Preamble: This shortie is my attempt to come up with an example that makes the problem underlying immortal time bias as obvious as possible. 

In Japan, people who reach the age of 100 are handed a commemorative silver sake cup from the prime minister. Nothing says “we’re delighted you’re still with us” like presenting someone with tableware they are, actuarially speaking, unlikely to get much use out of.

(In 2015, according to widespread English–language reporting, this started to get too expensive, and so cheaper options were considered. Apparently, the solution was to make the cup from silver-plated nickel silver^[1]Nickel silver, for the record, contains no silver, and still is a less confusing label than immortal time bias. rather than solid silver, cutting the cost per centenarian considerably. Kanpai!)

Imagine you have data from the Japanese population and  calculate how long those who received the cup and those who never received the cup live, on average. Unsurprisingly, those who received the cup have a life expectancy over 100 — because they needed to turn 100 to receive the cup in the first place. Those who never received the cup have a considerably lower life expectancy, maybe somewhere around 80 — the usual Japanese life expectancy, minus every cupper.

It is very unlikely that from this, you’d conclude that the commemorative cup somehow extends life expectancy by 20 years. That’s because you’re not stupid. But also, there’s no variation in who receives the cup, so we’re lacking the counterfactual centenarian who has to drink their sake from an inferior cup. So, for the sake of sake, let’s turn this into a randomized experiment: whenever somebody turns 100, the prime minister flips a silver-plated nickel silver coin to decide who gets a cup and who doesn’t. (This would be a great alternative way to save costs because in expectation you need per 2 centenarians 1 cup.)

Once again, we can calculate life expectancy for those who did receive a cup, and those who didn’t. But instead of doing a reasonable analysis – more  on that later – we once again analyze the whole population of Japan, not just those who reached the age of 100. In this manner, the cup group still ends up with an average life expectancy above 100. The group of those who never received a cup will be a mix of people who never reached the age of 100 plus some unlucky centenarians, so life expectancy will again be much lower, again closer to 80 (albeit slightly higher than in the scenario above, thanks to the unlucky centenarians).

Again, you wouldn’t conclude that receiving the (now randomized) cup extended life expectancy by 20 years. The comparison is rigged because to potentially receive the cup, you have to make it to 100 to begin with. Everybody who dies earlier than that couldn’t have received the cup treatment and ends up in the control group. 

This scenario is an (extreme) example of immortal time bias. What is happening here is that the “outcome clock” starts before the treatment occurs: Our outcome is life expectancy and we count that starting from birth; the treatment occurs a century later. Essentially, everybody in the treatment group gets 100 years “for free” before they are even given the treatment. Looking back from the treatment, they actually were immortal for the last 100 years. They couldn’t have possibly died, because if they had, they would not have ended up in the treatment group. Think of it as wearing plot armor that protects them from any danger up to the treatment. 

Admittedly, that isn’t really how we usually think of immortality, and that’s also why I find the label rather confusing. It’s like saying that the readers of this blog post have been immortal up to this point in time.^[2] Despite our efforts to make our writing accessible to a large audience, we have not yet discovered a way to reach the dead. Congratulations, here is a commemorative blog post on immortal time bias! Alternatively, we may just as well frame the bias the other way around, thinking of the people who did not live to see the silver lining cup (or the poor souls who died before they could read this blog post). Because they died before they could receive the treatment and subsequently die in the treatment group, let us honor their memory by calling it the “what is dead may never die” bias.^[3]I’m aware that a bias only truly arrives once it has a sufficiently ominous name. This is the closest methodologists get to having a metal band.

In any case, in our example, the solution is really simple: we have to align the outcome clock with the treatment. In our silver cup experiment, we achieve this by simply checking how much longer people live starting from the point in time they were randomized into the two groups. We can then simply compare the two randomized groups to estimate the effect of the silver cup on life expectancy, without having to worry about immortal time. That being said, this is not a feature of the randomized intervention but rather of the analysis – it’s entirely possible to analyze data from a randomized trial so that immortal time bias is introduced along the way.^[4]For example, maybe some people die before the prime minister manages to give them their treatment. Naturally, those people couldn’t possibly benefit from the life-extending properties of the cup, so we may be tempted to exclude them from the treatment group. But if we do that, we once again ensure that people who are included in the treatment group enjoy some stint of immortality, this time between randomization and actual treatment.

The rest of the owl

Apparently, immortal time bias is a really common problem in health research. That is not because the researchers involved are stupid, but because in the usual scenario, it is a lot less obvious – the relevant bits of information to spot it aren’t handed to you in a silver cup – and harder to avoid.

It usually affects longitudinal observational studies in which information about treatment status is not collected with a simple checkbox at the beginning of data collection, but possibly at different points in time and retrospectively, and the treatment of interest may need to be pieced together from multiple pieces of information. Then, usually, a survival analysis is conducted. The description of how the treatment variable was generated may sound perfectly reasonable when considered in isolation; likewise, the analysis itself may not sound problematic at all. It’s when these two things are combined in such a way that the outcome clock starts before the treatment that the problem arises.

The fact that those studies are observational studies probably also adds some degree of confusion about the nature of the bias – after all, the people who potentially could have received treatment (if they hadn’t died before) might have been very different from those who couldn’t to begin with. That’s true, but it is not what immortal time bias is about. Likewise, the possibility of selective attrition may lead to confusion – after all, the people who survived to treatment may be different on many variables, including unobserved ones. That, as well, is likely true. But it’s not necessary for immortal time bias to arise. For immortal time bias, you really just need the misaligned design in which the outcome clock starts before the treatment happens. Think of it as an accounting error with a very exciting  name that can happen in the most boring of all imaginable worlds.

I’m not sure I could confidently spot immortal time bias in the wild. In fact, I am writing this blog post to manifest an understanding of the whole matter. I feel like I came up with a somewhat satisfying example to get intuitions going, so my work here is done and I am going to leave the rest to the pros.  Suissa and Azoulay (2012) discuss multiple instances of immortal time bias in research on the effects of metformin on the risk of cancer; their examples very nicely showcase the many faces of the problem.^[5] Many thanks to Saloni Dattani, who pointed me to this helpful resource. Ellie Murray has written a helpful blog post on the matter which covers the canonical type of graph used to illustrate the phenomenon (which, honestly, I find a bit confusing) and also discusses the second type of immortal time bias – when immortal time is improperly excluded.

And that’s about as far as my understanding will safely carry me. If you’ve read this far, your reading clock has finally caught up with my writing clock, the two aligned at last.