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I can't seem to find a model to solve the following problem. It seems to be closely related to IRT, but not exactly.

I have a list of users and items, I know the preference between users and items, in the form of:

[user_1, item_1],
[user_2, item_1],
[user_2, item_4],
...

The data is fairly sparse; each user only has preference for a very small number of items.

I'd like to recover a mapping for both users and items on a number line, preferably with confidence intervals. The difference from IRT seems to be twofold:

Difference 1

I imagine the formulation to be some linear combination of user and item (or item-cutoff) mappings, for example:

$$ logit(P_{prefer}) = x_i - \delta_j $$

Where $x_i$ is the value of user $i$, and $\delta_j$ is the cutoff for item $j$. So if $x_i < \delta_j$, the probability is lower, and if $x_i > \delta_j$, the probability is higher.

But I really just want to minimize how "close" they are. I want to minimize $(x_i - \delta_j)^2$, for example.

Difference 2

The $\delta_j$ recovered from IRT are point estimates (in the 3PL model). Preferably, I'd like to recover both user and item mappings with confidence intervals.

If not IRT, I wonder if a mixed effects model could be used here? Any help would be appreciated. Thank you.

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You could try using the R package ltm that fits this type of models. An example analysis of dichotomous outcomes can be found here. The model you describe seems to be the classical Rasch model covered in the example mentioned above.

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