22 Lecture 13

22.1 Binomial

Outcome is a count from zero to some known upper bound

Binomial events with N trials large/unknown and probability of event is small

Poisson with varied exposure/offset

Example of oceanic tool complexity

Modeled with a poisson link

The model outcomes are terrible - though they fit the data, the intercepts don’t pass through origin

Wouldn’t we expect zero population = zero tools?

Solution is a scientific model

The relationship can be thought of as a change in tools per unit time

Change in time = alpha P ^ beta

Alpha: innovation rate, P: population per person = each person has some change of inventing something

Beta: diminishing returns, saturation effect, “someone else will invent it for you”

TODO: read more about this, highlight it, etc

The resulting model using this function based in the scientific model is not perfect, but meanings are clearer, the intercept actually goes through 0

This is an ad hoc function, not a link

Estimate rates by modeling time-to-event

Can’t ignore censored cases

Example cats

Time to adoption for observed adoptions is simplest, an exponential function

For censored cats

Specialized complex distributions

eg. ordered categories, ranks

Blends of stochastic processes

eg. varying means, probabilities, rates

eg. zero-inflation, hurdles

Example monks

Number of manuscripts per day

Can we infer the number of days they get drunk?

Drunkenness is a hidden state

There is a probability that they drink or work, and within the work, a probability that they produce 0 or 1+ manuscripts