A Beta Asymmetric Unfolding Model for Continuous Bounded Responses

Application to the Modeling of Behavior Change Dynamics

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

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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- 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 $\theta$.

- 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: \[\begin{cases} \frac{dx}{d\theta}=\left[\alpha x(\theta)-\gamma y(\theta-\delta) \color{#0587c4}{x(\theta)}\right]\color{#0587c4}{[1-x(\theta)]}\\ \frac{dy}{d\theta}=\beta y(\theta)\color{#0587c4}{[1-y(\theta)]} \end{cases}\]

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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

\[
\tiny x(\theta)=\frac{\exp\left[\alpha(\theta-\delta)+\lambda\right]}{\left\{
\exp\left[\beta(\theta-\delta)\right]+1\right\}
^{\frac{\gamma}{\beta}}+\exp\left[\alpha(\theta-\delta)+\lambda\right]}
\]

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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

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

(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 Model**^{2}: \[ E(X_{ij}|\theta_{i})=\frac{\exp\lambda_{j}}{\exp\lambda_{j}+\exp\left[\theta_{i}-\delta_{j}\right]+\exp\left[-\left(\theta_{i}-\delta_{j}\right)\right]\color{#0587c4}{+2}} \](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.

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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- 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**.

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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I. Precontemplation (non motivation) |
II. Contemplation (expect change within 6mo) |
III. Preparation (expect change within 30d) |
IV. Action (quit, last 6mo) |
V. Maintenance (quit, more than 6mo ago) |
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01 - Social liberation (Perception of non-smokers' behavior) |
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02 - Environmental reevaluation (reassess impact on environment) |
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03 - Emotional relief (express negative feelings) |
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04 - Consciousness raising (taking information on quitting smoking) |
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05 - Sef-reevaluation (reassess one's behavior and values) |
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06 - Self-liberation (Decision, will power) |
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07 - Stimulus control (Remove any cue or incentive) |
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08 - Helping relationships (Rely on significant others) |
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09 - Reinforcement management (Find alternative sources of satisfaction) |
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10 - Counter-conditioning (Find replacements and substitutes) |

(DiClemente and Prochaska, 1982; DiClemente and Prochaska, 1985; DiClemente et al., 1991; Prochaska et al., 1988)

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

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Noel, Y. (1999). Recovering Latent Unimodal Patterns of Change by Unfolding Analysis : Application to
Smoking Cessation. *Psychological Methods*, 4(2), 173-191.

- $N=614$ 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).

Note: Figures are median locations by stage.

Thank you for your attention.

yvonnick.noel@univ-rennes2.fr

**A multi-process response mechanism****A Beta Asymmetric Unfolding Model (BAUM)****Application: Modeling behavior change dynamics**

•

•

•

- 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 $\theta$.

- 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.

- 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).

Noel, Y. (2022, July). *A dynamical framework for the derivation of cumulative response models*. 87th
International Meeting of the Psychometric Society, Bologna, Italy.

$$\begin{cases}
\color{#0587c4}{f_p(\theta)}&=1-\frac{1}{1+\exp\left[\alpha\theta+C\right]}\\
\color{#D54100}{f_c(\theta)}&=1-\frac{D}{\left[1+\exp\left[\alpha(\theta-\delta)\right]\right]^{\frac{\beta}{\alpha}}}
\end{cases}$$

*A dynamical framework for the derivation of cumulative response models*. 87th
International Meeting of the Psychometric Society, Bologna, Italy.

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

- 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.