statisticalrethinkin..

234 7. INTERACTIONSbs -38.91 34.94sigma 57.34 46.23bws NA -39.5nobs 27 27Now consider these estimates and try to figure out what the models are telling us about theinfluence of water and shade on the blooms. First, consider the intercepts a, α. e estimateof the intercept changes a lot from one model to the next, from 53 to −84. What do thesevalues mean? Remember, the intercept is the expected value of the outcome when all of thepredictor variables take the value zero. In this case, neither of the predictor variables evertakes the value zero within the data. As a result, these intercept estimates are very hard tointerpret. is is a very common issue, and most of the regression examples so far in thisbook have suffered from it.Now, consider the slope parameters. In the main-effect-only model, m7.6, the MAPvalue for the main effect of water is positive and the main effect for shade is negative. Takea look at the standard deviations and intervals in precis(m7.6) to verify that both posteriordistributions are reliably on one side of zero. You might infer that these posterior distributionssuggest that water increases blooms while shade reduces them. For every additionallevel of soil moisture, blooms increase by 76, on average. For every addition unit of shade,blooms decrease by 42, on average. ose sound reasonable.But the analogous posterior distributions from the interaction model, m7.7, are quitedifferent. First, assure yourself that the interaction model is indeed a much better model:R code7.22compare(m7.6,m7.7)DIC pD dDIC weightm7.7 293.41 4.87 0.00 0.99m7.6 303.88 4.29 10.47 0.01is is a slam dunk for m7.7. So let’s seriously consider the posterior distribution from m7.7.Now both main effects are positive, but the new interaction posterior mean is negative. Areyou to conclude now that the main effect of shade is to help the tulips? And the negativeinteraction itself implies that as shade increases, water has a reduced impact on blooms. Butreduced by how much?Sampling from the posterior now and plotting the model’s predictions would help immenselywith interpretation. And that’s the course I want to encourage and provide codefor. In general, I don’t think it’s ever safe to interpret interactions without plotting them.But for the moment, let’s instead look at the value of centering the predictor variables andre-estimating the models.7.3.3. Center and re-estimate. To center a variable means to create a new variable that containsthe same information as the original, but has a new mean of zero. For example, tomake centered versions of shade and water, just subtract the mean of the original fromeach value:R code7.23d$shade.c