statisticalrethinkin..

12 Monsters and Mixturesis chapter is about building more complex statistical models, by piecing together thesimpler tools of previous chapters. e tension is always between the power we derive fromthe models and the dangers we face when we misunderstand or misapply them. Inferencebecomes simultaneously harder, riskier, and potentially more powerful.Monsters are models that are pieced together from simpler, oen exponential family,pieces to form specialized outcome distributions. Ordered categories and ranks are verycommon kinds of data that don’t really suit any of the previous distributions. ese kinds ofdata need their own, somewhat monstrous, probability densities.Mixtures are models that comprise multiple stochastic processes, usually embedded withinone another. Measurement error and other stochastic processes can be blended with the outcomedistributions you’ve already seen to produce models that can estimate and cope withthe outcomes that depend upon more than one stochastic process.12.1. Ordered Categorical OutcomesDescribe why we need a density for ordered categorical outcomes. Show examples. Verycommon in social sciences. In natural sciences, appears in ecological field data, for example.12.1.1. An ordered probability density. e probability distribution we seek here will beborn of stating how we’d like to be able to interpret it. e main criterion is that we’d like tooperate with LOG-ODDS. Recall from the last chapter, when you met binomial GLM’s, thatthe log-odds of any event E is just:logPr(E)1 − Pr(E) .is is simply the logarithm of the odds. Back there (page 277), I explained the advantages ofmodeling the log-odds as an additive combination of parameters and variables. All of thosereasons matter here, in the same ways.So we want to work with log-odds, again. But there’s also another desirable property oflog-odds, which will help us now that we have, in effect, a multinomial distribution with morethan two mutually exclusive outcomes: We should focus on the CUMULATIVE LOG-ODDS ofeach possible value. Let me explain why. e probability density we are aer has a numberof discrete, ordered observable values. Our goal is to get a formula—a probability density—that tells us the likelihood of each value. is function will be like the dnorm and dbinomand dpois of earlier GLM’s. It will help us in getting to this goal if we start with cumulativeprobability, though, instead of the discrete probability of each observable value. Cumulativeprobability is just the probability of any value or smaller value. Why go cumulative at the start303