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I've recently been learning about the Rasch model. Previously I've used various kinds of generalized regression, including linear as well as logistic and "vanilla" fixed-effects models as well as models with random effects.

What I still haven't been able to understand from my reading is what distinguishes the Rasch model from an ordinary mixed-effects logistic regression which includes random effects for person and item, and includes no fixed effects. The mathematical formulation for the two appears to be essentially the same, except the parameters in the exponent in the logistic function are shuffled and relabeled. Almost all the stuff I've read about the Rasch model spends a lot of time talking about its conceptual underpinnings and its applications in test design, and very little time talking about its technical details and how they are the same as or different from other statistical techniques.

So, what is the difference? If I have a table of results like this:

Person    Item   Result
A         1      Right
A         2      Wrong
...
B         1      Wrong
B         2      Right
...etc.

What is the actual difference between feeding this data to a Rasch model, versus feeding it to a mixed-effects logistic regression and interpreting the random effect weights as "person ability" and "item difficulty"?

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Indeed, a Rasch model is a special case of mixed effects logistic regression with a random effect per person and a fixed effect per item. In the psychometrics literature there have been three main estimation procedures developed to estimate its parameters. Marginal maximum likelihood in which the random effects are integrated out, conditional maximum in which estimation proceeds by conditioning on sufficient statistics of the random effects, and joint maximum likelihood that simultaneously estimates the fixed and random effects.

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  • $\begingroup$ A fixed effect per item? That seems very strange. I would have expected item to be a random effect just like person. Also, you say it is a "special case", but can you give any further details? What would be the difference in interpretation of the Rasch results versus the random effect weights on a vanilla logistic regression? $\endgroup$ – BrenBarn Aug 30 '18 at 18:15
  • $\begingroup$ By a fixed effect per item, I mean that you put the items indicator as a factor in your mixed-effects logistic regression, and no intercept. For example, have a look at Section 2 of this paper: jstatsoft.org/article/view/v017i05 $\endgroup$ – Dimitris Rizopoulos Aug 30 '18 at 18:31
  • $\begingroup$ I don't see how that has any connection with what "fixed effect" means. A fixed effect is an effect that is not a random effect. $\endgroup$ – BrenBarn Sep 1 '18 at 7:02
  • $\begingroup$ The standard version of the Rasch model has the following parameterization $Pr(y_{ij} = 1 | b_i) = \sigma(b_i - \beta_j)$, where $y_{ij}$ is the binary outcome for subject $i$ at item $j$, $b_i$ is the random effect for subject $i$, $\sigma$ the sd of the random effects (aka discrimination parameter), and $\beta_j$ is the difficulty parameter which is a fixed per item. The classic formulation of a mixed effect logistic regression is additive, i.e., $Pr(y_{ij} = 1 | b_i) = \alpha_j + b_i$, but of course you can reparameterize to get $\beta_j$. $\endgroup$ – Dimitris Rizopoulos Sep 1 '18 at 8:18

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