What do “dichotomous” and “polytomous” mean in IRT?

Item response theory (IRT) is the dominant psychometric paradigm for constructing, scoring, and analyzing assessments.  Virtually all large-scale assessments utilize IRT because of its well-documented advantages.  In many cases, however, it is referred to as a single way of analyzing data.  However, IRT is actually a family of models, and a growing family at that.

I often hear the question: “What is the difference between dichotomous and polytomous?”  Well, these terms represent two subfamilies within the IRT family.

Dichotomous IRT Models

Dichotomous IRT models are those where there are two possible item scores.  Note that I say “item scores” and not “item responses” – the most common example of a dichotomous item is multiple choice, which typically has 4 to 5 options, but only two possible scores (correct/incorrect).  True/False or Yes/No items are also obvious examples, and are more likely to appear in surveys or inventories, as opposed to the ubiquity of the multiple choice item in achievement/aptitude testing.  Other item types that can be dichotomous are Scored Short Answer and Multiple Response (all or nothing scoring).  Learn more about item types here.

What models are dichotomous?

The three most common are the 1PL/Rasch, the 2PL, and the 3PL.  Which one to use depends on the type of data you have, as well as your doctrine of course.  A great example is Scored Short Answer items: there should be no effect of guessing on such an item, so the 2PL is a logical choice.  Here is a broad overgeneralization:

  • 1PL/Rasch: Uses only the difficulty (b) parameter and does not take into account guessing effects or the possibility that some items might be more discriminating than others; however, can be useful with small samples and other situations
  • 2PL: Uses difficulty (b) and discrimination (a) parameters, but no guessing (c); relevant for the many types of assessment where there is no guessing
  • 3PL: Uses all three parameters, typically relevant for achievement/aptitude testing.

What do dichotomous models look like?Dichotomous IRF from FastTest

Dichotomous models, graphically, will have one S-shaped curve with a positive slope, as seen here.  This models that the probability of responding in the keyed direction increases with higher levels of the trait or ability.  Technically, there is also a line for the probability of an incorrect response, which goes down, but this is obviously the 1-P complement, so it is rarely drawn in graphs.  It is, however, used in scoring algorithms (check out this free software download and this white paper).  In the example, a student with theta = -3 has about a 0.28 chance of responding correctly, while theta = 0 has about 0.72 and theta = 2 has about 0.94.

Polytomous IRT Models

Polytomous models are for items that have more than two possible scores.  The most common examples are Likert-type items (Rate on a scale of 1 to 5) and partial credit items (score on an Essay might be 0 to 5 points).  IRT models typically assume that the item scores are integers.

What models are polytomous?

Unsurprisingly, the most common polytomous models use names like rating scale and partial credit.

  • Rating Scale Model (Andrich, 1978)
  • Partial Credit Model (Masters, 1982)
  • Generalized Rating Scale Model (Muraki, 1990)
  • Generalized Partial Credit Model (Muraki, 1992)
  • Graded Response Model (Samejima, 1972)
  • Nominal Response Model (Bock, 1972)

What do polytomous models look like?Polytomous IRF from FastTest

Polytomous models have a line that models each possible response.  The line for the highest point value is typically S-shaped like a dichotomous curve.  The line for the lowest point value is typically sloped down like the 1-P dichotomous curve.  Point values in the middle typically have a bell-shaped curve.  The example on the right is for an Essay scored 0 to 5 points.  Only students with theta >2 are likely to get the full points (blue), while students 1<theta<2 are likely to receive 4 points (green).

I’ve seen “polychotomous.”  What does that mean?

It means the same as polytomous.  For an interesting discussion by my graduate advisor and co-founder of ASC, David J. Weiss, please read http://conservancy.umn.edu/handle/11299//117413.

How is IRT used in our platform?

We use it to support the test development cycle, including form assembly, scoring, and adaptive testing.  You can learn more at this page.

How can I analyze my tests with IRT?

You need specially designed software, like Xcalibre.  Classical test theory is so simple that you can do it with Excel functions.

Recommended Readings

Item Response Theory for Psychologists by Embretson and Riese (2000).  These authors also both graduated from the psychometrics program at Minnesota.

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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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