A differential equation framework for the derivation of item response functions
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Yvonnick Noël
LP3C, Rennes 2 University, France
International Meeting of the Psychometric Society, Prague, July 18th 2024
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http://yvonnick.noel.free.fr/papiers/imps2024
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
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Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
•
•
•
The basic idea
- We will consider that response/success $Y$ on an item as
determined by interacting hidden subprocesses $X_{1},X_{2},\ldots,X_{K}$ ($k=1,\ldots,K$):
- Subprocesses have intensity functions of the form $X_{k}(\theta)$ , i.e. are dependent upon some latent attitude/ability $\theta$.
- Subprocesses have potentially activating or inhibiting influences on each other's variation.
- The resulting expected response function $Y(\theta)$ is modeled through a differential equation model.
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
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•
•
A dynamical multi-process model
- In classical IRT (Rasch, 2PL...) models, responses are considered determined by a single latent dimension, summarized as attitude or ability ($\theta$).
- We consider here a situation where two latent
response processes $x_1$ and $x_2$ are dynamically related, for example
($\alpha,\beta > 0$):
$$\begin{cases} \frac{dx_1}{d\theta}=\alpha x_1(\theta)\\ \frac{dx_2}{d\theta}=\beta x_1(\theta-\delta) \end{cases}$$ - Interpretation : With positive change rates $\alpha$ and $\beta$, we get a chained dependency where $x_1$ increases with $\theta$, and $x_2$ increases with $x_1$, possibly with some delay/shifting $\delta$, along a common dimension $\theta$.
Boundary constraints
- Solution functions for this type of differential equations are well-known to be linear combinations of exponentials (e.g. compartment models).
- This is not suited to the modeling of bounded data (i.e. binary data or continuous bounded scales).
- We simply introduce logistic penalty factors, as ($x > 0$): $$\begin{cases} \frac{dx_1}{d\theta}=\alpha x_1(\theta){\color{#0587c4}{[1-x_1(\theta)]}}\\ \frac{dx_2}{d\theta}=\beta x_1(\theta-\delta){\color{#0587c4}{[1-x_2(\theta)]}} \end{cases}$$ to ensure both $x_1$ and $x_2$ (and $y$) remain in the $[0;1]$ range.
Solution functions
- Upon integration of the previous system, we get: $$\begin{cases} x_1(\theta)=1-\frac{1}{1+\exp(\alpha\theta+C)}\\ x_2(\theta)=1-\frac{D}{\left[1+\exp[\alpha(\theta-\delta)]\right]^{\frac{\beta}{\alpha}}} \end{cases}$$
- The first equation is a standard logistic, while the second one is a generalized logistic with an exponent (induction) parameter (Richards, 1959). It includes the Rasch, 2PL, 3PL and reflected Logistic Positive Exponent Model (Samejima, 1972; Bolfarine & Bazan, 2010) as special cases.
- It may be considered as a generic model of a process growth causally induced by another process (potentially also a new interpretation for classical models).
A two-process cumulative causal chain
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
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•
•
A dynamical multi-process model with inhibition
- We now consider the situation where both processes
increases, but one exerts some inhibition on the second:
\[\begin{cases} \frac{dx_1}{d\theta}=\alpha x_1(\theta)\color{#0587c4}{-\gamma x_2(\theta-\delta)}\\ \frac{dx_2}{d\theta}=\beta x_2(\theta) \end{cases}\] $\alpha,\beta,\gamma>0$ - Interpretation : $x_1$ and $x_2$ are increasing with $\theta$, $x_2$ exerting an inhibitory action over $x_1$, possibly with some delay/shifting $\delta$, along a common dimension $\theta$.
Boundary constraints
- Solution functions for this type of differential equations are well-known to be linear combinations of exponentials (e.g. compartment models).
- This is not suited to the modeling of bounded data (i.e. binary data or continuous bounded scales).
- Boundary constraints are imposed to ensure that the result remains in the $[0;1]$ range: \[\begin{cases} \frac{dx_1}{d\theta}=\left[\alpha x_1(\theta)-\gamma x_2(\theta-\delta)\color{#0587c4}{x_1(\theta)}\right]\color{#0587c4}{[1-x_1(\theta)]}\\ \frac{dx_2}{d\theta}=\beta x_2(\theta)\color{#0587c4}{[1-x_2(\theta)]} \end{cases}\]
Solution functions
- Upon integrating this system, we get
(Noel, 2017) :
$$\begin{cases} x_1(\theta)=\frac{\exp\left[\alpha\theta+C_1\right]}{\color{#0587c4}{\left\{ \exp\left[\beta(\theta-\delta)\right]+1\right\}^{\frac{\gamma}{\beta}}}+\exp\left[\alpha\theta+C_1\right]}\\ x_2(\theta)=\frac{\exp\left[\beta\theta+C_2\right]}{1+\exp\left[\beta\theta+C_2\right]} \end{cases}$$
- A more IRT-like expression for $x_1(\theta)$ is obtained by
introducing a new $\lambda$ parameter with $C_1=-\alpha\delta+\lambda$:
$$ x(\theta)=\frac{\exp\left[\alpha(\theta-\delta)+\lambda\right]}{\color{#0587c4}{\left\{ \exp[\beta(\theta-\delta)]+1\right\} ^{\frac{\gamma}{\beta}}}+\exp\left[\alpha(\theta-\delta)+\lambda\right]} $$
Parameter interpretation
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
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Proficiency in Arabic language
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
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Discrete stage vs. continuous model of change
Predictive power
End
Thank you for your attention.
yvonnick.noel@univ-rennes2.fr
Contents
- A multi-process response mechanism
- Cumulative models
- Unfolding models
- Applications
•
•
•