B Aggregation of Convex-Combination Influence Models

Frequent visitors to the website, Rotten Tomatoes, may be surprised to discover that a movie with a 15% positive rating is not necessarily better reviewed than than one which gets, say, 45% positive. This is because dichotomizing reviews as either just good or bad, period, discards lots of information. A strong consensus that a film is not great (say, mostly C-) will result in a low percentage of positives. On the other hand, a movie with a lot of terrible reviews but a strong lukewarm contingent (say, 55% Fs and Ds, 45% C+) can have a much higher “positive” percentage. While the subject-matter is much less important, this example has exactly the structure of voters located on a one-dimensional latent political space being forced to chose between two candidates. More generally, this points to the importance of uncertainty assessments. Even if one were to accept all of Groseclose’s modeling assumptions, the reader of Left Turn has no idea of what the views of the hypothetical unbiased American voter are. The reader knows Groseclose’s point estimate (≈ 25, as we said), but this estimate, like every other, is uncertain and surrounded by some margin of error. Since it is a function of earlier estimates, themselves imprecise7, the obvious and correct thing to do is to propagate the uncertainty. But Left Turn gives no sense of how much uncertainty attaches to its claims, even of a purely statistical sort. Equally seriously, Nyhan [2005] and Gasper [2011] have shown that estimates from Groseclose’s first, ideal-point model are highly sensitive to details of which data are included in the analysis, adding systematic to statistical uncertainty. We do not know how big the margin of error for Groseclose’s estimate of the location of Unbiased America should be, but it surely matters if a good confidence interval is [22, 28] or [10, 90], or indeed [0, 100].