# How to calculate the parameter value for a test with Item Response Theory?

Given a set of responses to a test with multiple choice I wish to analyse it with Item Response Theory. I am planing to use the 3PL (3 parameters) in the Item response function. How do I infer the values of these parameters to the model from this data? I have never used this method before and from what I understand is that each respondent has their own $$\theta$$ value (has 3 values parameter values $$\alpha,b,c$$). So if there are $$N$$ people who took the test, I will need to calculate the $$\theta_{1...N}$$? What approaches can be used to obtain these values?

The general idea of parameter estimation in IRT models is that you can either use a conditional approach, that is you estimate items parameters by conditioning on the latent trait ($$\theta$$, person's ability), or a marginal approach, much like in mixed-effects models. Since in your case there are more than one item parameter, you can see the ability as a weighted score for each individual (which technically is derived from an ML estimator or, e.g., posterior MAP or EAP). There was a special issue in the Journal of Statistical Software: Psychometrics in R. Despite the title, it is not only about the R software, but it provides a nice overview of almost all estimation approaches.
Note that in the Rasch model (1-PL), the sum score is a sufficient statistic, so that you won't get as many $$\theta_i$$ value as the sample size, but a distinct $$\theta_i$$ for each response pattern.