Item response theory has "graded" models for Likert items. Is there an IRT model for if some items that are "Likert" and some that are "binary"?

Question

Item Response Theory is most conventionally applied to binary data, but can also be applied to Likert data with "graded" models. But what if you have data that is a mixture of both Likert and binary? In other words, not data of exclusively one type? Is there any type of Item Response Theory model for that type of dataset?

In the documentation for the IRT package in R, "mirt", these are all the models:

Example R code for generating the data:

set.seed(5)
df <- data.frame(i1=round(runif(100, 1, 7)),
                 i2=round(runif(100, 1, 7)),,
                 i3=round(runif(100, 1, 7)),
                 i4=rbinom(100, 1, .5))

library(mirt)

Answer 1

Common to all item response models is the assumption that - controlling for the latent trait - the residuals are uncorrelated. This implies that each item is estimated according to its item type without regard to the other items; so you can have multiple items of different item type.

In mirt you can specify an item-specific type with a vector-based itemtype option. E.g., continuing your example:

mod1 <- mirt(df, 1, itemtype = c("graded", "graded", "graded", "Rasch"))
coef(mod1)

Two comments to your question:

• there are other than the graded response model available for likert-type items (e.g. the gpcm)
• for many IRT models that are developed with binary data in mind there is a generalization towards polychotomous data (e.g. Rasch -> [g]pcm).

This implies that the following might by a valid IRT model given your data

mod2 <- mirt(df, 1, itemtype = "Rasch")
coef(mod2)

For greater detail, study the options of the itemtype-argument and the IRT Models section in the help page for the mirt method (e.g., mirt manual).