# Little letters are also important

In the PISA 2012 Technical Report on page 312 (chapter 16), the formula (16.3) for the Partial Credit Model is printed for computing scale parameters. Looking at it twice, one can see that something must be wrong with the subscribs used for the sumations.

$P_{x_i}(\theta_n) = \frac{\displaystyle\exp\sum_{k=0}^x(\theta_n-\delta_i+\tau_{ij})}{\displaystyle\sum_{h=0}^{m_i}\exp\displaystyle\sum_{k=0}^h(\theta_n-\delta_i+\tau_{ik})}$

By consulting the chapter “Polytomous Rasch Models and their Estimation” in the book “Rasch Models” (Fischer, 1995), and combining the formulas (15.3) with (15.8) there, we get

$P(X_{vi}=h) = \frac{\displaystyle\exp(\phi_h\theta_v + \sum_{l=0}^{h}\alpha_{il})}{\displaystyle\sum_{l=0}^m \exp(\phi_h\theta_v + \sum_{j=0}^{l}\alpha_{ij})}$

Now by comparing the meaning of all the symbols and applying it on the formula from PISA Technical Report – by respecting as much as possible of the notations used by them – we get

$P_{x_i}(\theta_n) = \frac{\displaystyle\exp\sum_{k=0}^{x_i}(\theta_n-\delta_i+\tau_{ik})}{\displaystyle\sum_{h=0}^{m_i}\exp\displaystyle\sum_{k=0}^h(\theta_n-\delta_i+\tau_{ik})}$

We see that we have to change $\tau_{ij}$ in the numerator of the very first formula above to $\tau_{ik}$, and $x$ to $x_i$.

That makes sense!