The Median is half the message

If you're a fan of the late science writer (and paleontologist) Stephen J Gould, you might remember a piece called "The Median is not the Message" in which he discusses his response to being diagnosed with a rare cancer. Although his doctor tells him not to go looking for information, he of course reads the scientific literature and discovers that the median life expectancy for someone with his disease is six months i.e. half the patients live for less than six months. More reading reveals that there's a long tail to the distribution i.e. beyond six months, some people survive for years. Then he lists the positive factors in his case that would put him in the tail of the distribution, ending up saying that he has a positive attitude and everything to live for.

It's on the last two points that I diverge. Jen has a lot to live for, but I can't see that that makes any difference to the tumour cells.The mantra of a positive attitude requires a separate blogorave - suffice to say that I don't know it makes any difference to the biology. But Gould has an important point - especially in natural systems, variability is the rule and central measures like the median are abstractions. So by itself, the median survival time is not that useful - it's only half the message.

I've been hesitant to write about Jen's overall prospects, not least for fear of misunderstanding. It's easy for people to only hear one number (e.g. the median survival) and see that as a prediction. I think that's why our oncologist is so reticent about giving any prognosis. I also know that mathematics is anathema to many people (Jen still has dreams about failing year 12 maths in high school, when in fact she passed!). Whereas for myself, being a mathematician at heart, I was strangely relieved when I finally found a journal article with a graph of survival and hazard curves - it's like finding a person who can explain what's happening in your language.

One important thing I learnt about curves of overall survival is that they have very long tails - roughly, this means a small proportion of people survive a very long time. (For the highly numerate, the hazard rates have a log-normal distribution i.e. the logarithm of the rate is normally distributed). The practical consequence is that the variability is huge, and it might be more useful to talk about a range of uncertainty rather than one "central" number such as the median. Nevertheless, the typical description is to quote the median survival and then the survival rate at some interval (1, 5, 10 years etc).

It's interesting to contrast this with, say, the statistics of height distribution. For white American males of about 25 years old, the median height is 179 cm (and the average is nearly the same). Only 5% are shorter than 168 cm, and only 5% are taller than 190 cm. So here the spread either side is only about 6% of the median value. In this context, the median height is a useful measure.

Now in Jen's case, we were told that after chemotherapy finishes, the median time to progression (i.e. the time before the tumours start to grow again, and we resume chemo) is 7-9 months. But when I looked more closely at the data in journal papers, I could see that for 20% of people it was less than 3-4 months, and for 20% of people it was more than 18 months. If one took the bottom and top 5% the range would be even larger. So in this case, it might be better to quote the range (3-18 months) rather than the median. (Mathematically, you need at least two parameters for the distribution). The take-home message is not to get too fixed on a particular number - no-one knows. The uncertainty in the time frame is one of the more difficult aspects of having cancer. Since diagnosis she's had five months so far - 161 days today, and we're grateful for every one.


To anticipate a common response: Jen isn't "just a statistic", but the statistics are a grim summary of what has happened to a lot of real people. So why is there such a big uncertainty? Firstly because the extent of people's cancers at the time of treatment varies greatly. Secondly, current understanding of cancer types is too limited to predict the response to particular chemotherapy treatments.

I'm sorry if the starkness of the statistics is overwhelming for you (as it is for us). We haven't given up hope because the real focus of our hope is not upon mere survival but on life with God, now and beyond death. I'm grateful for the statistics because they give us a sense of how much time Jen might have left with us, and help us to plan how we might use it for the best. I hope that it can help our friends in the same way. If you want to see Jen, then sooner is better than later, because there are no guarantees.