# How is a bifactor IRT model different from a factor analysis?

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!

• 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. Commented Jul 13, 2013 at 8:26
• 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? Commented Jul 13, 2013 at 16:17
• Thank you everyone for your comments! I feel like I better understand the similarities and differences between the two concepts now. Commented Jul 13, 2013 at 18:27
• @philchalmers, No, I meant what I wrote. The analogue for the discrimination is the loading. Commented Jul 14, 2013 at 1:51
• @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. Commented Jul 14, 2013 at 1:58