I have an epidemiology question with logs 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!
 A: Armitage and Doll assumed that some total number of mutations are required for a cancer to develop. It's simplest to explain if the probability of any mutation occurring over a short time interval is low, independent of age, and equal for all cancer-causing mutations. I find it simpler to understand if you drop the Armitage-Doll requirement of a particular required order of mutations, but the relation between the slope of your plot and the number of mutations will be the same in either case.
If only 1 mutation were needed for a cancer, then the incidence rate would be constant over age, equal to the mutation probability. That would be a slope of 0 on your plot.
If 2 mutations are needed, the age-dependent issue becomes the probability that 1 prior mutation occurred before the (age-independent) second mutation. The probability of the cancer developing at any age is then the probability of already having the first mutation (first power of age) times the (age-independent) probability of getting the second mutation. Slope of 1 on your plot. And so on for larger numbers of required mutations: the slope of incidence rate versus age represents the probability of already having all except the last mutation, with the probability of the last mutation independent of age. So a slope of n means that n+1 mutations are ultimately needed.
The value of 5.5 on your plot represents the difference between cancers at young ages and older ages. Childhood and youth cancers typically occur when an individual is already born with a cancer-related mutation, so that fewer additional mutations are required for a cancer to develop. In adults, cancer incidence represents the accumulation of acquired mutations. Armitage and Doll reviewed literature suggesting that analysis should be restricted to individuals over 25 years of age, in which case your plot would have a slope closer to 6.
You should know that the simple interpretation in terms of "mutations" has changed somewhat over the 60 years since that paper appeared. It's probably best to think about cancer-related events rather than mutations in DNA as we understand them today. (The Watson-Crick DNA-structure paper was less than 1 year old when Armitage and Doll wrote.) The nature of these events is nicely explained by Hanahan and Weinberg in a 2011 review.
