Continuous time, discrete event models

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Recently, I’ve been exposed to situations where I am trying to model discrete, binary events (i.e., 0 or 1 like heads-or-tails). My knee-jerk response has been: use a logistic regression or another model with a binomial outcome. The jack-of-all trades generalize linear model usually servers me well in these situations. However, my recent events have had continuous-time predictors. Although Cox proportional hazards model can be used if the event is something like survival, this did not seem appropriate for my situation because I had multiple events occurring per individual. Enter in continuous-time, discrete events.

A Poisson regression is similar to a binomial if the probability of an even occurring is small enough. Enter in a Poisson regression as a method for modeling animal behavior. I first saw this in a mathematical statistics paper describing models animal movements, but found another paper by some of the co-authors that was more accessible. From this, I learned I needed to use the following version of the Poisson regression:

y ~ Poisson(μ)

μ = τ exp( β x’).

I was able to program this in Stan, by adopting code I found online. This model can also be modified to treat individuals a random effect (and prevent pseudo-replication) if the data allows or requires it.

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