Using data on Irish education transitions (http://lib.stat.cmu.edu/datasets/irish.ed) compare logit and probit regression using the augmented data method

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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

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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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