statisticalrethinkin..

14 Multilevel Models II: Slopes14.1. Everything can vary and probably shouldReasons to use varying slopes.14.1.1. Varying slopes and correlated effects. Another reason for the growing popularityof multilevel models is to improve inference about predictor variables. Even if you don’thypothesize that the effect of a variable is different across clusters, accounting for variationin the overall responses among clusters can help you get better estimates of even the nonvaryingeffects.e easiest way to appreciate this fact is to recall the UCBadmit data from Chapter 11.In those data, failing to model the varying means across departments lead to exactly theopposite inference of the truth. I addressed the problem back then by using naive no-poolingestimates derived from adding a dummy variable for each department. But since the varyingeffect estimates are on average superior to such no-pooling estimates, we could have doneeven better by using a multilevel model.In fact, let’s quickly return to those data and do just that. is will also allow me to showyou some more aspects of the notation and of these models. Unlike in the tadpole data,now our clusters (departments) will span the rows in the data frame. So we’re going to needanother index, j, to indicate clusters, while we keep i to indicate rows within them. Let’s loadthe data and add these i and j labels, to make this point clear.data(UCBadmit)d