Open and Distance Learning



The Formula of the Rasch Model

The Rasch model is a probabilistic version of the scalogram. It was introduced in 1960 by the Danish statistician Georg Rasch. As starting point, we take the function $f\; in\; Equation5,\; which\; characterizes\; the\; scalogram.\; To\; refresh\; our\; memory,\; this\; function\; is\; repeated\; here:$

Ypi = f(Tp, Di)	= 1 if Tp > Di
	= 0 if Tp < Di

The scalogram is a deterministic model because, for a give T_p and D_i, the response (i.c., correct or incorrect) is fixed. The probabilistic version of the scalogram involves that, for a given T_p and D_i, a probability of a correct response is specified. In other words, the scalogram is characterized by a function f that has Y_pi (correct/incorrect) as a result, whereas the Rasch model is characterized by a function that specifies the probability of the event Y_pi=1. This probability is denoted by P(Y_pi=1).

The value of P(Y_pi=1) (a number between 0 and 1) is determined by T_p and D_i. Contrary to the scalogram, for which the size of the difference between T_p and D_i is of no importance (if it is positive, a correct response is given, and if it is negative, an incorrect response), the size of this difference is of importance for the Rasch model: The larger the difference between T_p and D_i, the larger the probability of a correct response. Thus, the probability of a correct response is large for persons that are much more able than the difficulty if the item, and the probability is small for persons for which the reverse holds. These probabilities are obtained by taking a particular function of the difference (T_p-D_i). This function is the following:

 

This is the formula of the Rasch model.

The numerator and the denominator of the right-hand side of Equation 3 involves a power with base e (e=2.718). The function in Equation 3 is called the logistic function and its course is shown in Figure 7. The function approaches 1 as (T_p-D_i) increases (the person is much more able than the difficulty of the item) and it approaches 0 if (T_p -D_i) decreases (the item is much more difficult than the ability of the person).

 
Figure 7: The course of the logistic function.

Because only two events can happen, a correct (Y_pi=1) or an incorrect (Y_pi=0) response, it holds that P(Y_pi=1)+P(Y_pi=0)=1, from which follows that

=

=

= (4)

Thus, the total amount of probability (by definition equal to 1) is distributed over two events: One part for the correct response (equal to the right-hand side of Equation 3), and another part for the incorrect response (equal to right-hand side of Equation 4). This is why we call it a probability distribution.