A Beta Asymmetric Unfolding Model for Continuous Bounded Responses

Application to the Modeling of Behavior Change Dynamics

Yvonnick Noel

LP3C, University of Brittany, Rennes 2, France

56th Annual Meeting of the Society for Mathematical Psychology, Amsterdam, July 20th, 2023

http://yvonnick.noel.free.fr/papiers/baum

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

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 $x$ and $y$ are dynamically related, in the following way:

  \[\begin{cases} \frac{dx}{d\theta}=\alpha x(\theta)\color{#0587c4}{-\gamma y(\theta-\delta)}\\ \frac{dy}{d\theta}=\beta y(\theta) \end{cases}\] with $\alpha$, $\beta$ et $\gamma$ positive factors, and $\delta$ a shifting parameter.

• Interpretation : With positive change rates $\alpha$ and $\beta$, $x$ and $y$ are increasing with $\theta$, $y$ exerting an inhibitory action over $x$, 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: \[\begin{cases} \frac{dx}{d\theta}=\left[\alpha x(\theta)-\gamma y(\theta-\delta) \color{#0587c4}{x(\theta)}\right]\color{#0587c4}{[1-x(\theta)]}\\ \frac{dy}{d\theta}=\beta y(\theta)\color{#0587c4}{[1-y(\theta)]} \end{cases}\]

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

Solution functions

• Upon integrating this system, we get (Noel, 2017) :

  $$\begin{cases} x(\theta)=\frac{\exp\left[\alpha\theta+C_1\right]}{\color{#0587c4}{\left\{ \exp\left[\beta(\theta-\delta)\right]+1\right\}^{\frac{\gamma}{\beta}}}+\exp\left[\alpha\theta+C_1\right]}\\ y(\theta)=\frac{\exp\left[\beta\theta+C_2\right]}{1+\exp\left[\beta\theta+C_2\right]} \end{cases}$$

  Noël, Y. (2017). Item Response Models for Continuous Bounded Responses, with applications in the analysis of emotion, personality and behavior change. Senior habilitation thesis, University of Brittany, Rennes 2, France.

• A more IRT-like expression is obtained by introducing a new $\lambda$ parameter with $C_1=-\alpha\delta+\lambda$ and $C_2=\delta$:

  $$ x(\theta)=\frac{\exp\left[\alpha(\theta-\delta)+\lambda\right]}{\color{#0587c4}{\left\{ \exp[\beta(\theta-\delta)]+1\right\} ^{\frac{\gamma}{\beta}}}+\exp\left[\alpha(\theta-\delta)+\lambda\right]} $$

Parameter interpretation

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

The Beta Asymmetric Unfolding Model (BAUM)

• A Beta response model may be defined as $X_{ij}\sim Beta(m_{ij},n_{ij})$, for subject $i$ and item $j$, with:

  $$\begin{cases} m_{ij}&=\exp\left[\alpha_j(\theta_i-\delta_j)+\lambda_j\right] \\ n_{ij}&=\left\{ \exp\left[\beta_j(\theta_i-\delta_j)\right]+1\right\}^{\frac{\gamma_j}{\beta_j}} \end{cases} $$

• The expected response function (ICC) is then given by:

  $$ E(X_{ij}|\theta_{i})=\frac{m_{ij}}{m_{ij}+n_{ij}}=\frac{\exp\left[\alpha_{j}(\theta_{i}-\delta_{j})+\lambda_{j}\right]}{\left\{ \exp\left[\beta_{j}(\theta_{i}-\delta_{j})\right]+1\right\}^{\frac{\gamma_{j}}{\beta_{j}}}+\exp\left[\alpha_{j}(\theta_{i}-\delta_{j})+\lambda\right]} $$

A generic model

• Within the generic model: \[ E(X_{ij}|\theta_{i})=\frac{\exp\left[\alpha(\theta-\delta)+\lambda\right]}{\left\{ \exp\left[\beta(\theta-\delta)\right]+1\right\} ^{\frac{\gamma}{\beta}}+\exp\left[\alpha(\theta-\delta)+\lambda\right]} \]
• ... fixing $\alpha=\beta=1$ and $\gamma=0$ brings us back to the Beta Response Model¹:

  \[ E(X_{ij}|\theta_{i})=\frac{\exp\left(\theta_{i}-\delta_{j}+\lambda_{j}\right)}{1+\exp\left(\theta_{i}-\delta_{j}+\lambda_{j}\right)} \]

  (1) Noel, Y. and Dauvier, B. (2007). A beta item response model for continuous bounded responses, Applied Psychological Measurement, 31(1), 47-73.

• ... fixing $\alpha=\beta=1$ and $\gamma=2$ brings us back (almost) to the Beta Unfolding Model² : \[ E(X_{ij}|\theta_{i})=\frac{\exp\lambda_{j}}{\exp\lambda_{j}+\exp\left[\theta_{i}-\delta_{j}\right]+\exp\left[-\left(\theta_{i}-\delta_{j}\right)\right]\color{#0587c4}{+2}} \]

  (1) Noel, Y. and Dauvier, B. (2007). A beta item response model for continuous bounded responses, Applied Psychological Measurement, 31(1), 47-73.

  (2) Noel, Y. (2014). A beta unfolding model for continuous bounded responses, Psychometrika, 79(4), 647-674.

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

Modeling the variance

Shaping the variance

• In a $Beta(m,n)$, the variance is inversely related to the sum of its parameters:

  \[ V(X;m,n)=\frac{mn}{(m+n)^{2}(m+n+1)}=\mu(1-\mu)\left[\frac{1}{\color{#0587c4}{m+n}+1}\right] \]

• The model can be modified as a $Beta(m',n')$ such that:

  \[ \begin{cases} m'(\theta)=\color{#0587c4}{w(\theta)}\exp\left[\alpha(\theta-\delta)+\lambda\right]\\ n'(\theta)=\color{#0587c4}{w(\theta)}\left\{ \exp[\beta(\theta-\delta)]+1\right\}^{\frac{\gamma}{\beta}} \end{cases}\]

  with the variance inflating function ($v>0$):

  \[ w(\theta)=\exp\left[-v(\theta-\delta)+\tau\right] \]

• We note that this leaves the expectation unchanged.

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

Processes and stages of behavioral change

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

Cumulative vs. unfolding models of change

An empirical study

• $N=614$ students from Schools of Nursing in France responded to the Self-Change Strategies questionnaire (Etter & al., 2000) at the beginning of their first year.

  Noël, Y., Molimard, R. & Martin, C. (2004, October). A longitudinal study in Schools of Nursing. 19th Meeting of the French Tobbacology Society, Paris.

• The questionnaire includes 8 cognitive items et 12 behavioral items.
• Students responded to items on a 0-100% agreement visual analogue scale.
• Three models have been tested : The cumulative BRM (AIC=-8478.144), the symmetric BUM (AIC=-10401.1) and the asymmetric BAUM (AIC=-10754.2).

Discrete vs. continuous models of change

Measuring change readiness

Note: Figures are median locations by stage.

End

Thank you for your attention.

yvonnick.noel@univ-rennes2.fr

Contents

1. A multi-process response mechanism
A differential equation framework
•
Solution function
2. A Beta Asymmetric Unfolding Model (BAUM)
The expected response model
•
The variance model
3. Application: Modeling behavior change dynamics
The processes and stages model of change
•
A continuous IRT reformulation

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 $x$ and $y$ are dynamically related, for example ($\alpha,\beta > 0$): $$\begin{cases} \frac{dx}{d\theta}=\alpha x(\theta)\\ \frac{dy}{d\theta}=\beta x(\theta-\delta) \end{cases}$$
• Interpretation : With positive change rates $\alpha$ and $\beta$, we get a chained dependency where $x$ increases with $\theta$, and $y$ increases with $x$, 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 ($x > 0$): $$\begin{cases} \frac{dx}{d\theta}=\alpha x(\theta){\color{#0587c4}{[1-x(\theta)]}}\\ \frac{dy}{d\theta}=\beta x(\theta-\delta){\color{#0587c4}{[1-y(\theta)]}} \end{cases}$$ to ensure both $x$ and $y$ remain in the $[0;1]$ range.

Solution functions

• Upon integration of the previous system, we get: $$\begin{cases} x(\theta)=1-\frac{1}{1+\exp(\alpha\theta+C)}\\ y(\theta)=1-\frac{D}{\left[1+\exp[\alpha(\theta-\delta)]\right]^{\frac{\beta}{\alpha}}} \end{cases}$$
• 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

The Beta distribution

Moments

• The expected response in the Beta is: \[E(X;m,n)=\mu=\frac{m}{m+n} \]
• The modal response is: \[Mo(X;m,n)=\frac{m-1}{m+n-2} \]
• Variance is given by: \[ V(X;m,n)=\frac{mn}{(m+n)^{2}(m+n+1)}=\mu(1-\mu)\left[\frac{1}{\color{#0587c4}{m+n}+1}\right] \]

Constraints on the variance function

• It can be shown that for $\alpha=\beta=1$ and $\gamma=0$ (Beta Response Model), $v=\frac{1}{2}$ gives us the exact variance function of the BRM.
• Similarly, for $\alpha=\beta=1$ and $\gamma=2$ (Beta Unfolding Model), the choice $v=1$ gives us the exact variance function of the BUM.
• Finally, constraining $v$ such that:

  \[ v_{\gamma}=\left(\frac{\alpha+\beta}{2}\right)\left(\frac{\gamma+2}{4}\right) \] will fit both cases while interpolating between them.