## 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