# Oncology: how to estimate the survival curve?

I have the age at diagnosis for 58 cancer patients and I want to build a survival model. I want to know at a given age with no prior cancer diagnoses, what is the cancer free survival?

I know that the Kaplan Meier estimate is a nonparametric method, and for parametric models there are exponential, weibull, log-normal, etc. However, how do I know which one to use? How do I know my data follows an exponential distribution? A weibull distribution? Or something else?

How were people recruited and how did you follow them? Are you considering age or calendar year to be the time denominator for survival? Survival as a function of age is okay, except that it does not preclude the cohort or period effects upon survival.

I can't imagine that you will get much useful information from 58 observations aside from, perhaps, summarizing the distribution. For that, a kaplan meier curve is a fine visual summary of those survival data.

Have you excluded people whom you followed who did not get a cancer diagnosis? They are censored observations and it is important that they are included in the analysis so that your survival probabilities are not biased. 58 patients who get cancer gives you a 100% cancer risk over a short time span.

Parametric survival models are not often used for such analyses because the assumptions are difficult to test or even speculate at.

• hey Adam, I'm not sure how the people were recruited and followed. I'm considering age to be the time. My data set does not contain any controls, so everyone in the data set has been diagnosed with cancer (so I have no censored observations). And as for your last point (about parametric survival models), so I should just stick to KM estimate? – Adrian Aug 5 '15 at 20:01
• @Adrian you need to know that information, otherwise I'd strongly suggest omitting the analysis since it will be very misleading. The KM curve is biased if you restrict to cases who have a positive diagnosis within $X$ number of years, especially if $X$ is unknown. – AdamO Aug 5 '15 at 20:05

The truth is that you never really know if your data follows an exponential, Weibull, etc. (with the exception of data you simulated yourself). In fact, it's a very safe assumption that your real world data does not exactly follow any parametric distribution. As such, you are picking a model which best resembles the patterns you typically see in that type of data and assume that this is a good approximation of the actual distribution.

A fairly simple method for assessing whether the parametric model you assumed is appropriate is to compare to the Kaplan Meier curves themselves with the fitted model; i.e. plot the fitted KM survival curve and the fitted parametric curve on the same plot. If the parametric model is appropriate, then the KM survival curve should basically resemble parametric curve, but with some random noise. If it is a bad fit, you may see some systematic differences between the two curves.

That being said, in most cases I would greatly prefer the KM curves for estimation, mostly due to them being more robust and basically as efficient (the survival estimates for which using a parametric model has much lower standard error than KM curves are the estimates which rely heavily on the parametric assumption, so the gain you get is from making assumptions for which you cannot properly assess the appropriateness). However, there are some cases in which you may want a more complex model and so you must use parametric models, such as cure-rate models.