statisticalrethinkin..

342 13. MULTILEVEL MODELSprobability of survival in tank0.2 0.4 0.6 0.8 1.010 25 25 351 16 32 48tankFIGURE 13.1. Empirical proportions of survivors in each tadpole tank,shown by the filled blue points, plotted with the 48 per-tank estimates fromthe multilevel model, shown by the black circles. e dashed line locates theoverall average proportion of survivors across all tanks. e vertical lines dividetanks with different initial counts of tadpoles, 10 (le) and 25 (middle)and 35 (right). Each value on the horizontal axis is the index of a tank (row)in the data. In every tank, the estimate from the multilevel model is closerto the dashed line than the empirical proportion is. is reflects the poolingof information across tanks, to help with inference about each tank.13.2.1. Pooling. All three of these phenomena arise from a common cause: pooling informationacross clusters (tanks) to improve estimates. What POOLING means here is thateach tank provides information that can be used to improve the estimates for all of the othertanks. Each tank helps in this way, because we made an assumption about how the varyinglog-odds in each tank, α j , related to all of the others. We assumed a distribution, the normaldistribution in this case. Once we have a distributional assumption, we can use Bayes’theorem to optimally share information among the clusters.ink of it this way. Suppose you encounter the data for each tank of tadpoles in sequence.Before you see the outcome of the first tank, tank 1, you can have only a naivenotion of what the observed survival proportion will be. But once you see the outcome oftank 1, you’ve gained information and your guess has improved. Now the second tank comesalong. Before you realize the outcome of the second tank, can you say anything about its proportionthat will survive? Of course you can, because you have the outcome of the first tankto improve your prediction. You won’t be too confident in your guess, perhaps. But it’ll bebetter on average than your truly naive guess was for tank 1. And once the second tank’s dataare realized, you can improve your guess from generalizing the first tank’s outcome, using