I am a graduate student in biochemistry working on cancer. I am currently looking at epidemiology models of the disease. Epidemiologists have developed formulas that predict the frequency of various cancers as a function of age. The disease incidence and mortality rates occurs as a function of $a^4$ to $a^7$. I am reading some papers, but I don't understand what the authors are saying:
1) If the probability of an outcome is indicated by $a^n$, this means that $n+1$ independent events, each occurring randomly and with comparable probability per unit time, must take place before the ultimate outcome (e.g. lethal cancer). I understand that one multiplies the probabilities together to follow $a^n$ (e.g. flipping a coin three times to get a sequence). However, why is it $n +1$ and not simply $n$?
2) The author introduces the log-log plot where the slope of the regression represents $n$. Basically, you take the log of the age (x-axis) and the log of the incidence rate and put a regression line. For example, I have used Canadian statistics data on cancer mortality in men and generated a log-log plot.
So, the slope represents $n$, so there should be 5.5 events that occurred to create a lethal cancer, no? I would greatly appreciate the community's feedback. Thank you!