# “Dichotomous” Vs “Polytomous” 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. But, IRT is actually a family of fast-growing models.

I often hear the question: “What is the difference between *dichotomous* and *polytomous*?” Well, these terms represent two subcategories within item response theory, which we will discuss here.

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## Dichotomous IRT Models

Dichotomous IRT models are those with 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).

### What models are dichotomous?

The three most common dichotomous models 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 models, graphically, will have one S-shaped curve with a positive slope, as seen here. This model 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 white paper).

In the example, a student with theta = -3 has about a 0.28 chance of responding correctly, while theta = 0 has about 0.60 and theta = 1 has about 0.90.

## 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 models have a line that dictates 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 is for an Essay that 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.

## 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 on 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).

#### Nathan Thompson, PhD

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