On the importance of specifying explicit response functions in preference, emotion and behavior measurement
Yvonnick Noel
LP3C, University Rennes 2, Brittany, France
17th Sensometrics Conference, Paris, June 4th, 2024

http://yvonnick.noel.free.fr/papiers/respfunc
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Noel, Y. (2015, September). Five factors but one dimension: An alternative view at the Big Five Factor model of personality. International Conference of the AFERTP (Association Francophone d'Etude et de Recherche sur les Troubles de la Personnalité), Tours; université François Rabelais.
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Fan, J., Guo, J., & Zheng, S. (2020). Estimating Number of Factors by Adjusted Eigenvalues Thresholding. Journal of the American Statistical Association, 117 (538), 852–861.
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$$E(X)=\alpha(\theta\delta)$$
where $\theta$ and $\delta$ are unknown person (state or attitude) and item (threshold or mean) parameters, and $\alpha$ a scale (or loading) parameter.
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Noel, Y., & Dauvier, B. (2007). A Beta Item Response Model for Continuous Bounded Responses. Applied Psychological Measurement, 31(1), 47–73.
Plot of principal component vectors as a function of the true $\theta$ values, from both the observed data (dots) and the true expected responses functions (lines).
Dony, R. (2000). KarhunenLoève transform. In K. R. Rao & P. Yip (Eds.), The transform and data compression handbook. Boca Raton, FL, USA: CRC Press, Inc.
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\[\begin{cases} \frac{dx}{d\theta}=\alpha x(\theta)\color{#0587c4}{\gamma y(\theta\delta)}\\ \frac{dy}{d\theta}=\beta y(\theta) \end{cases}\] with $\alpha$, $\beta$ et $\gamma$ positive factors, and $\delta$ a shifting parameter.
$$ x(\theta)=\frac{\exp\left[\alpha(\theta\delta)+\lambda\right]}{\color{#0587c4}{\left\{ \exp[\beta(\theta\delta)]+1\right\} ^{\frac{\gamma}{\beta}}}+\exp\left[\alpha(\theta\delta)+\lambda\right]} $$
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.
$$\begin{cases} m_{ij}&=\exp\left[\alpha_j(\theta_i\delta_j)+\lambda_j\right] \\ n_{ij}&=\left\{ \exp\left[\beta_j(\theta_i\delta_j)\right]+1\right\}^{\frac{\gamma_j}{\beta_j}} \end{cases} $$
$$ E(X_{ij}\theta_{i})=\frac{m_{ij}}{m_{ij}+n_{ij}}=\frac{\exp\left[\alpha_{j}(\theta_{i}\delta_{j})+\lambda_{j}\right]}{\left\{ \exp\left[\beta_{j}(\theta_{i}\delta_{j})\right]+1\right\}^{\frac{\gamma_{j}}{\beta_{j}}}+\exp\left[\alpha_{j}(\theta_{i}\delta_{j})+\lambda\right]} $$
\[ E(X_{ij}\theta_{i})=\frac{\exp\left(\theta_{i}\delta_{j}+\lambda_{j}\right)}{1+\exp\left(\theta_{i}\delta_{j}+\lambda_{j}\right)} \]
(1) Noel, Y. and Dauvier, B. (2007). A beta item response model for continuous bounded responses, Applied Psychological Measurement, 31(1), 4773.
(1) Noel, Y. and Dauvier, B. (2007). A beta item response model for continuous bounded responses, Applied Psychological Measurement, 31(1), 4773.
(2) Noel, Y. (2014). A beta unfolding model for continuous bounded responses, Psychometrika, 79(4), 647674.
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Noel, Y. (2014). A beta unfolding model for continuous bounded responses, Psychometrika, 79(4), 647674.
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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) 

01  Social liberation (Perception of nonsmokers' behavior) 

02  Environmental reevaluation (reassess impact on environment) 

03  Emotional relief (express negative feelings) 

04  Consciousness raising (taking information on quitting smoking) 

05  Sefreevaluation (reassess one's behavior and values) 

06  Selfliberation (Decision, will power) 

07  Stimulus control (Remove any cue or incentive) 

08  Helping relationships (Rely on significant others) 

09  Reinforcement management (Find alternative sources of satisfaction) 

10  Counterconditioning (Find replacements and substitutes) 
(DiClemente and Prochaska, 1982; DiClemente and Prochaska, 1985; DiClemente et al., 1991; Prochaska et al., 1988)
Noel, Y. (1999). Recovering Latent Unimodal Patterns of Change by Unfolding Analysis : Application to Smoking Cessation. Psychological Methods, 4(2), 173191.
Noël, Y., Molimard, R. & Martin, C. (2004, October). A longitudinal study in Schools of Nursing. 19th Meeting of the French Tobbacology Society, Paris.
Note: Figures are median locations by stage.
Thank you for your attention.
yvonnick.noel@univrennes2.fr
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