Which of the above procedures refers to polytomous models? Seems like all are dichotomous only or mostly.

Thanks for a very informative and easy-to-follow post!

These graphs always seem to show item response curves as a function of participant abilities. Is there a way to reverse that and show participant response curves as a function of item difficulty?

I am not aware of any package that provides such functionality. However, you can create such a plot yourself with the available information. For example, the code plot(betas, plogis(pp_ml[1] - betas), ylim = 0:1) plots the probabilities of the first person for solving all the 24 items.

https://link.springer.com/c...
the book《Item Response Theory Equating using R 》maybe useful , thanks for your help.

do u konw how to do Fixed Item Parameter Calibration by using R?

You can fix parameters to specific values, for example, in the ltm package by using the constraint argument of the rasch() function. Furthermore, there are general IRT package that have functions for linking/equating (e.g., mirt, TAM, sirt) and packages directly dedicated to linking/equating (equateIRT, equate, plink). But to be honest, I am not too much into this topic, and I am therefore not aware of packages that directly implement the algorithms described in Kim (2006, JEM); but as a starting point, I would take a closer look at the mentioned packages.

Thank you so much for your help！i have looked into TAM ,but the description of constraint argument is different between TAM and LTM，constraint is descript as "Set sum constraint for parameter identification for items or cases” in TAM, so that i think it cannot set item parameter. I don't know the thougt is right or wrong.hahaha
anyway, i will look into the paper and packsges that you suggested. your answer is very useful for me! thanks again！Happy Friday!

You're most welcome.
Constraining item parameters in TAM::tam.mml() is done via the xsi.fixed argument. See "Model 1a" in the help file for an example.

http://www.edmeasurementsur...
xsi.fixed is described in the TAM tutorials, The author's aim at building Score Equivalence Table. in my opinion ,i think the method is different between the method of Fixed Item Parameter Calibration because xsi.fixed cannot fixed the parameter of assigned items.

I've tried this method before, but it doesn't fix the parameters of the anchor items, only the difficulty of all the items.I'll try the other methods, and if i find it, I'll share it with you.thanks again.

or do u konw which papers about the implementation of Fixed Item Parameter Calibration？thanks！

The paper by Kim (2006, https://doi.org/10.1111/j.1... ) is probably a good starting point.

What are the formulas to find difficulty, discrimination and theta values ??

The code for estimating the thetas is given in the section person parameters and for the item parameters in the section 2 PL. Is that what you are looking for?

res_2pl_1 <- ltm(dat_1 ~ z1)
after this command you will get the all discrimination and difficulty values but how you will calculate those values manually or by using specific formula to calculate those values ?? or i there any way to find the back end mathematics

"Coefficients:
Dffclt Dscrmn"

This above values of difficulty and discrimination.
I'm not looking for code, I just wanted to know the formula behind how to calculate those values manually.

for e.g Probability of 2 PL model = e^(a(t-b)/1+ (e^(a(t-b))

so,a = discrimination
b = difficulty
t = theta
Formula to find difficulty,discrimination,theta ?

The formula you give for a single item is multiplied across all items and persons, and this gives the likelihood. This (log)likelihood is then used to estimate (not calculate) the parameter values given a data set. Usually, the item parameters are estimated first (CML or MML) and the person parameters second (EAP, MAP, ML, etc.). You will find details about parameter estimation, for example, in Embretson and Reise (2000, chap 8 + 7), in Johnson (doi: 10.18637/jss.v020.i10), and in many other IRT resources.
But note again, you cannot "calculate" the values by hand, they need to be estimated given the data.

I have tried this all in Rstudio but I wanted to implement this in python. So as u said we cant calculate the values by hand, then how do I implement the same thing without knowing math behind such commands in R, like ltm package.
Suggest if you have any other options.

To estimate the parameters, you need two things. First, the (log)likelihood function is needed, which is given by multiplying across items and persons. This is probably a function/statement in Python that you would have to implement yourself. Second, an algorithm is required that finds the maximum of this function given a data set. I suspect that such algorithms (or at least related examples) exist in Python, try to search for "Python maximum likelihood estimation". (Maybe scipy.optimize?)
Furthermore, a quick Google search makes me believe that some attempts have already been made to implement IRT models in Python.