Estimation of Separable Representations in Psychophysical Experiments

Estimation of Separable Representations in Psychophysical Experiments Michele Bernasconi (mbernasconi@eco.uninsubria.it) Christine Choirat (cchoirat@eco.uninsubria.it) Raffaello Seri (rseri@eco.uninsubria.it) Universit degli Studi dell’Insubria; Dipartimento di Economia; Via Ravasi 2 Varese, 21100 (VA) Italy W r for any real r); this will lead us to impose a restric- tion in our empirical investigation. Abstract Studying how individuals compare two given quantita- tive stimuli, say d 1 and d 2 , is a fundamental problem. One very common way to address it is through ratio magnitude estimation, that is to ask individuals not to give values to d 1 and d 2 but rather to give their es- timates of the ratio p = d 1 /d 2 . Several psychophysi- cal theories (the most known being Stevens’ power-law) claim that this ratio cannot be known directly and that there are cognitive distortions on the apprehension of the different quantities. These theories result in the so- called separable representations (which include Stevens’ model). In this paper, we propose a general statisti- cal framework that allows for testing in a rigorous way whether the separable representation theory is grounded or not. We conclude in favour of it, but strongly reject Stevens’ model. As a byproduct, we provide estimates of the psychometric functions of interest. Keywords: Psychophysical experiments; model; separable representation. Several relations of this kind have been proposed in the literature. The original Stevens’ model reduces simply to the case in which ψ and W are power functions (see e.g. Luce, 2001a, Section “Relations among the prop- erties”). Remark that, as explained above, this case is undistinguishable from the case in which W is the identity function so that Stevens’ model can be recov- ered as ψ (d) = d κ and W (p) = p (or alternatively as ψ (d) = d and W (p) = p κ ; this second formulation will be used for our empirical investigation because of its computational advantages). When criticizing what he calls Stevens’s Assumptions (Narens, 1996, p. 109), Narens (1996) states (pp. 110-111) that the fact that W (p) = p seems to be “anything more than a coinci- dence”. Steven’s Starting from the representation (1), Luce (2001b) proposes a function of W based on hypotheses similar to Prelec’s (1998) ones: The Theoretical Framework We propose to a subject two stimuli d 1 and d 2 out of a set of stimuli D and we ask him to state in what proportion p they are with respect to each other. According to Stevens’ (1946) psychophysical framework, we can see this as a problem of magnitude estimation. 1 We say that a separable representation holds if a psychophysical function ψ and a subjective weighting function W exist such that the ratio p is in the following relation with d 1 and d 2 : ψ (d 1 ) = W (p) . ψ (d 2 ) W (p) = exp [−ρ h (− ln p) i ] exp ρ (ln p) p ∈ ]0, 1] p ∈ ]1, ∞[ Other examples of functions W are derived in Prelec (1998) and discussed in Luce (2001b). Luce (2002, pp. 526-528) proposes some other forms for ψ and W . The objective of this paper is to test whether Stevens’ power law model is appropriate or wether the separable representation holds in a simple ratio magnitude estima- tion experiment. Several papers have dealt with tests of behavioral properties (Ellermeier and Faulhammer, 2000, Zimmer, Luce and Ellermeier, 2001, Steingrims- son and Luce, 2003a, 2003b, based on the framework of Luce, 2002, 2004) or tests of particular functional forms (Hollands and Dyre, 2002). However, it seems to us that this is the first time in the literature that a formal and direct test of the formula (1) is conducted. Equation (1) corresponds to Narens’ (1996) model apart from the fact that he does not necessarily suppose that p is a number (but a numeral). Usually the equality W (1) = 1 is supposed to hold, that is the individu- als are able to correctly estimate ratios of equal stimuli; moreover, ψ is defined up to a multiplicative constant so that we can suppose that ψ (1) = 1. At last, it is easily seen that both ψ and W are defined up to a power transformation (i.e., if ψ and W are functions for which a separable representation holds, then so are also ψ r and In the next Section we describe an experiment of ratio magnitude estimation involving 69 subjects. Then, we will show how the functions W (·) and ψ (·) can be es- timated nonparametrically using polynomial regression and how the representation (1) can be tested. At last, the results are described. Sometimes magnitude estimation denotes the case in which a stimulus d 1 and a ratio p are known and the sub- ject has to choose another stimulus d 2 such that the ratio of the two stimuli is given by p, even if this should be more correctly called magnitude production.