I.3.4.1: Rasch Model

In the item analysis model proposed by Rasch, we observe a binary $n\times m$ matrix $Y=\{y_{ij}\}$. The (unconditional) log-likelihood is $$\mathcal{L}(x,z)=\sum_{i=1}^n\sum_{j=1}^m y_{ij}\log\pi_{ij}(x,z)+(1-y_{ij})\log(1-\pi_{ij}(x,z)),$$ with $$\pi_{ij}(x,z)=\frac{\exp(x_i+z_j)}{1+\exp(x_i+z_j)}.$$

The negative log-likelihood can be written as $$f(x,z)=\sum_{i=1}^n\sum_{j=1}^m\log\{1+\exp(x_i+z_j)\} -\sum_{i=1}^n y_{i\star}x_i-\sum_{j=1}^m y_{\star j}z_j,$$ where $\star$ indicates summation over an index. The stationary equations have the elegant form $$\begin{align*} \pi_{i\star}(x,z)&=y_{i\star}\\ \pi_{\star j}(x,z)&=y_{\star j}. \end{align*}$$ The standard algorithm for the unconditional maximum likelihood problem [Wainer, Morgan, Gustafsson [1980] cycles through these two blocks of equations, using Newton's method at each substep. In this case Newton's method turns out to be $$x_i^{(k+1)}=x_i^{(k)}-\frac{\pi_{i\star}(x^{(k)},z^{(k)})-y_{i\star}} {\sum_{j=1}^m{\pi_{ij}(x^{(k)},z^{(k)})(1-\pi_{ij}(x^{(k)},z^{(k)}))}},$$ and similarly for $z_j^{(k+1)}.$