# What is "cohort likelihood" and where is it from?

this is my very first post and I would consider myself a beginner in statistics, but I couldn't find anything about this in other asks.

I am learning about the Self-controlled case series (SCCS) model and in the derivation of its likelihood the author begins with a "cohort likelihood" formula (s. image).

I have spent a lot of time searching for "cohort likelihood" or this formula which vaguely resembles the probability function of a Poisson process, but I couldn't find anything that matches.

(Assumption 1 refers to the assumption in SCCS models that the event that we are looking at needs to be rare OR if one individual can have more than one event these events need to happen independently from one another.)

Can someone tell me where this likelihood comes from so I can make sense of what its role is in the later part of the derivation?

## 1 Answer

If you had a person with a single event in a non-homogenous Poisson process, the probability density that it was at $$t$$ would be proportional to $$\lambda_i(t)$$. By assumption 1 we can say different events on the same person are independent, either because multiple events don't happen or because they are independent. That's the first term. The exponential term is the probability of having zero events in the interval $$(a_i,b_i]$$, and it's not in the product.

I think the is called the cohort likelihood because it's the appropriate likelihood for an arbitrary member of the full cohort from which events are drawn, rather than the likelihood conditional on selection into the case sample.

The likelihood for the full cohort will be the product of terms like the first one for people who have $$n_i>0$$ events and like the exponential one for people with no events.