Latent variable model with threshold:
Suppose that there is underlying normal response \(Y^*\) and we observe (WLOG, \(\tau=0\)) \[ y_i=\begin{cases}0, & \mbox{ if }y_i^*\le \tau \mbox{ (threshold) }\\ 1, & \mbox{ if }y_i^* > \tau.\end{cases} \]
Suppose \(y_i^*=\alpha+\beta x_i+\epsilon_i\), where \(\epsilon_i\) iid \(N(0,1)\). Then,
\[\begin{eqnarray*} P(Y_i=1) &=& P(Y_i^*>0) \mbox{ }\mbox{ }\mbox{ (여기서 } P(Y_i^*>0)\mbox{이 아닌 } P(Y_i^*>\tau)\mbox{더라도 } \alpha^*=\alpha-\tau \mbox{로서 상쇄된다.}) \\ &=& P(\epsilon_i>-(\alpha+\beta x_i))\\ &=& 1-\Phi(-(\alpha+\beta x_i))\\ &=& \Phi(\alpha+\beta x_i), \mbox{ where } \beta= \mbox{change in } E(Y^*) \mbox{ for 1-unit increase in } x_i. \end{eqnarray*}\]
Normal tolerance distribution(Bliss, 1935):
It has been used frequently in toxicology.
\((x_i,y_i)=(\)dosage, response\()\) for subject \(i\), where \(y_i=1\) if \(i\)-th subject death, 0 otherwise.
Suppose tolerance \(T\sim N(\mu,\sigma^2)\). Then, \(Y_i=1 \iff T_i \le x_i\).
Note that
\[\begin{eqnarray*} \pi_i&=&P(Y_i=1)\\ &=&P(T_i\le x_i)\\ &=&P\left(\frac{T_i-\mu}{\sigma}\le \frac{x_i-\mu}{\sigma}\right)\\ &=&\Phi\left(\frac{x_i-\mu}{\sigma}\right)=\Phi(\alpha+\beta x_i) \end{eqnarray*}\]
where \(\alpha=-\frac{\mu}{\sigma}\), \(\beta=\frac{1}{\sigma}\).
Recall likelihood equations for binary GLM are \[ \sum_{i=1}^n \frac{n_i(y_i-\pi_i)x_{ij} f(\eta_i)}{\pi_i(1-\pi_i)}=0, \mbox{ }\mbox{ }j=0,1,\ldots,p, \] where \(\pi_i=F(\sum_j \beta_jx_{ij})\) and \(f(\eta_i)=F'(\eta_i)\) \((n_iY_i\sim \text{Bin}(n_i,\pi_i)\).
For probit model, the likelihood equation is \[ \sum_{i=1}^n \frac{n_i(y_i-\pi_i)x_{ij} \phi(\sum_j\beta_jx_{ij})}{\pi_i(1-\pi_i)}=0, \mbox{ }\mbox{ }j=0,1,\ldots,p, \] where \(\phi(u)=\frac{1}{\sqrt{2\pi}}e^{-u^2/2}.\)