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Just wondering if there were any examples of how to fit a many facet Rasch model in Mplus? This would be used when trying to fit a model that would adjust for rater effects on the scale for an individual (e.g., when multiple experts observe an individual on the job and rate their performance). 


Can you show me in formulas what the manyfacet Rasch model looks like? 


Hi Bengt, Linacre (1994) defines the model as: ln(P_{nijk}/P_{nijk1}) = B_{n}  D_{i}  C_{j}  F_{k} Where P_{nijk} is the probability of the nth examinee earning the kth rating from the jth judge on the ith item P_{nijk1} is the probability of the nth examinee earning the kth  1 rating from the jth judge on the ith item B_{n} is the ability (Theta) of the nth examinee D_{i} is the difficulty of the ith item C_{j} is the severity of the jth judge F_{k} is the difficulty of the step up from the kth1 category k and k=1,M (p 1) Linacre, J. M. (1994). ManyFacet Rasch Measurement. Chicago, IL: MESA Press. 


In Engelhard (2013) the model is presented as: Psi_nmik = exp(Theta_n  Lambda_m  Delta_i  Tau_k) / 1 + exp(Theta_n  Lambda_m  Delta_i  Tau_k) The category response function is written as: PI_nmij = exp(Sigma_{j=0}^{k}(Theta_n  Lambda_m  Delta_i  Tau_k)) / Sigma_{r=0}^{mi}exp(Sigma_{j=0}^{r}(Theta_n  Lambda_m  Delta_i  Tau_k)) Where Theta_n is the judged ability of the nth person, Lambda_m is the severity of the mth rater, Delta_i is the judged difficulty of the ith domain, and Tau_k is the difficulty of scoring in the kth category relative to the k1 category (p. 200). Engelhard, G. (2013) Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences. New York City, NY: Routledge. 


There's a bit more detail available in this article without having to find/dig through a book if it helps: https://www.researchgate.net/publication/228465956_Manyfacet_Rasch_measurement 


I would think this is doable in Mplus but I have not done it. Take a look at Section 4 of our IRT note: http://www.statmodel.com/download/MplusIRT.pdf where we discuss the (Generalized) Partial Credit model (your model has some similarities to it). Here we give a reference to a 2015 article by HugginsManly & Algina which discussed how Mplus' nominal outcome modeling can be arranged to handle GPC (although GPC is now automated in Mplus) using Model Constraint. Similarly, your model should be doable. Perhaps Lambda is a fixed effect  if so, it could be handled by dummy covariates. Maybe someone on SEMNET has tried this model in Mplus. 


I think in the Facets software lamba is treated as a fixed effect. There also seems to be more flexible ways to recover the parameters using different types of mixed effects models discussed here: https://www.researchgate.net/profile/Matthew_Gaertner/publication/289527493_Crossclassified_Random_Effects_Models_for_Assessing_Rater_Severity_and_Differential_Rater_Functioning/links/572ab0e108ae2efbfdbc244d.pdf?origin=publication_list The random effects models seem like they would provide more options for estimating different types of rater effects based on the specific model used. For these types of models I'm not exactly sure how the data need to be structured to estimate the parameters of interest (e.g., does the data need to be normalized to the point where each record represents each possible choice per item for each item and rater combination with a series of dummy indicators for the selected response per item and rater combination). 

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