The Rasch Equation
For person $n$ with ability $\theta_n$ on item $i$ with difficulty $b_i$, the log-odds (logit) of success is:
\[\log\frac{P_{ni}}{1 - P_{ni}} = \theta_n - b_i\]Solving for the probability:
\[P_{ni} = \frac{e^{\theta_n - b_i}}{1 + e^{\theta_n - b_i}}\]When $\theta_n = b_i$, $P_{ni} = 0.5$. Both $\theta$ and $b$ are measured in logits on an unbounded scale, typically centred by setting $\bar{b} = 0$.
Parameter Separation
A key property of the Rasch model is that the raw score $r_n = \sum_i u_{ni}$ is a sufficient statistic for $\theta_n$, and the item total $s_i = \sum_n u_{ni}$ is sufficient for $b_i$. This allows conditional maximum likelihood (CML) estimation of item difficulties free of person parameters.
Fit Statistics
| Statistic | Formula | Expectation |
|---|---|---|
| Outfit MSQ | $\sum_n z_{ni}^2 / N$ weighted by extreme persons | $\approx 1.0$ |
| Infit MSQ | Variance-weighted residuals | $\approx 1.0$ |
Values between 0.7 and 1.3 are generally acceptable. Outfit is sensitive to unexpected responses far from the item difficulty; infit is more sensitive to unexpected responses near the item difficulty.