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.
assume \(Y \sim Ber\) and \(p_i = P(Y_i =1)\), the success probability.
consider 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 |