statisticalrethinkin..

314 12. MONSTERS AND MIXTURESResults in FIGURE 12.1. In each plot, the blue lines indicate the boundaries betweenresponse values, numbered 1 through 7, bottom to top. e thickness of the blue lines correspondsto the variation in predictions due to variation in samples from the posterior. Sincethere is so much data in this example, the path of the predicted boundaries is quite certain.e horizontal axis represents values of intention, either zero or one. e change inheight of each boundary going from le to right in each plot indicates the predicted impactof changing a story from non-intention to intention. Finally, each plot sets the other twopredictor variables, action and contact, to either zero or one. In the upper-le, both areset to zero. is plot shows the predicted effect of taking a story with no-action, no-contact,and no-intention and adding intention to it. In the upper-right, action is now set to one.is plot shows the predicted impact of taking a story with action and no-intention (actionand contact never go together in this experiment, recall) and adding intention. is upperrightplot demonstrates the interaction between action and intention. Finally, in thelower-le, contact is set to one. is plot shows the predicted impact of taking a story withcontact and no-intention and adding intention to it. is plot shows the large interactioneffect between contact and intention, the largest estimated effect in the model.12.1.3.3. What about repeat measures? You might have realized at the start of this examplethat each participant responded to multiple (more than 30) different scenarios (“case”in the data frame). In other words, we have repeat measures on each participant. Not onlythat, but each participant responded to different versions of the same story (“story” in thedata frame). So not only are there repeat measures by participant, but there are also repeatmeasures by story.ese data are crying out for a multilevel analysis, using random effects. Many readers,whether they’ve learned about random effects before or not, may have heard of a concerncalled pseudo-replication, which is related. It stands to reason that all of the responses froma single participant are likely correlated with one another, if for no other reason than somepeople like to give high responses and others give low responses. What we’re really interestedin is how the factors action, intention and contact change a person’s response, accountingfor these clustering effects. We might also be interested in the strength and patterning ofthe clustering itself, as well. Whether you’re interested in the clustering itself or not, failureto model it can result in imprecise or misleading estimates.ere are some good ways to deal with this kind of issue. e best approach is to usean actual multilevel (also known as hierarchical) model. You’ll have to wait a until the nextchapter for that solution. Another approach, quite common still in fields like economics, isto just add a categorical variable to the model, labeling each cluster of possibly-correlatedoutcomes with a different dummy variable. is is known as the fixed effects approach, forreasons you can worry about when you get to Chapter 13. In this case, the fixed effectsapproach is pretty expensive, computationally. ere are 331 different participants in thedata, so that implies 330 new independent intercept parameters! ese parameters alloweach participant to have their own unique average response. en the slope parameters willindicate changes within participants, rather than changes relative to an average statisticalparticipant.Asking you to fit such a model, using map, would be an act of torture. You’d have toconstruct 330 new dummy variables and then type in an absurdly long formula for phi, withan equally absurd start list. So before we leave ordered categorical outcomes behind, the