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I am sorry for a more basic question, but I was unable to find any good sources on google. How is a bifactor IRT model different from a factor analysis?

How would you describe their key differences?

Any references you could direct me to would be very helpful.

Thank you for sharing your knowledge!

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    $\begingroup$ Since IRT or latent trait model is "FA for categorical data" the latent variable drives not the magnitude of the observed characteristic but the probability of its observation. It is logistic FA. The difference is like between OLS regression and logistic regression. The difficulty parameter of IRT is analogous and not homologous to the uniqueness parameter of FA. Thus the theory is slightly different. If you need a procedure on categorical data which is exactly homologous to FA, that would be nonlinear FA (= FA after so called optimal scaling; see "CatPCA"), not IRT. $\endgroup$ – ttnphns Jul 13 '13 at 8:26
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    $\begingroup$ I think you meant to say that the 'discrimination' parameter is analogous to the uniqueness parameter in FA, not the difficulty. Also, if IRT (more specifically, the 2PL model) is the categorical analogue for a single factor FA since logit(P) = a + b*theta, then wouldn't MIRT be the categorical analogue to exploratory and confirmatory FA since the logit transformation is still the same; logit(P) = a + b1*theta1 + b2*theta2? $\endgroup$ – philchalmers Jul 13 '13 at 16:17
  • $\begingroup$ Thank you everyone for your comments! I feel like I better understand the similarities and differences between the two concepts now. $\endgroup$ – Xander Jul 13 '13 at 18:27
  • $\begingroup$ @philchalmers, No, I meant what I wrote. The analogue for the discrimination is the loading. $\endgroup$ – ttnphns Jul 14 '13 at 1:51
  • $\begingroup$ @ttnphns but the difficulty is just an intercept and has little to do with uniqueness. High or low difficulty tells you nothing about the correlation that the item has with the latent trait. Discrimination parameters are, however, related to the commonality directly (1 - uniqueness). Larger discrimination parameters indicator more of a correlation with the latent trait, which translate into larger commonalities and hence smaller uniqueness values. $\endgroup$ – philchalmers Jul 14 '13 at 1:58
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In several ways, they really aren't any different. A confirmatory bifactor model using IRT is really just a full-information method to deal with categorical data directly, rather than the limited information methods found in structural equation modeling (SEM). Had the observations been continuous, a structural equation modeling approach would be the correct way to analyze the model anyway. Bifactor models in the IRT literature tend to be very popular since they are widely applicable in ability testing situations, and have had special treatment of how the models are estimated, but fundamentally are the same as traditional factor analysis methods (IRT is best thought of a non-linear factor analysis of categorical data, but there are other benefits to it as well).

The limited information SEM approach when exploring these kinds of models for categorical data is still beneficial though since they aren't so computer intensive (they only deal with estimating the first and second moments, rather than all moments as IRT does). However, they require special kinds of estimation and correlation matrices, such as a tetrachoric or polychoric matrix, along with special kinds of estimators such as the WLSMV algorithm.

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