Using data on Irish education transitions (http://lib.stat.cmu.edu/datasets/irish.ed) compare logit and probit regression using the augmented data method and the dbern.aux function. Take \(P=2\) predictors, namely sex and DVRT (Drumcondra Verbal Reasoning Test), with response \(y=0\) if leaving certificate not taken, and \(y=1\) if taken. Specifically obtain the number of subjects with probabilities \(\operatorname{Pr}\left(z_{\text {rep }, i})\) either under 0.1 or over 0.9 . Also obtain the number of cases with elevated residuals \(\varepsilon_{i}=z_{i}-X_{i} \beta\), namely with \(\operatorname{Pr}\left(\mid \varepsilon_{i} \| y\right)>1.96\) under a probit regression, and \(\operatorname{Pr}\left(\left|\varepsilon_{i}\right| / \kappa^{0.5}\right)>2.02\) under a logistic regression. The basic code (not including these checks) for a probit regression using the dbern.aux function is

Also using the dbern.aux function assess the best subset of predictors (for probit regression only) via the jump.lin.pred function. In this case the basic code omits specifying the eta[] regression term, or a prior on the regression coefficients, though a prior precision on the coefficients (tau.beta) is included as a known input. Thus one has

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model for (i in 1:n) (y[i] dbern. aux (z [i]); z[i] ~dnorm (eta [i], 1); eta [i]