Bayesian IRT

These functions allow to calibrate item parameters or estimate abilities under a unidimensional 2PL IRT model by using the PolyaGamma distribution approach introduced by ^[Polson] and adapted to IRT models by ^{[JiangTemplin]}.

The documentation starts with an introduction on the basic tools to work on examinees and on items. It follows a section with estimation/calibration tools for latents and item parameters, separately. At the end, an example of simultaneous estimation of item parameters and abilities is provided.

Basic Tools for Examinees

An examinee is a physical or virtual person who is intended to answer the questions of a test (the items) for measuring an underlying personal latent trait. The purposes of the test may be various, from ability assessment, to clinical diagnosis. In this example, a set of virtual examinees is generated and their properties and dedicated methods are described.

First of all, we generate a groundtruth set of $N = 100$ examinees with 1-dimensional latent ability sampled from a standardized Normal distribution. We use the default factory $Examinee()$ to create a virtual respondent:

N = 100
true_examinees = [Examinee(n, "examinee_$(n)") for n = 1 : N]

We can look at the examinees' latent traits by analysing the field latent. In this case, the field latent is an instance of the Latent1D mutable struct (i.e. its subfields can be modified). It has the following subfields:

• val::Float64
• bounds::Vector{Float64}
• prior::Distributions.ContinuousUnivariateDistribution
• posterior::Distributions.ContinuousUnivariateDistribution
• likelihood::Float64
• chain::Vector{Float64}
• expected_information::Float64

Any latent object of such a type has structural properties, such as value and bounds, but also Bayesian properties, such as prior and posterior distributions (which, in this case, are specified as univariate thanks to Distributions.jl).

The default factory for an examinee returns an examinee with a Distributions.Normal(0,1) as prior (and posterior) for their ability. The value of their ability is randomly extracted from the posterior distribution. In our example, the examinees in true_examinees are all default examinees. Thus, we can see that their latent properties have been set to the defaults:

println(true_examinees[1].latent.prior)
println(true_examinees[1].latent.posterior)
println(true_examinees[1].latent.val)

The method get_latents() can be used to extract the latents of all the examinees in a vector. While, get_latents_vals() produces a matrix with $N$ rows and $L$ columns, where $L$ is the maximum latent dimension among the examinees (1 in our example, since all the examinees have a latent of type Latent1D). Let's look at how they work on the first 10 respondents:

println(get_latents.(true_examinees[1:10]))
println(get_latents_vals.(true_examinees[1:10]))

Basic Tools for Items

Examinee Assessment

Item Calibration

Simultaneous Assessment and Calibration