statisticalrethinkin..

13.4. CROSS-CLASSIFIED MODELS 349the bias-variance (underfitting-overfitting) tradeoff. When individual ponds are very large,pooling in this way does hardly anything to improve estimates, because the estimates don’thave far to go. But in that case, they also don’t do any harm, and the information pooledfrom them can substantially help prediction in smaller ponds.e partially pooled estimates provide better average prediction. And they do so becausethey adjust individual cluster (pond) estimates to negotiate the tradeoff between underfittingand overfitting.13.4. Cross-classified modelsWe can use more than one type of cluster in the same model. is is especially commonwhen units in the data possess MULTIPLE MEMBERSHIP in more than one kind of cluster.For example, the observations in data(chimpanzees) are lever pulls. Each pull iswithin a cluster of pulls belonging to an individual chimpanzee. But each pull is also withinan experimental block done at the same time. So each observed pull has membership in bothan actor (1 to 7) and a block (1 to 6).Re-estimate chimpanzee model, using varying effects and now being able to include theexperiment block:L ijk ∼ Binomial(1, p ijk )p ijklog = α + α Aj + α Bk + (β P + β PC C ijk )P ijk1 − p ijkα Aj ∼ Normal(0, σ A )α Bk ∼ Normal(0, σ B )α ∼ Normal(0, 1)β P ∼ Normal(0, 1)β PC ∼ Normal(0, 1)σ A ∼ Cauchy(0, 1)σ B ∼ Cauchy(0, 1)where L ijk is pulled.left for case i and actor j and block k, α Aj is the varying effect of actorj, α Bk is the varying effect of experiment block k, and σ αA and σ αB are the standard deviationsthat correspond to each varying effect type. e predictors P ijk and C ijk are prosoc.leftand condition. To load the data and estimate the model:library(rethinking)data(chimpanzees)d