# Are discrimination parameters in two-parameter IRT models only specific to items?

Discrimination parameters in two-parameter model from IRT are usually considered item parameters.

But I've come to doubt it. Think about psychophysics; for example, detecting luminance. I don't think anyone would say that discrimination (i.e., slope) parameter applies to luminance itself. It comes from human perception, so I figure it's a person parameter and it could be different for each person.

What about test situation? Is it without any doubt that the discrimination parameter is considered as an item parameter? I doubt it is, especially given the fact that there is room for differential item functioning (DIF).

So I'm concerned with the fact there's any theoretical basis for saying that discrimination parameters are item-parameters.

I think I could model person-specific discrimination parameter model, in which item-discrimination parameter is determined by person-specific discrimination parameters in the group. That would justify DIF.

So, my question would sum up like this:

1. Is that if there's any theoretical/empirical basis for saying that discrimination parameters are item-parameters? How can one be sure of it?

2. Is there any related reference for person-discrimination parameter model mentioned above?

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@KH Kim, I believe there is some difficulty when any mathematical model meets the real world. In the IRT model, items and item parameters are invariant to the pool of individuals who answer those items - that's the theoretical building block of the model. This is quite a deep conversation in which you are immersing yourself, but I would suggest it is a building block issue - do you accept the mathematical model or not...

The issue might be sample size. A large response group is usually required for stable item estimates with 2-parameter models (300-500). It is also assumed that the sample used to achieve stable estimates reflect the population and are not a subset. If you have discrimination parameters that appear to be different for your sample, then the problem might not be with the model, but applying the exam/survey to an inappropriate group.

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The discrimination parameter is an item parameter because of how it is specified in the model

$resp_{ip} \sim \alpha_i ( \theta_p - \beta_i)$

where i is for item and p is for person. In this model, the person (represented only by $\theta$) has nothing to do with the discrimination parameter.

IRT models the interaction between persons and items, and the item parameters attempt to describe their influence on that interaction.

IRT models can be extended to include the types of questions you are asking. For example if you had a randomized treatment and a control group, you could specify something like

$resp_{ip} \sim (\alpha_1 + \alpha_2 I_{p \in control} )\times ( \theta_p - \beta_i)$

where $\alpha_2$ could be interpreted as the difference in discrimination between the treatment and control group. You could extend the model in many other ways that are interesting for your situation. These models are easy to fit in a flexible modeling software like Jags or Stan.

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This is more a comment that grew too long and had to be edited than an answer buy anyway…

I don't know these models very well but it might help to remember that they were developed with personal attributes (traits like abilities, attitudes or personality) in mind. The model then describes responses to a particular item as a function of an individual's score on these latent traits. Of course, each person responds differently to an item but these individual differences are captured by the person's level on the relevant traits, not by the other parameters. That the functional form and item parameters are the same for everybody is a necessary assumption to make it all tractable.

Also, situational variables or transient states (emotion, fatigue…) are nuisances. By definition, traits are stable properties of a person and everything should be done to minimize the impact of other variables on the measurement.

Extending this to psychophysics, it could be some detection threshold (and not luminance itself) that would be interpreted as a trait and that's what discrimination and the item characteristic curve would be applied to. Does it make sense?

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