The Rasch Model

The Rasch model for item analysis says that that the probability that person $i$ gives a correct response to item $j$ is $$\pi_{ij}=\frac{\exp(\theta_i+\epsilon_j)}{1+\exp(\theta_i+\epsilon_j)}.$$ The likelihood is $$L=\prod_{i=1}^n\prod_{j=1}^m \pi_{ij}^{y_{ij}}(1-\pi_{ij})^{1-y_{ij}},$$ which means that the negative log-likelihood has the form $$\mathcal{L}=\sum_{i=1}^n\sum_{j=1}^m\log[1+\exp(\theta_i+\epsilon_j)]- \sum_{i=1}^n y_{i\star}\theta_i+\sum_{j=1}^m y_{\star j}\epsilon_j.$$ Now consider $g(x)=\log(1+e^x),$ which we can write as $g(x)=\log(1-\pi(x)).$ We find $$\begin{align*} g'(x)&=\frac{e^x}{1+e^x}=\pi(x),\\ g''(x)&=\frac{e^x}{(1+e^x)^2}=\pi(x)(1-\pi(x)), \end{align*}$$ thus $0\leq g''(x)\leq\frac{1}{4},$ which shows we can apply quadratic majorization in this case.

We leave the details of the majorization, which are just ``completing the square'' as usual, to the reader. The algorithm forms the matrix $H$ with elements $$h_{ij}=(\theta_i+\epsilon_j)-2(\pi_{ij}+y_{ij}),$$ and it updates the parameter estimates by computing row and column averages of this matrix.