Simulating Item Responses with the Partial Credit and Rasch Models

Using the sim.PCM Function

Introduction

In this tutorial, we’re going to explore how to simulate item responses using the sim.PCM function. This function is handy for generating polytomous item responses, allowing us to simulate responses for models like the Partial Credit Model (PCM) or even a Rasch model when dealing with dichotomous items.

What’s great about sim.PCM is its flexibility—it lets you play around with different item levels, thresholds, and person abilities. Whether you want a simple Rasch model or a more complex situation where items have different numbers of categories, this function has you covered!

Setting Up the Simulation

Let’s start by simulating some basic data. We’ll begin with a simple setup: 100 respondents answering 5 items, each with 3 response levels (think of it like a Likert scale with three options).

Example 1: Basic Simulation

Here, we’re simulating responses for 100 people across 5 items, each with 3 response options. It’s as easy as that!

simulated_data <- sim.PCM(pN = 100, iN = 5, lN = 3)

Just like that, you’ve got your data! This gives you a simple matrix of simulated responses that match our configuration: 100 respondents answering 5 items, each with 3 response levels.

Example 2: Customizing Item Thresholds

What if you want more control over the item difficulty thresholds? You can pass in your own custom thresholds instead of letting the function generate them for you. Here’s how:

custom_delta <- matrix(c(-0.5, -1, -2, 1, 0.5, 1), nrow = 3)
simulated_data_custom <- sim.PCM(pN = 50, iN = 3, lN = 3, delta = custom_delta)

In this example, we’re specifying thresholds for 3 items. The custom_delta matrix defines how difficult each threshold is for the respective items. If you look at the output, you’ll see the responses simulated based on these difficulty levels.

Example 3: Simulating with Custom Person Abilities

What about the people? Maybe you don’t want a standard normal distribution for their abilities. Let’s say you want a little more variety in the abilities of your respondents—no problem! You can also specify your own custom person abilities.

custom_theta <- rnorm(100, mean = 0, sd = 1.5)
simulated_data_theta <- sim.PCM(pN = 100, iN = 5, lN = 3, theta = custom_theta)

Here, we’re generating a set of 100 person abilities (custom_theta) with a mean of 0 and a standard deviation of 1.5. This gives you more control over how the simulated people interact with the items in your test.

Example 4: Simulating a Rasch Model

Now let’s move from polytomous items to a Rasch model with dichotomous items. In this case, each item only has two possible responses. We’ll simulate data for 50 people, with abilities ranging from -3 to +3, responding to 10 items.

rasch_theta <- seq(-3, 3, length.out = 50)
rasch_delta <- matrix(c(0,1.5,0,-1,0,0.5,0,-0.5,0,1,0,-1.5,0,2,0,-2,0,0.8,0,-0.8), ncol = 2, byrow = TRUE)
simulated_rasch <- sim.PCM(pN = 50, iN = 10, lN = 2, delta = rasch_delta, theta = rasch_theta)

In this simulation, we have a Rasch model with 10 items and 50 persons. The rasch_theta values span from -3 to 3, representing a range of person abilities. The rasch_delta matrix specifies the difficulty of each item.

Example 5: Simulating Items with Different Numbers of Levels

What if your items don’t all have the same number of response options? That’s where the itemLevels argument comes in handy. For example, let’s simulate a mix of items: some have 5 levels, some are dichotomous, and some have 3 levels.

simulated_mixed_levels <- sim.PCM(pN = 50, iN = 10, lN = 5, 
                                  itemLevels = c(5, 5, 5, 5, 2, 2, 2, 3, 3, 3))

Here we’re simulating responses for 10 items, with 4 items having 5 levels, 3 items being dichotomous, and 3 items having 3 levels. This kind of flexibility is great for simulating more realistic data, where items can have different response structures.

Wrapping Up

The sim.PCM function is a powerful tool for simulating item responses for both polytomous and dichotomous items. Whether you’re setting up a simple Rasch model, simulating items with multiple levels, or adding custom thresholds and person abilities, this function gives you the flexibility to generate realistic data tailored to your needs.

Now that you’ve seen a few examples, you can start experimenting with different configurations to see what works best for your research or testing scenarios. Happy simulating!