statisticalrethinkin..

236 7. INTERACTIONSe primary reason is that, when the predictors are centered, the MAP value for α isthe just the empirical mean for the outcome, mean(d$blooms). at is also the start valuegiven to map. With un-centered predictors, the MAP value for α lies a long way from theempirical mean. Hence, the long search that failed.7.3.3.2. Estimates changed less across models. Why did centering the predictor variablesresult in the main effect posterior means remaining the same across the models with andwithout the interaction? In the un-centered models, the interaction effect is applied to everycase, and so none of the parameters in µ makes sense alone. is is because neither ofthe predictors in those models, shade and water, are ever zero. As a result the interactionparameter always factors into generating a prediction. Consider for example a tulip at theaverage moisture and shade levels, 2 in each case. e expected blooms for such a tulip is:µ i | si =2,w i =2 = α + β w (2) + β s (2) + β ws (2 × 2)So to figure out the effect of increasing water by 1 unit, you have to use all of the β parameters.Plugging in the MAP values for the un-centered interaction model, m7.7, we get:µ i | si =2,w i =2 = −150.8 + 181.5(2) + 64.1(2) − 52.9 × 2 × 2You can compute the prediction in R:R code7.25k