RASCH AND OTHER MODELS

**Question:**

*How does RUMM2030 compare with the Two- and
Three-Parameter IRT Models?*

**Explanation:**

The language of the two- and three- parameter
models for dichotomous responses in IRT is mixed up with the general idea of
parameters in models. That is, in IRT, when one says *two *parameter,
there is an immediate association with the Birnbaum model which has a
discrimination parameter and a location parameter for each item. However, one
can have more than one parameter in other models.

In a Rasch analysis using the Rasch model for dichotomous items, that is,
*two* ordered categories, there is only the one parameter for an item, its
location. In the psychometric literature concerned with psychophysical scaling,
this entity is called a threshold. It is the point at which each of the
response has a 50% chance of occurring.

In the model for three ordered categories, it is
possible to estimate *two* independent parameters. These are the
thresholds dividing the continuum into three categories. They can EITHER be
parameterized as two separate values, normalized to all the item-threshold
parameters (that is sum to zero across all item-threshold combinations) OR they
can be parameterized as the average location of these thresholds, and then have
each value deviate from this average and sum to zero. In this way, one
parameter summarises the location of the item, and a second summarises the
distance between the thresholds (that is, the spread).

With four categories entered into the model, it is possible to get the average of the thresholds, then the spread of the thresholds, and then their skewness, and so on.

RUMM2030 also makes this re-parameterization of the thresholds. Instead of modeling the thresholds directly (e.g., for a five category item: threshold1, threshold2, threshold3, threshold4) RUMM2030 estimates moments of the distribution of these thresholds. So, in this case, you have the mean of the thresholds, their variance (spread), and the higher moments skewness and kurtosis. Thus, instead of four thresholds, you estimate four moments, and then recover the thresholds from these parameters. This process is done for convenience of conditional estimation.

However, in the case of five ordered categories, it is possible to constrain the re-parameterization of the thresholds so that only the location and spread are involved. In that case, all the thresholds within an item will be equally spaced. Thus, we now have only two parameters for the item, when in fact it is possible to have more.

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**Question:**

*Does RUMM2030 provide analyses for the ‘Rating’
and ‘Partial Credit’ Models?*

**Explanation:**

When conducting a Rasch analysis with the Rasch Model, all items should be considered as polytomous. Items with 2 categories [usually called dichotomous] are treated simply as a special case of items with 3 or more categories. Merely specify the maximum score possible for the item and RUMM2030 does the rest.

RUMM2030 nomenclature tries to avoid terms like ‘the partial credit model’ because there is only ONE unidimensional Rasch model for ordered categories. The situation with dichotomously scored items is just a special case. It is true that by calling:

- the case with all thresholds the same across the items the ‘rating model’ and
- the case with different thresholds across items the ‘partial credit model’

it has given the impression that these are different models.

Both the ‘rating’ and ‘partial credit’ situations are different in the parameterization and the number of parameters only and NOT in the structure of the response at the level of the response of a person to an item.