. Landmark analysis is simple, but it drops everyone who has the event before the landmark, counts anyone who starts treatment after it as untreated, and its answer can change with the landmark chosen.
- Clone, censor and weight. When a treatment strategy allows a grace period to start, each person can be copied into every strategy they are compatible with at time zero, censored when they deviate from it, and reweighted for that censoring. The weights remove the bias from this censoring only if the factors that drive starting treatment and also predict the outcome are measured, and because each person appears in more than one arm, confidence intervals usually come from bootstrapping.