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Kaplan-Meier Survival Curves
Can a datagraphic used in cancer research be adapted to other situations?
You're conducting a drug trial. You have an experimental group (these are the patients to whom you've given the drug that you're testing). And you have a control group (these patients got the placebo). And you observe the two groups over time, gathering data to see if you get a better outcome from the experimental group than from the control group.
With cancer drug treatments the outcome you are often looking for is longer survival. You want the patients who took the drug to live longer than the patients who didn't take the drug. And if they did live longer, you want to know how much longer, and is the difference in survival outcomes significant (i.e. could the difference just have occurred by chance or was it—instead—likely that it was the result of the intervention).
A graphical tool that is used to describe drug trial results is called the Kaplan-Meier Survival Curve. It is an eye-catching way of making sense of survival data.
What does a Kaplan-Meier Survival Curve look like?
Let's suppose that you have a sample of 100 patients who were given the drug and another sample of 100 patients who were given a placebo. Those given the drug are the experimental group (the blue line); those given the placebo are the control group (the black line).
The horizontal axis measures time. And because we've decided in advance of the trial that we want to measure survival up to three years, we've plotted time in months along the horizontal axis, so there are 36 tick-marks, one for each month.
The vertical axis measures the number of patients who survive as time elapses.
For example, suppose that two of the patients in the control group (the black line) die in the first month. The blue line will start off at 100 but will drop by two—to 98—at Month 1. Suppose another two people in the control group die in the second month: the line will drop to 96 for Month Two. If a further two patients were to die in Month 3, then the blue line will drop by another two to 94 in Month 3. And so on.
We've juxtaposed the experimental group (the blue line) over this control group line. And of course what we are looking for (and what we have got) are shallower drops in the blue line than we got in the black line. We want our experimental blue line to stay above the control group black line and we want to see as much space as possible between the two lines. If the two lines are indistinguishable from each other then we will have found no treatment effect and we'll have to go back to the drawing board.
Can we apply this to other NHS situations?
The application of Kaplan-Meier that sprang immediately to the mind of Kurtosis is in describing the impact of allied health professional (AHP) interventions on length of stay. AHPs have long been preoccupied with trying to find ways of describing the impact of what they do. In particular, physiotherapists and occupational therapists frequently talk about how their interventions can, for example, reduce the length of hospital stay for elderly patients.
With this in mind Kurtosis hunted down some real data from a real Scottish hospital where there were two acute elderly wards. One of the wards was—anecdotally—better supplied with physiotherapy and occupational therapy staff than the other. But the casemix of the two wards was pretty much identical. The only factor governing which of the two wards a patients was admitted to was which one happened to have an empty bed at the time it was needed.
Ward 1 was the ward that had the least amount of physiotherapy and occupational therapy time; Ward 2 was the ward with the most.
Here is the Kaplan-Meier survival curve that tracks patients admitted to these two wards in the ten-month period from April 2006 to January 2007.
We need to take care when we look at this graph. For one thing, if we want to demonstrate that enhanced physiotherapy and occupational therapy decreases length of stay then we want our blue line to be below the black line (which is of course the other way around from the previous drug trial graph).
Another thing to take note of is that although we are still measuring survival, it is survival slightly strangely defined. We are looking at how long—and how many—patients stayed in hospital following admission.
And it goes without saying that the study design of this may leave something to be desired. But that is another topic for another day.
To make absolutely sure we understand what this graph is saying, look at the right-hand end points of the two lines at 42 days (or seven weeks) following admission. 16% of the Ward 1 patients were still in hospital after seven weeks; whereas only 7% of the Ward 2 patients were still occupying a hospital bed.
The graph tells us other things, too. It tells us that there wasn't really much difference between the two wards after seven days had elapsed. After seven days, 80% of patients in Ward 1 were still in hospital compared with 79% of Ward 2 patients. If our outcome measure had been concerned with getting more patients home in less than a week, this graph would send us back to the drawing board.
However, after two weeks have elapsed we can see a difference: after two weeks, 55% of Ward 1's patients were still in hospital but only 47% of Ward 2's patients were.
And after three weeks the gap had widened even more: 38% compared with 28%.
The Kaplan-Meier curve shows this data extremely well. It's easy to understand, relatively easy to construct, and there isn't a huge amount of data-collection involved. Moreover, the data can be supplemented with confidence interval intelligence. We can calculate the 95% confidence intervals for the differences between the two groups at any stage on the 'survival' timeline.
Yes, there's a difference in outcome but is the difference significant?
At the top of the page we said that there were three things we'd want to know if we were testing a drug. Firstly, do patients live longer? Secondly, if so, how much longer? And, thirdly, is the difference in survival between the experimental and control groups statistically significant?
It's worth taking a quick look at significance. After all, it's no use churning out an impressive-looking Kaplan-Meier chart if it turns out that the space between the lines could just have arisen by chance.
To address the significance problem we can draw a chart that shows the confidence intervals of the differences between the two acute elderly wards at each point along the horizontal axis. If you're not familiar with confidence intervals, it can be tricky to make sense of it, so let's show it first then explain it afterwards:
Look at the green dots first. Ignore the up-down lines; just look at the green dots. These represent the differences between the Ward 1 length of stay figures and the Ward 2 length of stay figures. For example, if you look at the left-most green dot you'll see that it is exactly zero. Both Ward 1 and Ward 2 started off with no difference in their length of stay. Before the patients had stayed one day it made no difference whether the patients received loads of physiotherapy and occupational therapy or not, they would still be in hospital.
Now look at the next green dot, the one that's second from the left. This is showing a value of -0.4% because after one day's stay the Ward 1 patients actually did better than the Ward 2 patients. It's the same story for the next green dot. More patients in Ward 2 were discharged after two days than in Ward 2.
But from now on, as we move to the right on the chart, it becomes a different story. All of the green dots are above zero. If the advantages of enhanced physiotherapy and occupational therapy effectiveness only manifest themselves once a couple of days have elapsed then there is a story here. And the story gets better as we move rightwards because the differences become greater until it stabilises around the 10% mark.
To cut a long story short, the impact of enhanced physiotherapy and occupational therapy is significant on patients who are going to be in hospital for longer than a week. For those patients their length of stay is significantly shorter if they receive the enhanced service.
On Sunday 9th May 2010 Kurtosis went to see Malcolm Gladwell speak at Edinburgh's Festival Theatre.
His topic for the evening was serendipity. More specifically, scientific serendipity. The history of science is full of examples of serendipity. One of the most famous is probably Fleming's discovery of penicillin in 1928. But Gladwell explained that there were three types of serendipity: Columbian serendipity, Archimedean serendipity and Galilean serendipity.
Columbian serendipity (named after Christopher Columbus) is when you stumble across something—America, for example—completely by accident. Archimedean serendipity is when an idea that you had suddenly—Eureka!—makes sense because something else happens; Galilean serendipity is about a leap into the unknown: you prepare yourself for discovery even though you don't actually know what you are going to find.
Gladwell puts these three different types of serendipity into a league table. Columbian serendipity is at the bottom. Columbus actually just made a mistake—the only thing he had going for him was that he was resourceful enough to set sail. Archimedes comes next. He had the framework of a theory to go on, he was just 'waiting' for the critical moment that helped him make sense of it all. At the top of the league table comes Galilean serendipity. Galileo built a telescope without really knowing what he would see through it. Having built it he just observed and waited and watched. Maybe something good would happen; maybe not.
The case study of Galilean serendipity that Gladwell used concerned cancer research. A company called Synta Pharmaceuticals was trialling a drug that they thought might improve outcomes for melanoma. It turns out they were wrong (so far, at least). You can get a better synopsis of Gladwell's argument here.
Anyway, the thing that most struck Kurtosis about Gladwell's talk was the fact that he explained what a Kaplan-Meier survival curve was without the need for any visual aids. He waved his arms around a little bit and just described—verbally—the kind of graph you'd expect to see for a successful trail and the sort of curve you'd expect to see for an unsuccessful trail.
If Kaplan-Meier can made accessible in this way then we thought that this might be a way of making sense of some other day-to-day NHS problems and topics...
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