statisticalrethinkin..

13.2. MULTILEVEL TADPOLES 339and even non-additive stochastic effects, but also because they make predictions at differentlevels of the data. In many cases, we are interested in only one or a few of those levels, and asa consequence, model comparison using metrics like AIC and DIC becomes more difficult.e basic logic remains unchanged, but now we have to make more decisions about whichparameters in the model we wish to focus on.ere’s a lot to accomplish in this chapter. First, let’s explore the advantages of the multilevelapproach, in the context of a familiar data example from the previous chapter.13.2. Multilevel tadpolesIn the previous chapter, I used the tadpole predation (data(reedfrogs)) data to illustratethe value of explicitly modeling the heterogeneity among units in the data. In that case,the units were separate aquarium tanks with tadpoles in them. e outcome of interest wasthe number of survivors aer a fixed period. Even aer accounting for differing treatmenteffects among the tanks, there was still substantial variation in survival rate across tanks.As a result, the beta-binomial model that explicitly modeled that variation could produce abetter description of the data.at model, recall, was specified as:y i ∼ Binomial(p i , n i ),p i ∼ Beta(¯p, θ),where y i is the number of survivors in tank i, p i is the probability of each tadpole in tanki surviving, n i is the number of tadpoles in tank i at the beginning of the experiment, andfinally the parameters to be estimated are ¯p and θ. ese are the parameters of the betadistribution, which estimates the shape of the variation among tanks.But what is conspicuously missing from our work with this model are estimates of thesurvival probability, p i , in each tank. We’ve defined those parameters, but we never estimatedthem. We just estimate their distribution. Wouldn’t it be nice to have the individual tankestimates, in addition to the estimates defining the distribution of which they are members?If we ever hope to make decent predictions on a per-tank basis, knowing the p i ’s would be ahuge help.Of course, maybe in this case you don’t care much about predicting anything in eachtank, because it was an experiment, and so the tanks are regarded as ephemeral collections.But in many cases, we do care about the specific collections, because they are lasting entities.Suppose for example that the “tanks” were streams or ponds in the natural world.Now making accurate predictions is suddenly more important, and in addition identifyingstreams with unusually high mortality may lead to useful measurement that identifies additionalcausal variables. But even when we don’t care about the unique clusters in our sample,it helps to estimate the per-cluster parameters, because doing so can provide better estimatesof any treatment effects in the model.Call these p i parameters, all 48 of them, VARYING EFFECTS. 114 ey are parameters likeany other, when used to make predictions. But they are varying among clusters of observationsin the data. e clusters here are tanks, and the observations are individual survive/dieevents of tadpoles. Varying effects provide the needed per-cluster parameters to aidin inference about both unique clusters and predictor variables. ese parameters are alsocommonly called random effects. ey stand in contrast to the so-called fixed effects or constanteffects parameters of a single-level model, in which no assumption is made about thedistribution from which the parameters arise.