Rasch Model - The Mathematical Form of The Rasch Model For Dichotomous Data

The Mathematical Form of The Rasch Model For Dichotomous Data

Let be a dichotomous random variable where, for example, denotes a correct response and an incorrect response to a given assessment item. In the Rasch model for dichotomous data, the probability of the outcome is given by:

\Pr \{X_{ni}=1\} =\frac{e^{{\beta_n} - {\delta_i}}}{1 + e^{{\beta_n} - {\delta_i}}},

where is the ability of person and is the difficulty of item . Thus, in the case of a dichotomous attainment item, is the probability of success upon interaction between the relevant person and assessment item. It is readily shown that the log odds, or logit, of correct response by a person to an item, based on the model, is equal to . It can be shown that the log odds of a correct response by a person to one item, conditional on a correct response to one of two items, is equal to the difference between the item locations. For example,

\operatorname{log-odds} \{X_{n1}=1 \mid \ r_n=1\} = \delta_2-\delta_1,\,

where is the total score of person n over the two items, which implies a correct response to one or other of the items (Andersen, 1977; Rasch, 1960; Andrich, 2010). Hence, the conditional log odds does not involve the person parameter, which can therefore be eliminated by conditioning on the total score . That is, by partitioning the responses according to raw scores and calculating the log odds of a correct response, an estimate is obtained without involvement of . More generally, a number of item parameters can be estimated iteratively through application of a process such as Conditional Maximum Likelihood estimation (see Rasch model estimation). While more involved, the same fundamental principle applies in such estimations.

The ICC of the Rasch model for dichotomous data is shown in Figure 4. The grey line maps a person with a location of approximately 0.2 on the latent continuum, to the probability of the discrete outcome for items with different locations on the latent continuum. The location of an item is, by definition, that location at which the probability that is equal to 0.5. In figure 4, the black circles represent the actual or observed proportions of persons within Class Intervals for which the outcome was observed. For example, in the case of an assessment item used in the context of educational psychology, these could represent the proportions of persons who answered the item correctly. Persons are ordered by the estimates of their locations on the latent continuum and classified into Class Intervals on this basis in order to graphically inspect the accordance of observations with the model. There is a close conformity of the data with the model. In addition to graphical inspection of data, a range of statistical tests of fit are used to evaluate whether departures of observations from the model can be attributed to random effects alone, as required, or whether there are systematic departures from the model.

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