C2/M2/M2* in longitudinal IRT model

[Charlie_Rutgers]

Hi Phil,

I hope you’re well – I have a question about how we can estimate the M2/M2*/C2 statistic for longitudinal IRT models.

 

The C2 statistic compares the model-implied (expected) proportions of item response patterns against the observed proportions. However, it seems like the observed proportions and the expected proportions would be different at each timepoint. After reviewing the code, I can’t figure out how the function handles the multiple timepoints.

 

-         Are the observed proportions computed using both timepoints? The observed data is (of course) different at the two timepoints, so how do we handle that?

-         Are the expected (model-implied) proportions computed using “Theta” variable from both timepoints? Again, I expect theta to be different at both timepoints, which one do I use?

 

-         Finally, the N used to compute the statistic, if we are using data from multiple timepoints, does that impact the N value used?

That’s hard to articulate, so let me refer to the code in the function in hopes of clarifying: https://github.com/philchalmers/mirt/blob/master/R/M2.R

 

-         lines 175 and 176 are using variable “Theta” to compute the probability trace

 

-         “EIs” and “prob” are then used later to compute the fit statistics

-         Is Theta a single dimension or multidimensional? Do you compute a **unique** expected probability at each timepoint?

 

Would greatly appreciate any input on this, as I’m trying to write out the C2 statistic for a CDM/DCM and it’s giving me fits.

 

Thanks again for all the work you do on the mirt library and hope to thank you in person once conferences are back up and running.

 

Charlie 

[Phil Chalmers]

Hi Charlie,

If it helps, the setup for the C2 statistic is actually the same as how structural equation modeling GOF statistics are formulated; based on the univariate moments for each category in each, and the (marginalized) covariance estimates between the items. The model implied proportions are constructed from collapsing the expected probability functions into univariate proportions and bivariate moments, so conceptually speaking the dimensionality of the model is not really an issue as they are just collapsed to single expected probability values (hence, in the code on line 176, the object will have a constant dimension, where for a dichotomous item the object prob will always have nrow(Theta) rows and 2 columns, which are then collapsed to single expected values by marginalizing across the columns via the expectation operation).  Of course, multidimensionality is a problem from a numerical integration perspective, but it's not different with the M2 family of statistics). Does that help, or have I just confused you further? :-)

Phil

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[Charlie Iaconangelo]

Hi Phil, 

What you're saying makes sense for the computation of the observed vs expected proportions, but in the actual computation of the M2/C2/whatever statistic there's an "N" that is leading to my confusion. 

Equation 25 here (pg 15 of the pdf) - all of the statistics have the same form, and that value of N is what I'm hung up on. 

https://files.eric.ed.gov/fulltext/ED555702.pdf

Line 104 in my code, I use the sample size N to compute the C2 value. 

https://github.com/CJangelo/CLCM/blob/main/R/C2_clcm.R

What am I missing here?

Hope you're doing well. 

Charlie