Excuses and Confusion With Respect to Survival Analysis
Written by Ben van Hout, Prof. PhD
An exponential model assumes that the probability to die at any point in time is constant, conditional upon whether one has lived up till that point (the hazard rate). Cancers grow; the bigger the tumour load, the higher the probability of dying and as such one would expect the hazard rate to increase over time. Thus, an exponential model may not make sense. One may have heard me saying something along those lines during ISPOR conferences. I remember one occasion when a junior researcher proudly showed me that an exponential model fitted the data quite well, and I asked, a bit tongue-in-cheek, whether this made sense. She should have humiliated me, with all respect, of course. She should have responded by telling me, “Each individual may have his/her own cancer growth curve, and indeed for each individual, the probability of dying may increase with time. And yes, an exponential model for cancer survival may not make much sense on an individual level. However, Professor van Hout, that does not mean that it cannot be appropriate on a population level.” The latter struck me once more when I was playing with my computer, simulating Gompertz growth curves for the tumour cell growth and assuming a proportional hazards model relating tumour load to the hazard rate. And to my surprise, when assuming some beta distribution to growth parameters in the tumour load, with each individual having a different survival curve, I ended up with an overall exponential survival curve at the population level. So, yes, an exponential model may make perfect sense. In short, I need to apologise to the junior researcher and thank her for just looking puzzled rather than making a fool out of me.
Occasionally, I am questioned about the relationship between progression-free survival (PFS) and overall survival (OS) and the correlation between them. Most often the question comes in a discussion about whether PFS is a good intermediate endpoint. My confusion, which may be obvious, stems from the realization that PFS is the minimum of the time to progression and the time to death. So, the one is part of the other and one may realize that the more people die before progression, the higher the correlation between PFS and OS. Which is confusing, at least to me. But I am also confused when nobody dies before progression. In that case, when post-progression survival (PPS) is always, say, 5 years, there is a perfect relationship between PFS and OS, and the correlation coefficient will be 1. And indeed, PFS is a perfect predictor of OS. But what if PPS is completely uncorrelated to PFS (a correlation of zero)? In that case, the correlation between PFS and OS depends on the length of the PPS in comparison to the PFS. If PPS is minimal, the correlation is again 1; if PPS is the same magnitude as PFS, it is 0.5; and if PPS is on average longer than the PFS, the correlation will drop below 0.5. A negative correlation coefficient is of course bad news: a longer PFS means a shorter OS. But then again, part of this may be masked when there are many people dying before progression, getting the correlation above zero. Isn’t it far more informative to analyze whether the time to progression is correlated to PPS? If there is not a correlation, no worries—one may still state that PFS may be a good intermediate endpoint. If it is positive, great. If the correlation is negative, one may want to worry. Only, in that case, I might want to analyze the relationship between OS and PFS to see whether it is really bad news. Much simpler. Feel free to send me an email at BenvanHout@openhealthgroup.com with your thoughts. I may have to apologize again.
Ben van Hout is the Chief Scientific Officer in Modeling & Meta-Analyses and Real-World Evidence, located in the York, UK offices. He combines an appointment of professor of Health Economics at the School for Health and Related Studies of the University of Sheffield with a position of Chief Scientific Officer of OPEN Health. Prof van Hout has extensive experience in modeling and has contributed to the methodology of economic evaluation in various areas.