Errata: Figure 13.1, shrinkage and Jensen's inequality

[danmyles]

There is some text on p. 405 (2nd edition) that describes a pattern in Figure 13.1, it reads: “notice in every case, the multilevel estimate is closer to the dashed line than the raw empirical estimate”.

This statement is incorrect. Looking at Figure 13.1 it is clear that the multilevel estimates are pooled towards a different location, closer to 0.9 than 0.8. The dashed line is at about 0.8. For example, note that the raw empirical estimate for tank 30 is closer to the dashed line.

I’ve spent ages trying to debug one of my own models before returning to this textbook and finding the same issue.

I think this has to do with Jensen’s Inequality: the mean of the scaled multi-level estimates is not the same as the scaled value of the population mean. In short, because the link function is non-linear (both convex and concave at various locations), we don’t see the typical pattern of shrinkage of tank averages towards the population average on the probability scale. This pattern would occur on the log-odds scale, but the logit link distorts things a bit?

I might have this wrong and I’m sure Richard would do a much better job of explaining what is going on here (maybe an overthinking box for the 3rd edition?). But I wanted to make a note in case anyone else found themselves losing their mind falling down the same rabbit hole.