statisticalrethinkin..

11.1. BINOMIAL 279model this way:y i ∼ Binomial(n, p i )we define the log-odds instead:Equivalently, you can write:p i = α + βx iy i ∼ Binomial(n, p i )p ilog = α + βx i1 − p ilogit(p i ) = α + βx iSo it’s still true that we are using the GLM approach to model a unique probability foreach case i. But the additive model defines the log-odds for case i, not the probability. If wemade p i itself a linear model, then it would be easy to find parameter values that would makep i less than zero or greater than one. Since a probability should be between zero and one,that approach could cause problems. But the log-odds extend symmetrically around zero toboth −∞ and +∞, so making the log-odds into a linear model is a lot safer.Okay, so what does this imply about the probability itself, p i ? We have to care about theanswer to this question, because in order to fit the model to the data, we have to calculatelikelihoods. So we want to solve for p i . Suppose for example that the additive model of thelog-odds for case i is just called ϕ i . In math speak, this implies:p ilog = ϕ i . (11.1)1 − p iNow we solve for the probability itself. Do this by taking (11.1) and solving for p i . Aer alittle algebra, you get:p i = exp(ϕ i)1 + exp(ϕ i ) .You might recognize this probability as the LOGISTIC, with the log-odds sometimes calledthe LOGIT. 105 e logistic function pops up all over science. I first learned it in biology, asthe result of natural selection on the frequency of a favorable allele. But it has emerged herejust as a consequence of the desire to be able to make our model about log-odds.What we’ve done so far is establish a LOGIT LINK between the additive model and theprobability of an event. Now you need to see how to code this link.11.1.3. Example: Prosocial chimpanzees. e data for this example come from an experimentaimed at evaluating the prosocial tendencies of chimpanzees. 106 Each chimpanzee wasplaced in a room with two levers, one on the le and one on the right. Across from the focalanimal was a transparent divider, on the other side of which another chimpanzee would sit,in one of the treatment conditions. On each side of the divider, each lever was attached to aplate. When the focal animal pulled the le lever, the plates on the le side would move tobe close enough to each animal for him or her to retrieve anything on it. As a result, if thefocal animal pulled the le lever, whatever rested on the le plates became available to eachanimal. If he or she instead pulled the right lever, whatever was on the right plates becameavailable.ere were two treatment variables to be concerned with for now. First, one of the twosets of plates, le or right, always had a “prosocial” option that had food on both plates, while