statisticalrethinkin..

14.1. EVERYTHING CAN VARY AND PROBABLY SHOULD 357)applications=d$applications,male=d$male,dept=d$j) ,start=list(a=0,bm=0,a_dept=rep(0,6),bm_dept=rep(0,6),Sigma_dept=c(1,1),Rho_dept=diag(2))Now let’s look at the non-varying estimates:precis(m14.2)R code14.4Mean StdDev lower 0.95 upper 0.95a -0.51 0.79 -2.13 1.01bm -0.17 0.28 -0.71 0.39a_dept[1] 1.83 0.80 0.32 3.48a_dept[2] 1.27 0.81 -0.17 3.05a_dept[3] -0.14 0.79 -1.54 1.63a_dept[4] -0.11 0.79 -1.79 1.41a_dept[5] -0.63 0.79 -2.20 0.96a_dept[6] -2.09 0.80 -3.60 -0.42bm_dept[1] -0.64 0.37 -1.37 0.01bm_dept[2] -0.06 0.38 -0.84 0.73bm_dept[3] 0.25 0.29 -0.34 0.80bm_dept[4] 0.08 0.29 -0.49 0.69bm_dept[5] 0.29 0.31 -0.22 0.98bm_dept[6] 0.04 0.34 -0.69 0.67Sigma_dept[1] 1.76 0.71 0.81 3.17Sigma_dept[2] 0.55 0.36 0.07 1.16Rho_dept[1,1] 1.00 0.00 1.00 1.00Rho_dept[1,2] -0.28 0.37 -0.89 0.43Rho_dept[2,1] -0.28 0.37 -0.89 0.43Rho_dept[2,2] 1.00 0.00 1.00 1.00e first row of output is the estimate for α, the average log-odds of admission across alldepartments. Notice that is a highly imprecise estimate, which results from there being a lotof variation among departments in their admissions rates. e second row of output is theestimate of β, the average effect of being male on log-odds of admission. is is negative,but also highly imprecise. e last two rows of output are the estimates for σ α and σ β , thestandard deviations of the varying intercepts and slopes.What can you say from these estimates? Notice that the standard deviation of the interceptsis more than three times as large as the standard deviation of the slopes for male.A solid inference to draw from this difference is that more of the variation in log-odds ofadmission in the entire data arises from differences among departments in overall rates ofadmission than it does from differences in gender bias. at is, heterogeneity in overall admissionrates is more influential than heterogeneity in gender bias in admission.