# Using survival curves to represent distance-to-failure

Survival curves (sometimes called Kaplan–Meier curves) are a way of showing rates of survival in graphical form. Along the abscissa (x-axis) on our survival graph we would show time. On the ordinate (y-axis) we would show the percentage of people still surviving.

Consider a study of the effectiveness and risks associated with heart-transplant surgery. Time zero on a survival graph does not represent any particular calendar date. Instead, in this study, it would represent, for each individual, the day on which their surgery was performed. Assume that the earliest that anyone died post-operatively was after seven days, and that 10 per cent of people died exactly seven days after surgery. In that case, we would draw a line from the 100 per cent mark at time zero (0,100) to the 100 per cent mark at seven days (7,100). The line would then take a vertical step down, by 10 per cent to the point (7,90) and would then continue along until the time at which the next person dies. If 15 per cent of people suddenly drop dead on day 10, the line would continue from the point (7,90) to (10,90) and then take a step down to (10,75). Since everybody seems to die, we can assume that if the study went on long enough, the survival curve (the line we’ve been drawing) would eventually drop to zero.

Survival curves are ubiquitous in the medical literature. Frequently, the term “survival” does not relate to death but to a failure of some other kind. In a study of the transmission of Human Papilloma Virus (HPV), for example, survival time might mean the length of time between when a person first becomes sexually active and the time when (and if) they become infected with HPV.

Despite the ubiquity of survival curves, it seems that the x-axis is invariably used, at least in the literature related to health, to represent time. But, think of what one might learn if we used the x-axis to represent something else instead; distance, for example.

In a recent study published in Health Place, researchers related the socio-economic status of residential areas to the number of fast foot outlets that there were in each area. Surprise, surprise! There were more fast-food outlets in poorer areas. Now consider what a survival curve would look like, and what it could tell us. Survival would mean “what is the minimum distance a person has to go from their home before they run the risk of being tempted by high-fat fast food?” So for each person living in, say, Melbourne, Australia, we would determine how far they would have to go to reach their closest fast-food outlet. Now it might turn out (hypothetically) that (a) there are no fast food outlets in rich suburbs, (b) rich suburbs are very small; (c) poor suburbs are very very big, and (d) in the poorer, larger suburbs, all the fast food places are right on the suburb border. This would mean that people living in the middle of a poor suburb were actually a lot further away from the temptations of grease and fat, than the people living in the small rich suburb with multiple temptations located just over the border in poor man’s land.

Survival curves in which time was replaced by distance could be used to examine such things as the provision of health services to segments of the population (how far does each person have to go to see a doctor?). In an application not related to health, if time (and distance!) were replaced by “cost”, then one could produce survival curves describing how much people have to pay for a five minute telephone conversation with their parents.

Contributors: Daniel D. Reidpath