Rasch Model Estimation - Joint Maximum Likelihood

Joint Maximum Likelihood

Let 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

$\Lambda = \frac{\prod_{n} \prod_{i} \exp(x_{ni}(\beta_n-\delta_i))}{\prod_{n} \prod_{i}(1+\exp(\beta_n-\delta_i))}.$

The log-likelihood function is then

$\log \Lambda = \sum_n^N \beta_n r_n - \sum_i^I \delta_i s_i - \sum_n^N \sum_i^I \log(1+\exp(\beta_n-\delta_i))$

where is the total raw score for person n, 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 and and setting the result equal to 0. The JML solution equations are:

$s_i = \sum_n^N p_{ni},\quad i=1,\dots,I$

$r_n = \sum_i^I p_{ni},\quad n=1,\dots,N$

where . A more accurate estimate of each is obtained by multiplying the estimates by .

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