IRT to Rasch conversion

The 3-PL b values are roughly equivalent to the Rasch difficulties. The b difficulties are probably in probits, which could be converted to logits by multiplying them by 1.702. However, empirically, the conversion factor differs.

 

Do you have any of the 3-PL data? If so, use the 3-PL parameters to estimate the 3-PL thetas for those data. Then analyze the same data with Rasch and obtain the Rasch person measures (Rasch thetas). Cross-plot the two sets of thetas. The slope of the best-fit line gives the conversion between the empirical 3-PL scaling and the Rasch logit scaling.

 

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3-PL IRT or the Rasch Model?

 

Guessing and item discrimination alter the data. The Rasch measures are based on the data, and so guessing and discrimination alter the Rasch measures.

 

We do fit analysis to discover the extent to which aberrations in the data are altering the Rasch measures. In particular, Outfit is sensitive to guessing and infit is sensitive to item discrimination. Lucky guesses increase raw scores and so increase Rasch measures. High item discrimination increases logit differences and so increases test reliability. In the extreme, this is called the "attenuation paradox".

 

So, why do we use Rasch measures, instead of the 3-PL IRT model which parameterizes guessing and item discrimination?

 

If anyone actually applied the 3-PL IRT model rigorously, then every different person response string would produce a different person estimate ("pattern scoring"). This is impractical because it is too difficult to explain to non-specialists that the same raw score on the same items produces different thetas (ability estimates) for persons with different response strings. Accordingly even 3-PL advocates use the Rasch principle that the same raw score on the same items always produces the same person estimates!

 

Parameterizing item discriminations also has practical drawbacks. When item discriminations differ, then the perceived item difficulty hierarchy is different for persons of different abilities (because the item ICCs/IRFs cross). Again this is too difficult to explain to non-specialists. How can Addition be easier for low performers than Subtraction, but Addition can be more difficult than Subtraction for high performers? Accordingly a compromise item difficulty hierarchy (beta or delta) is presented as though it is the same for everyone. Again this is the Rasch principle that the item difficulties are the same for everyone!

 

So, though it is useful to know about guessing and item discrimination, in practice they are ignored when results are presented to non-specialists!

 

For a more thorough discussion of Rasch vs 3-PL, see https://www.rasch.org/rmt/rmt61a.htm

 

How about this as a summary?

"The Rasch model computes item difficulties and person abilities on a strictly linear scale based on item and person raw scores without requirements such as complete data and normal distributions. The effect is to limit the impact of guessing and item discrimination. This makes the Rasch model more convenient in terms of utility, inference and explanation to non-technical audiences compared to other IRT models."