Rasch model estimation

The Rasch model for dichotomous data takes the form:

${\displaystyle \Pr\{X_{ni}=1\}={\frac {\exp({\beta _{n}}-{\delta _{i}})}{1+\exp({\beta _{n}}-{\delta _{i}})}},}$

where ${\displaystyle \beta _{n}}$ is the ability of person ${\displaystyle n}$ and ${\displaystyle \delta _{i}}$ is the difficulty of item ${\displaystyle i}$.

Let ${\displaystyle x_{ni}}$ denote the observed response for person n on item i. The probability of the observed data matrix, which is the product of the probabilities of the individual responses, is given by the likelihood function

${\displaystyle \Lambda ={\frac {\prod _{n}\prod _{i}\exp(x_{ni}(\beta _{n}-\delta _{i}))}{\prod _{n}\prod _{i}(1+\exp(\beta _{n}-\delta _{i}))}}.}$

The log-likelihood function is then

${\displaystyle \log \Lambda =\sum _{n}^{N}\beta _{n}r_{n}-\sum _{i}^{I}\delta _{i}s_{i}-\sum _{n}^{N}\sum _{i}^{I}\log(1+\exp(\beta _{n}-\delta _{i}))}$

where ${\displaystyle r_{n}=\sum _{i}^{I}x_{ni}}$ is the total raw score for person n, ${\displaystyle s_{i}=\sum _{n}^{N}x_{ni}}$ is the total raw score for item i, N is the total number of persons and I is the total number of items.

Solution equations are obtained by taking partial derivatives with respect to ${\displaystyle \delta _{i}}$ and ${\displaystyle \beta _{n}}$ and setting the result equal to 0. The JML solution equations are:

${\displaystyle s_{i}=\sum _{n}^{N}p_{ni},\quad i=1,\dots ,I}$

${\displaystyle r_{n}=\sum _{i}^{I}p_{ni},\quad n=1,\dots ,N}$

where ${\displaystyle p_{ni}=\exp(\beta _{n}-\delta _{i})/(1+\exp(\beta _{n}-\delta _{i}))}$.

The resulting estimates are biased, and no finite estimates exist for persons with score 0 (no correct responses) or with 100% correct responses (perfect score). The same holds for items with extreme scores, no estimates exists for these as well. This bias is due to a well known effect described by Kiefer & Wolfowitz (1956). It is of the order ${\displaystyle (I-1)/I}$, and a more accurate (less biased) estimate of each ${\displaystyle \delta _{i}}$ is obtained by multiplying the estimates by ${\displaystyle (I-1)/I}$.

The conditional likelihood function is defined as

${\displaystyle \Lambda =\prod _{n}\Pr\{(x_{ni})\mid r_{n}\}={\frac {\exp(\sum _{i}-s_{i}\delta _{i})}{\prod _{n}\gamma _{r}}}}$

in which

${\displaystyle \gamma _{r}=\sum _{(x)\mid r}\exp(-\sum _{i}x_{ni}\delta _{i})}$

is the elementary symmetric function of order r, which represents the sum over all combinations of r items. For example, in the case of three items,

${\displaystyle \gamma _{2}=\exp(-\delta _{1}-\delta _{2})+\exp(-\delta _{1}-\delta _{3})+\exp(-\delta _{2}-\delta _{3}).}$

Some kind of expectation-maximization algorithm is used in the estimation of the parameters of Rasch models. Algorithms for implementing Maximum Likelihood estimation commonly employ Newton-Raphson iterations to solve for solution equations obtained from setting the partial derivatives of the log-likelihood functions equal to 0. Convergence criteria are used to determine when the iterations cease. For example, the criterion might be that the mean item estimate changes by less than a certain value, such as 0.001, between one iteration and another for all items.

• Expectation-maximization algorithm
• Rasch model

• Linacre, J.M. (2004). Estimation methods for Rasch measures. Chapter 2 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.
• Linacre, J.M. (2004). Rasch model estimation: further topics. Chapter 24 in E.V. Smith & R. M. Smith (Eds.) Introduction to Rasch Measurement. Maple Grove MN: JAM Press.