statisticalrethinkin..

340 13. MULTILEVEL MODELSSo how do we modify the tadpole beta-binomial model to estimate the p i parameters?I’m going to take a step to the side here to drop the beta distribution. Instead we’ll move forwardusing a normal distribution now for the log-odds of survival in each tank, rather than abeta distribution of the probability (p i ) in each tank. I do this, because this normal distributionassumption is the most common general approach to solving this problem, and it’s alsothe easiest to estimate. In the next chapter, I’ll show you how to bring the beta distributionback, if you want to. But you can get all of the conceptual lessons, either way.What we want to to do is define the log-odds of survival in each tank j as α + α j . eparameter α, with no j, is what it has usually been: the average log-odds in the entire sample.But each α j , with an j, is the deviation from this global average within each tank j. esewill be the varying intercepts for each tank. en we state that each varying intercept comesfrom a normal distribution with mean zero and standard deviation σ. In sum, the multileveltadpole survival model is:s ij ∼ Binomial(n i , p ij )p ijlog = α + α j1 − p ijα j ∼ Normal(0, σ)α ∼ Normal(0, 1)σ ∼ Cauchy(0, 1)You can fit this model, using techniques I’ll summarize later in this chapter, with:R code13.1library(rethinking)data(reedfrogs)d