7.2 Probit Regression

\[ E(Y_i) = p_i = \Phi(\mathbf{x_i'\theta}) \]

where \(\Phi()\) is the CDF of a N(0,1) random variable.

Other models (e..g, t–distribution; log-log; I complimentary log-log)

We let \(Y_i = 1\) success, \(Y_i =0\) no success. We assume \(Y \sim Ber\) and \(p_i = P(Y_i =1)\), the success probability. We cosnider a logistic regression with the response function \(logit(p_i) = x'_i \beta\)

Confusion matrix

Predicted
Truth 1 0
1 True Positive (TP) False Negative (FN)
0 False Positive (FP) True Negative (TN)

Sensitivity: ability to identify positive results

\[ \text{Sensitivity} = \frac{TP}{TP + FN} \]

Specificity: ability to identify negative results

\[ \text{Specificity} = \frac{TN}{TN + FP} \]

False positive rate: Type I error (1- specificity)

\[ \text{ False Positive Rate} = \frac{FP}{TN+ FP} \]

False Negative Rate: Type II error (1-sensitivity)

\[ \text{False Negative Rate} = \frac{FN}{TP + FN} \]

Predicted
Truth 1 0
1 Sensitivity False Negative Rate
0 False Positive Rate Specificity