ROUGH DRAFT authorea.com/98908

# The Rasch model

Consider a data set $$(X_{vi})$$ with responses from persons $$v$$ to items $$i$$

The Rasch model is defined by the probabilities $P(X_{vi}=x|\theta_v=\theta)=\frac{\exp(x(\theta-\beta_i))}{1+\exp(\theta-\beta_i)}$ for dichotomous responses, while the Rasch model for polytomous responses is defined by the probabilities $P(X_{vi}=h|\theta_v=\theta)=\frac{\exp(h\theta-\beta_{ih}))}{\sum_{k=0}^{m_1}\exp(k\theta-\beta_{ik})}$ with the restriction $$\beta_{i0}=0$$

Item parameters and their interpretation