Hazard and Survival

If I have a tumor that I’ve been told has a malignancy rate of 2% per year, does that compound? So after 5 years there’s a 10% chance it will turn malignant?

This turns out to be an interesting question, because the answer depends on what that 2% means. If we know that it’s the same for everyone, and it doesn’t vary over time, computing the compounded probability after 5 years is a relatively simple.

But if that 2% is an average across people with different probabilities, the computation is a little more complicated – and the answer turns out to be substantially different, so this is not a negligible effect.

To demonstrate both computations, I’ll assume that the probability for a given patient doesn’t change over time. This assumption is consistent with the multistage model of carcinogenesis, which posits that normal cells become cancerous through a series of mutations, where the probability of any of those mutations is constant over time.

Let’s start with the simpler calculation, where the probability that a tumor progresses to malignancy is known to be 2% per year and constant. In that case, we can answer OP’s question by making a constant hazard function and using it to compute a survival function.

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