statisticalrethinkin..

7.3. CONTINUOUS INTERACTIONS 237bws -51.87 12.95 -77.25 -26.49sigma 45.22 6.15 33.17 57.28And here are justifiable readings of each:• e estimate a, α, is the expected value of blooms when both water and shade areat their average values. eir average values are both zero (0), because they werecentered before fitting the model.• e estimate bw, β w , is the expected change in blooms when water increases byone unit and shade is at its average value (of zero). is parameter does not tell youthe expected rate of change for any other value of shade. is estimate suggeststhat when shade is at its average value, increasing water is highly beneficial toblooms.• e estimate bs, β s , is the expected change in blooms when shade increases by oneunit and water is at its average value (of zero). is parameter does not tell youthe expected rate of change for any other value of water. is estimate suggeststhat when water is at its average value, increasing shade is highly detrimental toblooms.• e estimate bws, β ws , is the interaction effect. Like all linear interactions, it can beexplained in more than one way. First, the estimate tells us the expected change inthe influence of water on blooms when increasing shade by one unit. Second, ittells us the expected change in the influence of shade on blooms when increasingwater by one unit.So why is the interaction estimate, bws, negative? e short answer is that water and shadehave opposite effects on blooms, but that each also makes the other more important to theoutcome. If you don’t see how to read that from the number −53, you are in good company.And that’s why the best thing to do is to plot implied predictions.7.3.4. Plotting implied predictions. Golems (models) have awesome powers of reason, butterrible people skills. e golem provides a posterior density, but for us humans to understandits implications, we need to decode the posterior into something else. Centered predictorsor not, plotting posterior predictions always tells you what the golem is thinking, onthe scale of the outcome. at’s why we’ve emphasized plotting so much. But in previouschapters, there were no interactions. As a result, when plotting model predictions as a functionof any one predictor, you could hold the other predictors constant at any value you liked.So the choice of which values to set the un-viewed predictor variables to wasn’t sensitive.Now that’ll be different. Once there are interactions in a model, the effect of changinga predictor depends upon the values of the other predictors. Maybe the simplest way to goabout plotting such interdependency is to make a frame of multiple bivariate plots. In eachplot, you choose different values for the un-viewed variables. en by comparing the plotsto one another, you can see how big of a difference the changes make.Here’s how you might accomplish this visualization, for the tulip data. I’m going to makethree plots in a single panel. Such a panel of three plots that are meant to be viewed togetheris a triptych, and triptych plots are very handy for understanding the impact of interactions.First, just draw samples from the quadratic posterior as usual:post