What is the two parameter IRT model (2PL)?

Item response theory is the predominant psychometric paradigm for mid or large scale assessment.  As noted in my introductory blog post, it is actually a family of models.  In this post, we discuss the two parameter IRT model (2PL).

The 2PL is described by the following equation (simplified from Hambleton & Swaminathan, 1985, Eq. 3.3):

This equation is predicting the probability of a certain response based on the examinee trait/ability level, the item discrimination parameter a, and the item difficulty/location parameter b.  If the examinee trait level is higher than the item location, the person has more than a 50% chance of responding in the keyed direction.

This phrase “in the keyed direction” is one you might often hear with the 2PL.  This is because it is not often used with education/knowledge/ability assessments, where items usually have a correct answer and guessing is often possible.  The 2PL is used more often in attitudinal or other psychological assessments, where guessing is irrelevant and there is no correct answer.  For example, consider an Extroversion scale, where examinees are responding Yes/No to statements like “I love to go to parties” or “I prefer to read books in my free time.”  There is not much to guess here, and the sense of “correct” is not relevant.  However, it is quite clear that the first statement is keyed in the direction of Extroversion, while the second statement is the reverse.  In fact, you would get the 1 point of response for saying No to that statement rather than Yes.  This is often called reverse-scored.

There are other aspects that go into whether you should use the 2PL model, but this is one of the most important.  In addition, you should also examine model fit indices and take sample size into account.

How do I implement the two parameter IRT model?

Like other IRT models, the 2PL requires specialized software.  Not all statistical packages will do it.  And while you can easily calculate classical statistics in Excel, there is no way to do IRT (well, unless you want to write your own VBA programs to do so).  As mentioned in this article on the three parameter model, there are a number of Irt software programs available but not all are created equal.  You should evaluate cost and functionality.  If you are a fan of R, there are packages to estimate IRT there.  However, I recommend our Xcalibre program for both newbies and professionals.  For newbies, it is much easier to use, which means you spend more time learning the concepts of IRT and not fighting command code that might be 30 years old.  For professionals,, Xcalibre saves you from having to create reports by copy and paste, which if you think about how much your hourly rate costs, that copy-and-paste time is incredibly expensive.

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Nathan Thompson, PhD

Chief Product Officer at ASC
I am a psychometrician, software developer, author, and researcher, currently serving as Chief Product Officer for Assessment Systems Corporation (ASC). My mission is to elevate the profession of psychometrics by using software (especially AI and machine learning elements) to automate the menial stuff like job analysis and Angoff studies, so we can focus on more innovative work. My core goal is to improve assessment throughout the world. I was originally trained as a psychometrician, doing an undergrad at Luther College in Math/Psych/Latin and then a PhD in Psychometrics at the University of Minnesota. I then worked multiple roles in the testing industry, including item writer, test development manager, essay test marker, consulting psychometrician, software developer, product owner, and business leader. Research and innovation are incredibly important to me. In addition to my own research, I am cofounder and Membership Director at the International Association for Computerized Adaptive Testing, You can often find me at other important conferences like ATP, ICE, CLEAR, and NCME. I've published many papers and presentations, and my favorite remains http://pareonline.net/getvn.asp?v=16&n=1.
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