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

Question

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?

Answer 1

All parameters --- difficulty, discrimination and guessing for each item, ability for each individual --- are estimated together using dedicated software. This includes R (ltm, eRm, among others), Stata (gllamm or the irt module which appears in Stata 14), Mplus, or SAS. For a complete list, see the Rasch Measurement Analysis Software Directory, but bear in mind that some of those packages are often specific of a particular OS and not as complete as the packages available in R. If you are looking for an open-source Java-based package, there's aslo Jmetrik, which can fit Rasch, 2PL, 3PL, and 4PL binary item response models and the partial credit (PCM), generalized partial credit (GPCM), and graded response (GRM) models

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.}