Monotonicity, convexity, and other qualitative psychophysical laws

JOURNAL

OF MATHEMATICAL

PSYCHOLOGY

31, 113-134

Monotonicity, Other Qualitative

(1987)

Convexity, and Psychophysical Laws

ERNEST W. ADAMS University

qf Callyornia,

Berkele)

AND

ROBERT F. FAGOT Universily

of Oregon

General methods are described for directly testing qualitative psychophysical laws, such as monotonicity and conuexily. These methods consist of estimating lower and higher orders of derivatives and procedures for verifying qualitative laws involving these derivatives. The quantitative power law hypothesis is compared for goodness of tit with the qualitative hypothesis that the psychophysical function is monotone and convex, in application to bisection brightness data. Other topics include discussion of a possible topological formulation of the distinction between qualitative and quantitative laws; the search for psychophysical invariants; and the failure of qualitative laws such as convexity to conform to Lute’s invariance-theoretic principles of theory construction. cr) 1987 Academic Press, Inc.

1. INTRODUCTION

The psychophysical laws that we call qualitative are described by inequalities involving psychophysical functions and their derivatives, in contrast to laws describable by equations involving these quantities (for present purposes it is not important to make this distinction precise, though Section 5.1 comments on a topological formulation). Monotonicity, described by an inequality in a psychophysical function’s first derivative, is an example of a qualitative psychophysical law, while the Power Law Hypothesis, describable by Plateau’s equation (Miller, 1964 and Plateau, 1874) is an example of a quantitative psychophysical law. That Monotonicity is “more qualitative” than the Power Law suggests that the former is

Request for reprints should be sent to either of the authors: Ernest W. Adams, Department of Philosophy, University of California, Berkeley, CA 94720 or Robert F. Fagot, Department of Psychology, University of Oregon, Eugene, OR 97403-1227. We are greatly indebted to Professor William Bade, University of California, Berkeley, for help in regard to topological formulations and problems commented on in Section 5.1. We are also indebted to J. T. Townsend, Editor, and anonymous reviewers for very helpful criticisms and suggestions.

113 0022-2496/87

$3.00

Copyright 0 1987 by Academtc Press, Ipc. All rights of reproductmn in any form reserved.

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less informative than the latter, one indication of which is that while solutions to Plateau’s equation can be described by two real parameters, that is not possible with Monotonicity and other qualitative laws. There are several reasons for studying qualitative psychophysical laws. One is that while monotonicity is the only commonly recognized qualitative law, it is by no means the only interesting one, and furthermore these laws have explanatory and predictive power which is worth examining. A case in point that will occupy much of our attention in the sections to follow is the Convexity Law: that psychophysical functions are upwardly or downwardly convex (it will be more convenient here to speak of upwards and downwards convexity rather than to use the more customary terms “convex” and “concave”), or equivalently that their second derivatives do not change sign.’ We will show in Section 3 that there are simple and natural tests for convexity and similar laws that do not require the construction of the psychophysical scale as a precondition of application, and in Section 4 we will see that the combination of the convexity and monotonicity laws has enough predictive power to be a serious competitor to the Power Law Hypothesis when the hypotheses are compared for predictive success in application to certain psychophysical data. A second topic of potential interest has to do with the fact that qualitative properties of psychophysical functions such as monotonicity and convexity are local (e.g., psychophysical functions can be monotone increasing and upwardly convex in certain intervals but not in others) and this suggests the fruitfulness of studying them locally. For instance, experimental study may establish that a psychophysical function is convex up (has positive second derivative) in one interval and convex down in another, with a point of inflection between them, and this in turn invites explanation in terms of local properties of the subjective scale (e.g., that it approaches sensory saturation above a certain level). This is related in turn to j.n.d.s. and to the methodological principle that determinations of small differences provide the basis for estimates of “smallest differences” (i.e., for estimates of derivatives) and these give grounds for qualitative generalizations concerning these differences. Aspects of this intuitive idea will be made precise in Section 3, where we will describe methods for estimating derivatives and operational procedures for verifying qualitative laws involving these derivatives. Since the power function (Stevens, 1957) appears to be firmly established as a (quantitative) psychophysical law, one might question the need for study of weaker qualitative laws. Apart from the intrinsic interest of the laws discussed above, it 1 Anderson (1977) is the only previous writer that we know of to consider convexity as a significant psychophysical assumption or law, though his point of view differs significantly from ours. He shows that his data on bisection of lengths is inconsistent with a generalized bisection scaling model (cf. Adams & Fagot, 1975), together with the assumption that the psychophysical function is convex. Given Anderson’s assumption that “The convexity assumption is hardly disputable,” he is led to reject the bisection model. From our point of view, according to which psychological scale values R(x) are “givens,” it is the convexity assumption that should be given up, at least assuming that the internal consistency assumptions of the bisection model are satisfied.

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should be recognized that the application of the Power Law to some psychophysical data raises many questions that have not yet been satisfactorily answered. One is that the Power Law only seems to be well established for ratio scaling methods (Ekman & Sjoberg, 1965), although Marks (1968) and Stevens (1971) have suggested that it holds for category scaling as well. It is also well established that the exponent of the power function, which can be interpreted as a measure of perceptual sensitivity, varies widely under a variety of experimental conditions (Poulton, 1968). Thus, the exponent varies with scaling method (Marks, 1974) the range and spacing of stimuli (Jones & Woskow, 1962; Poulton, 1968; Teghtsoonian, 1973), and standard and modulus (Beck & Shaw, 1961, 1965; Engen & Levy, 1955; Hellman & Zwisloki, 1961; Macmillan, Moschetto, Bialostozky & Engel, 1974; Stevens & Tulving, 1957; Stevens, 1956). There are also large individual differences in exponents for the same modality (e.g., McGill, 1960). A range of exponents of about 2 : 1 has been reported for heaviness, roughness, and area (Marks & Stevens, 1966), and also for loudness (Stevens & Guirao, 1964). Finally, Birnbaum and Elmasian (1977) concluded that their loudness data were incompatible with a power function, even under the assumption of a ratio model. Taken together the foregoing results invite consideration of “weaker” qualitative laws, and particularly monotonicity and convexity, both to study their “intrinsic empirical content,” and to study their relationship to the Power Law. When the Power Law gives a poor fit to data, convexity and monotonicity may lit and have significant predictive content. As Anderson (1977, p. 217) has put it: “The convexity assumption, which includes the log and power functions as special cases, hardly seems disputable.” Even when the Power Law fits, the question arises as to how much of its goodness of fit is due just to its being a special case of convexity and monotonicity. This question will be discussed in Section 4, where we will cite a possibly surprising finding. In the bisection of two intervals of brightness, the monotonicity-convexity combination gives better predictions than the Power Law in one case while the Power Law predicts better in the other. The remaining sections of this paper deal with the following topics. Section 2 gives formal definitions of psychophysical systems, functions, and laws, and characterizes certain laws that will concern us in what follows. Section 3 describes the procedure alluded to above for estimating higher and lower orders of derivatives of psychophysical functions. Section 4 discusses predictions based on the Monotonicity and Convexity Laws and their comparison with the Power Law, in application to data obtained in an experiment on the bisection of brightness intervals. Section 5 makes informal remarks on open problems and other issues connected with this study, including a possible topological formulation (Sect. 5.1), estimation and error (Sect. 5.2), the search for psychophysical invariants (Sect. 5.3) and the failure of qualitative laws such as Convexity to conform to certain invariance-theoretic principles of theory construction due to R. D. Lute (Sect. 5.4). Before starting we should emphasize that our aim is only to suggest that it may be worthwhile to study psychophysical laws from the qualitative point of view, and

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not to argue for the superioity of that to quantitative methods. In fact, one can argue (cf. Sect. 5.1) that quantitative laws are “ideals” which are approached by more and more strict qualitative laws, and it may be useful to examine the steps involved in this limiting process. In a sense this transfers the outlook of Fundamental Measurement Theory to psychophysics, which proceeds towards the quantitative by steps leading from the qualitative. 2. PSYCHOPHYSICAL SYSTEMS, FUNCTIONS,

AND LAWS

A psychophysical system will be an ordered triple, (A, S, R), in which A is a domain of stimuli, S is a real-valued function giving the measures of the stimuli in A on some physical scale, and R is another real-valued function over this domain giving the measures of its elements on some psychological scale. As will be commented on further below, we will here take S and R to be “givens” without inquiring into the manner in which the scale values S(x) and R(x) are measured or what theoretical models may be presupposed. Also, it will be important that A can be a finite set of stimuli, rather than the range of all “potential” stimuli in some modality. A psychophysical function consistent with (A, S, R) will be a real-valued function, f, whose range includes all values S(X) for x in A, and such that f(S(x)) = R(x) for all x in A, and a psychophysical law will be a constraint, L, either directly on psychophysical functions or indirectly on the psychophysical systems with which these functions are consistent. The class of psychophysical functions that satisfy constraint L will be denotedf(L), and the class of all psychophysical systems with which these functions are consistent will be denoted s(L). We will say that the function f satisfies law L if it is a member off(L), and that the system (A, S, R) satisfies L if it belongs to s(L). (A, S, R) belongs to s(L) if and only if there exists a psychophysical function, f, consistent with (A, S, R) belonging to f(L). Henceforth we will use letters x, y, etc., as variables over the stimulus domain, A, and use Greek letters cp, $, etc., as real variables. cp will usually denote a value on the physical scale and II/ the corresponding value on the psychological scale, while a, 8, etc., will denote values on either. We will assume that all psychophysical functions, f, are arbitrarily differentiable (hence when A is finite, the values S(x) will be a proper subset of the range off) and we will writef”’ for the rth derivative of J Our initial formulations of psychophysical laws will be in terms of these derivatives. This has the advantage of perspicuity, but it has the disadvantage that these are not directly transformable into system constraints which are susceptible to direct experimental test. The problem of finding integrated forms of psychophysical function constraints, which can be directly translated into system constraints, will be considered in the following section. There are three observations to make about our mathematical formulation. The first point has to do with the definition of R(x), and with the requirement that it should be exactly equal to f(S(x)). As noted, we are taking R(x) to be a given without inquiring into the scaling method used to determine it. For this reason we

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ignore the possibility that R(x) may itself be inferred from S(x) together with assumptions about the form of the psychophysical function. That is not to say that we regard this as the “right” point of view in psychophysics, but it has the advantage of simplicity and it can serve as the point of departure in developing modifications needed to accommodate complications involved in the construction of the psychological scale. The same point is to be made about the possibility of error, which would lead to weakening the requirement that R(x) should exactly equal f(S(x)), though some unsystematic remarks will be made concerning error variances in Section 5.2. The second point is that our approach would also have to be generalized in order to be able to accommodate laws such as those considered by Shepard (1981), in which R is a function of pairs or configurations rather than of single stimuli. We believe that our approach can be extended to appy to these generalizations, but we prefer to illustrate our methods in application to the simpler “classical” formulations. Finally, we must stress the distinction between psychophysical functions and psychophysical laws. It is not uncommon to regard psychophysical functions as laws (Lute, 1959, e.g., does not make the distinction). In the present formulation particular psychophysical functions must be regarded as psychophysical laws of a special kind: essentially as constraints that are satisfied by just one function. But that is the exceptional case and it is much more common for a law, such as monotonicity or convexity, to be satisfied by functions of a large class. We conclude this section by formulating the Monotonicity, Convexity, and Power Laws and generalizations in terms of constraints on the derivatives of psychophysical functions. An integrated form of the Power Law will also be given, while the integrated constraints corresponding to the qualitative Monotonicity and Convexity laws and certain generalizations will be given in Section 3. The Strict Increasing Monotonicity constraint is simply that f”’ must be strictly positive throughout its range. Strict Decreasing Monotonicity requires f(l) to be strictly negative, and Weak Increasing or Decreasing Monotonicity replaces strict positiveness and negativeness by weak positiveness and negativeness, respectively. For illustrative purposes we will confine attention mostly to Strict Increasing Monotonicity. Differential forms of the Convexity Laws simply replace first by second derivatives in the corresponding Monotonicity laws. Thus, Strict Upwards Convexity is the constraint that f (2’ should be strictly positive over its range of definition, Strict Downwards Convexity requires that f”’ should be strictly negative, and Weak Upwards and Downwards Convexity replace strict positiveness and negativeness by weak positiveness and negativeness. It is useful to formulate the Power Law together with its Limiting exponential and logarithmic cases as a third-order differential equation, which is essentially a generalization of Plateau’s Equation (Miller, 1964; Plateau, 1872). It should be stressed that while this equation may be of interest in its own right, suggesting the desirability of investigating possible psychological or neurological bases for it along

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the lines of Falmagne’s work on the Weber-Fechner Law (Falmagne, 1974, 1985); see also Baird and Noma (1978, Chap. 4), such a study lies beyond the scope of this paper (though some negative comments in Section 5.1 about the possibility of approximating this quantitative law by qualitative ones bear on the matter). Our present purpose in formulating the equation is simply to have an example of a well known quantitative law to illustrate our methodology of comparing such laws with qualitative laws for predictive success. The generalization of Plateau’s Equation we consider is

This is trivially equivalent to the form [ln(f(2))]“’ = k[ln(f”‘)]“‘, and except in the limiting cases k = 1 or k = 2 it simply says if the origin of the coordinate system is translated so that the inner and outer threshold parameters in Eq. (3) become equal to 0, then the log-log plot off is a straight line with slope 0 = (2 - k)/( 1 - k). The general solution to (1) is easily calculated, and it is given below in a form which shows how the “singular” exponential and logarithmic forms, (2b) and (2c), can be arrived at as limiting cases of the non-singular Power Law form (2a). (1 +(1-k)(acp+fi))‘2-“““-k’-1 (2-k) f(p)=

ex”+,“-

l +y

ln(1 -&p-j?) I Limiting

+y

-LY

+Y

for

k # 1 and k # 2

(2a)

(k-t

1)

(2b)

(k --f 2).

P)

case (2b) is derived from (2a) using the fact that (1 +(l -k)(ctcp+/?))““pk’

approaches ecarp+ PI as k approaches 1, and case (2~) is derived using 1’Hospital’s rule, by differentiating the numerator and denominator of (2a) with respect to k and then setting k = 2. Replacing the exponent (2 - k)/( 1 -k) in (2a) by 0 and making other obvious substitutions leads to a more familiar form of the Power Law

f(v) = a’(cp+ B’)” + Y’,

(3)

where 8’ and y’ are the inner and outer threshold parameters. Setting both equal to 0 yields the most commonly assumed form, while it is a controversial matter whether minimal sensory thresholds are best represented by non-zero values of 8’ or of y’ (Fagot, 1975; Marks & Stevens, 1968). Clearly the two threshold representations are not equivalent.

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With or without threshold parameters it is easy to see that all functions satisfying (2a)-(2c) are at least weakly monotone and convex, and therefore all solutions to (1) have these properties. Thus we can say that the Power Law is a special case of monotonicity and convexity. We can go further and assert that no solutions to (1) have derivatives of any positive order that change sign, which leads us to ask how nearly definitive this constant sign property is of this class of functions. This will be returned to in Section 5.1, where we will see that still further qualitative conditions must be imposed. That the Power Law constraint is strictly stronger than the monotonicity and convexity constraints suggests the desirability of considering how much of its predictive success is due just to being a special case of these two laws, and how much is due to the additional quantitative constraints it imposes. Section 4 will describe an experiment designed to investigate this. Finally, focussing on the Power Law as formulated in (3), it is useful to note how this integrated form can be translated into a psychophysical system constraint. This is accomplished simply by substituting S(x) for the physical scale value cp on the right, and substituting R(x) for the subjective scale value $ = f(cp) on the left, thus, R(x) = a’(S(x) + j?‘)” + y’.

(4)

It is the direct translatability of function constraints into system constraints in integrated forms of psychophysical laws that gives these forms their special significance, since the scale values S(x) and R(x) are subject to the most immediate observational determination. The next section shows how integrated forms may be obtained for qualitative laws like monotonicity and convexity, which permits them to be translated in turn into system constraints suitable for direct experimental test.

3. INTEGRATED

FORMS OF QUALITATIVE

LAWS

Let us now concentrate on the case in which the stimulus domain, A, contains just n elements, x, ,..., x,, which are ranged in increasing order on the physical continuum: i.e., such that S(x,) < . . . < S(x,). These determine a sequence of n data points, ( qpl, 11/,) = (S(X,), R(x,)), for i = l,..., n, which might be plotted graphically. Now focus on an r + 1 term consecutive subsequence of these data points, ( cpl, +, ), for i = m,..., m + r. There is a unique rth order polynomial containing these points which can be written (with a slight abuse of notation) ~,d(~)=a~,~

+b,.,cp + ... +c,,d.

The rth derivative of P&Y) has the constant value C,,, = r! c,.,, and this can be used as an estimate off”’ for any psychophysical function, f, that is consistent with those data. This follows from an easy generalization of the Mean Value Theorem (Courant, 1937, p. 102) which says that iffhas derivatives of arbitrary order and it passes through the given points then there must be some cp such that

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(PWZG(PG(Pt?l+r and f”‘(cp) = C . Small enough intervals [q,, (Pi+ ,] must pinch in the value 40 for which fcrTk) equals C,, and the succession of these rth derivative estimators gives considerable information about monotoninicty, convexity, and other psychophysical properties that are described in terms of derivatives. Given the data points (cp,, tii), for i = m ,..., m + r, the values of the coefficients in the polynomial P,,,~, and therefore c~,~ and C,,,, can be calculated readily. The following determinant form gives a useful representation of the latter,

(5)

Setting r = 1 in the above yields first derivative estimators, which are simply the slopes of the segments connecting the data points ((P,,,, $,) and ( qrn + , , I,,+, + , ), C tn.1 = II/ m+l -$m = m%z+,)-w,) (Pm+1 -Pm S(x m+1)-S(x,) That the estimators C,,, should be strictly positive is a necessary and sufficient condition for the system (A, S, R) to satisfy the Strict Increasing Monotonicity Law, and hence (6) gives us an integrated form of that law. Similarly, that the values C,,, should be strictly negative is a necessary and sufficient condition for (A, S, R) to satisfy the Strict Decreasing Monotonicity Law, while weak positiveness or negativeness of the C,,, are the conditions for (A, S, R) to satisfy the Weak Increasing or Decreasing Monotonicity Laws. Conditions for satisfying the different convexity laws may be stated in terms of the properties of the second derivative estimators Cm.2, and these are also derivable from (5). Applying that in the case r = 2 and making a substitution from (6) leads to the following, which shows an important relation between first and second derivative estimators C m,2 = qm+22pq,

(Cm+,,,

-Cm.,).

(7)

For (A, S, R) to satisfy Strict Upwards Convexity it is necessary and sufficient that C,,, should be strictly positive, which (7) shows is equivalent to the condition that the first derivative estimators should be strictly increasing. This is to be expec-

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ted of course, and we may say generally that for any r the necessary and sufficient condition for the r + 1 derivative estimators to be strictly positive is that the rth derivative estimators should be strictly increasing. Similarly the condition for the c ,n,r + 1 to be strictly negative is that the C,,, should strictly decrease, with analogous conditions holding for weak positiveness and negativeness. Of course (7) tells us not only whether the estimators Cm,* are positive or negative, but it gives us exact values as well, which would in turn permit us to say whether these estimators increase or decrease, and hence whether the third derivative estimators are positive or negative. We conclude this section with comments on the special cases in which the psychological scale values $i increase either arithmetically or geometrically. Arithmetic increase can be assumed when subjective values are obtained by an equisection method such as bisection (Torgerson, 1958, Chap. 6), to be returned to in Section 4, and geometric increase can be assumed when these values are obtained by fractionation methods (Torgerson, 1958, Chap. 5). In these cases the values of the estimators and hence convexity properties can be determined without prior construction of the psychological scale. In the equal-interval case we can assume that the values tii are equal to $, + ie, for some constant value E. Assuming that the tii are measured on an interval scale a zero and unit can be chosen arbitrarily, and we can set I,+~= 0 and set E equal to + 1 or -1, according as it is positive or negative. Substituting these values in the right-hand column in the determinant in the numerator of (5) we get rth derivative estimates that only involve the physical scale values cp,,..., (P~+~. Setting r = 1 we get

Cm.1

& = (Pm+1

-(Pm’

That the values C,+r should be strictly positive is equivalent to Strict Increasing Monotonicity, and the necessary and sufficient condition for that is that E should be positive: i.e., that the psychological scale values increase-which is obvious. Assuming Increasing Monotonicity, for the values C,,, to be strictly increasing, which is equivalent to Strict Upwards Convexity, it is necessary and sufficient that the succession of differences (P,,,+ 1 -(P,,, should be strictly decreasing, and for Strict Downwards Convexity the succession of differences should be strictly increasing. An empirical example of this will be discussed in the following section. Obviously local properties of the second derivative can also be studied, and these are reflected in rates of increase or decrease of these successive differences. When psychological scale values increase geometrically we can set ll/; = R't+b 1, where R is the ratio of each stimulus value $[ to the preceeding value $j- r. R is generally a “given,” since the subject is told to choose a stimulus that stands in a given ratio to the previous one. Assuming that these values are on a ratio scale, $, can be chosen arbitrarily, say as I,+~= 1. Again, these values can be substituted into the numerator of (5) and the result gives rth derivative estimates that involve only

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the physical scale values (P,,,,..., (P,,,+,. For instance, setting r = 1 and evaluating gives R" C RI.1= (Pm+1

-

rpm’

Assuming Increasing Monotonicity, the condition for Upwards Convexity is that the values of these C,,, should be increasing and the condition for Downwards Convexity is that they should be decreasing. Again, one can study local properties of the psychophysical function in terms of the succession of C,,,, but that cannot be pursued here. A final application of (5) will be be commented on briefly in Section 5.2, having to do with the estimation of variances in the estimators C,,, when subjective scale determinations are subject to independent errors. For now we will continue to ignore errors, and we will next describe a simple pilot experiment that has been carried out to compare the combination of the Monotonicity and Convexity Laws with the Power Law in application to data obtained in the bisection scaling of brightnesses.

4.

COMPARISON

OF THE CONVEXITY

AND POWER LAWS

The comparison we will focus on is that between the Convexity Law combined with Increasing Monotonicity (call this combination C), and the Power Law with positive exponent 0 (Eq. 3 with 8 > 0; call this P). Thus formulated, P is a special case of C, since all psychophysical functions that satisfy the Power Law with positive exponent are strictly monotone increasing and convex. The general question we are concerned with was raised in section 1: how much of the goodness of lit of law P in application to psychophysical data is due just to its being a special case of law C? Serious and as yet unresolved problems arise in devising appropriate measures of comparative goodness-of-fit, not the least of of which is that of specifying what it means for a convex curve to give a “closest lit” to data assumed to involve experimental error.’ We will largely side-step these problems here, and con’ If we generalize to all functions or to all continuous ones, rather than to ones with all orders of derivatives, and we define the “closest fitting” monotone or convex curve to a finite set of data points to be the one such that the total of the vertical distances from the points to the curve is a minimum, then elementary linear programming methods lead immediately to the following. The closest fitting monotone curve must always be a monotone step-function passing through a “maximum” number of data points, and the closest fitting convex curve will be part of the boundary of a convex polyhedron which passes through a maximum number of these points. Isotonic regression (cf. Barlow, Bartholomew, Bremner, and Brunk, 1972) and spline functions (cf. Greville, 1969) are other possible testing approaches. Isotonic regression deals with statistical inference under order conditions, which could yield procedures for evaluating our qualitative models. However, we know of no results in this literature which bear directly on our problems. Splines are like convex polyhedra boundaries in being “pieced-together” polynomials, and at the points at which they are pieced together they only have derivatives of orders one less than that of these polynomials. They are also useful for curve fitting, but they suffer from the same conceptual drawback as the polyhedra boundaries in not being infinitely differentiable. Of course, we recognize that in making these comments we have only taken a first step toward statistical solutions to the testing problem.

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fine ourselves to unsystematic comments that concern a particular comparison in which many of the problems do not arise and in which errors can be neglected. This comparison has to do with five stimuli, x1 - x5, that are equally spaced and in increasing order on the subjective scale, where subjective equal spacing is determined by bisection. Here the subject is given the two end-points, x1 and x5, and is first asked to locate a stimulus, xg, that bisects the interval between them. Having located x3 the subject is then asked to locate the two further stimuli x2 and x4 which bisect the subintervals (x1, xg ) and (x,, x,), respectively. Given the equal subjective spacing of x1 -x5, laws P and C both predict the physical scale values, (p2 = S(x,) and (p4 =S(x,) of x2 and x4, respectively, once initial physical scale values q1 = S(x, ), cp3 = S(x,), and (p5 = S(x,) are determined. The parameters in the Power Law can be uniquely determined from the values cp, , (p3, and (pS, and hence that law yields unique predictions of (p2 and (p4. In the Convexity case it is only predicted that (p2 and (p4 should lie within certain intervals, to be called the Convexity Innrervals for these stimuli. Nonetheless it is possible to ask how much closer the Power Law’s prediction is to the observed value than an arbitrary prediction within the Convexity Interval would be, and that affords one measure of the relative predictive accuracy of the two laws. Details will be described in a moment, but first it may be helpful to give a graphical representation of the Convexity Intervals in the predictions of ‘pz and (p4, assuming the downwards convexity which is found in the pilot experiment we will comment on. Figure 1 gives physical scale values on the abscissa and psychological ones on the ordinate, and the scale values for the three initial stimuli are represented by points with coordinates (cp,, G,>, CR, ti3), and ( (p5, $5 ). The psychological values, II/, , $3, and 11/s,are equally spaced, and the three points lie on a curve that is monotone increasing and downwardly convex. These points determine two triangles that are shown shaded in the figure, which outline the regions in which any monotone

$=R(x)

q$ ----------_----q$ -------------J:-----------I I % ------4------

1.

Construction

of convexity

interval

for bisection.

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increasing and downwardly convex curve that passes through these points must lie (thus, any power function that passes through the three points must lie inside these regions). If the psychological scale values 1,9~and ti4 of the stimuli to be predicted lie midway between Ic/, and ti3 and midway between I++~and $5, respectively, then the points ((p2, ti2) and (q4, $4) must lie along the horizontal segments bounded by the triangles, which are drawn as heavy lines in the figure. The physical scale values corresponding to these segments are the Convexity Intervals of (p2 and (p4, respectively, hence law C predicts that (p2 and (p4 must lie within these intervals. Symbolizing the Convexity Intervals of cpZand (p4 as CI((p,) and CI((p,), respectively, we can write

and CI(q,)

= [ 1.5q, -0.547,,

0.5cp, + 0.5cpJ.

Note that the left endpoint of CI((p,) is the maximum of the two values, (p, and 1.5~~ -0.5(ps. This is due to the fact that the horizontal midline of the lower triangle at height tiZ could intersect either the vertical side of the triangle or the upper sloping side. If the curve had happened to be convex upwards instead of downwards the lower endpoint of CI((p,) would have been unique, but the upper endpoint of CI((p,) would have been a minimum of two values. In any case, law C predicts that the observed values of q2 and (p4 must lie within the intervals CI((p,) and CI((p,), respectively. We now want to consider how well the Power Law, which makes unique predictions in the Convexity Intervals, compares with other possible predictions that could be made within the intervals. There are various ways of making this comparison, but the one on which we have experimental data is perhaps the simplest, though it is admittedly arbitrary. We simply extend the Convexity Law by adding the decision rule that selects the midpoint of the Convexity Interval as the “Convexity Law prediction,” which can then be compared for accuracy with the Power Law prediction. Details can be briefly summarized. To obtain Power Law predictions of (p2 and (p4 it is first necessary to estimate the exponent 0 in Eq (3) from (pr, (p3, and cps, given the assumed equal spacing of x, , x3, and x5 on the subjective scale. 8 is calculated by solving the equation:

cp;= 0.5(4 + cp;,. Given this, the Power Law predictions

(8)

of (pi, for i = 2,4, are (9)

QUALITATIVE

Convexity given by

Law predictions,

PSYCHOPHYSICAL

using the midpoints

pzj2 = max(0.75p,

125

LAWS

of the Convexity

+ 0.25~,, cpJ + 0.25~, - 0.25~~)

Intervals,

are

(10)

and &j4 = ‘p3 - 0.25~~ + 0.25~~.

(11)

Data from Peterson (1964) on the bisection of brightness intervals were used in the analysis. In this case the initial stimuli x1 and xg had physical intensities of cpr =0.5 and cps = 560 foot-lamberts, respectively. Each of six subjects then repeatedly estimated the bisectors x2, x3, and xq according to the method outlined above, first bisecting the interval (x,, xS) and then bisecting the subintervals (x,, x3) and (x,, x5). Each subject repeated this process 30 times. As a preliminary, the subjects’ responses were plotted graphically assuming equal spacing on the subjective scale, in order to ascertain the presence or absence of inflection points. Without exception it was found that the plotted data points fell on monotone increasing and strictly downwardly convex curves, and hence the data conformed exactly to Law C. Thus the problem of estimating convex curves giving closest fits to the data did not arise. For the purpose of comparing the Power Law prediction with the Convexity Interval midpoint prediction, the values cp,, cp3, and ‘pS were used to compute both the Power Law predictions p@z and p@d and the Convexity interval midpoint predictions ciz and c$j4 according to (S)(ll). The mean squared error of the Power Law prediction, p Vi, (computed as the mean square of the differences between the predicted value and the 30 bisection judgements made by a subject on the given interval) was then compared with the mean squared error of the Convexity Interval midpoint prediction, c Vi, by simply taking the ratios p I’,/= I’, for i = 2,4. These ratios should be less than 1 for the Power Law to yield better predictions than the arbitrary prediction of the midpoint of the Convexity Interval. Results are given in Table 1.

TABLE Ratios Subject 1 2 3 4 5 6

of mean squared

1 errors

p VJc V,

i=2

i=4

1.117 1.783 3.322 3.650 0.221 1.126

0.132 0.255 0.118 0.346 0.995 0.998

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What these results seem to suggest is that the superiority of the Power Law to convexity and monotonicity is itself local, the Power Law yielding markedly better predictions of (p4 than the midpoint of the Convexity Interval for four of the six subjects, but yielding markedly worse predictions of (p2 for three subjects, with no appreciable difference in the remaining cases. Of course these results are extremely partial and they are only meant to be suggestive, both of the direction that experimental studies might take and of the sorts of results that one might look for. However, this is a matter that cannot be pursued here, and we conclude with unsystematic comments on various issues raised by the present study.

5. OPEN PROBLEMS AND NEW TOPICS

5.1. Topology and the Convergence of Laws We will comment briefly on work in progress having to do with a topological approach to giving a precise characterization of our informal distinction between qualitative and quantitative psychophysical laws. This involves introducing a suitable topology on the space of possible psychophysical functions, which can be illustrated in the artifically simplified case in which we are just concerned with functions with continuous derivatives fci’ for i= l,..., r, defined only on a closed interval, which can for convenience be chosen to be the unit interval [0, 11. Generalizations are possible, but they are too complicated to be considered here. This class of functions forms a Banach Algebra, for which the distance between functionsfand g, given by the metric:

defines a natural topology (cf. Rickart, 1960, p. 300).3 If we consider generalized functions F(;(f) = F(f, f(l),..., f”‘) in a functionfand its derivatives f” ),..., ftr) (say F(f) = f’“), the strict qualitative constraint, F(f) > 0, defines an open set in this topology (i.e., the class of functions satisfying the constraints is an open set), while the corresponding weak constraint, F(f) 2 0, defines a closed set. Interestingly, though this does not hold in general, when F(f) is linear inf” ‘,..., f”’ the closed set defined by the weak constraint is the closure of the open

’ The Editor has pointed out that this metric also determines Bore1 sets which can serve as the basis for defining probability measures over the space. This in turn might serve as the starting point for a statistical treatment of error, about which we have very little to say here (except cf. the end of Sect. 5.2). As with most of the other subjects canvassed in this section this is very much unfinished business, but Gardner and Pfeffer’s paper (1984) makes it appear that the mathematics of this development could be very difficult.

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set defined by the strict constraint.4 For instance, the class of functions f such that f(*’ 2 0 is the closure of the set of all f such that fc2)> 0: i.e., the class of weakly upwardly convex functions is the topological closure of the class of strictly upwardly convex functions. One of the interesting things about the topological approach is that it makes it possible to formulate ideas about the convergence of laws in a precise manner. For example, we can regard the quantitative law f (*)=O as the limit of qualitative laws of the form 1S”‘j O, which are the differentiable strictly monotone increasing functions.

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128 5.2. Estimation

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and Error

It would be both theoretically and empirically useful to be able to generalize the methods of Section 3 to yield estimates not only of the values of different orders of derivatives of psychophysical functions, but also those of important functions involving these derivatives. For instance, it would be very useful to be able to estimate the value of the parameter k that enters into Eq. (1). This not only determines the Power Law exponent when k is different from 1 and 2, but it also determines whether f is exponential (k = 1) or logarithmic (k = 2). However, difficulties arise in directly estimating the values of parameters like k, which we will note since they lead to important open problems. The method used for estimating f"'(q) for q in some interval, given the value of f (cp,) at r + 1 data points ‘pi for i = m,..., nz + r, was to determine the unique curve p(q) with constant value p (I) that passes through these points, and then prove that fcr) must equal p”’ at some point in the interval [q,, cp,,,.]. This follows from the fact that if both f and p pass through the same data points then f(q) - p(q) must equal zero at all of these r + 1 points, from which it follows by the Generalized Mean Value Theorem that the rth derivative of this difference must equal 0 at at least one point in the interval. Since p (I) has a constant value, fcr) must have that value at at least one point. But generalizing this, for instance to estimate the ratio k in (1) leads to the following difficulty. Given that we are attempting to evaluate an expression involving third derivatives, it is plausible to do this on the basis of values off(cpi) at four points cp,, for i= m,..., m + 3. There is no difficulty in characterizing the unique function p satisfying (1) which passes through these four points. The problem is to prove that the ratio k’ determined by p must equal the ratio k determined by f at some point in the interval [q,,, (P,,,+~]. The M ean Value Theorem tells us that for i = 1, 2, 3, fci’ = p(j) at some points in the interval, but it does not tell us that these equations hold at the same point, and therefore that k = k’ at some point in the interval. We surmise that this can be shown, but lacking the proof the problem of estimating k remains open. A question that is closely related to estimation is that of the probable errors of these estimates. Equation (5) gives information about errors in the rth derivative estimates, C, r, in the special case in which the physical scale values are exact and the psychological scale values are independently distributed with means ccli and variances 0;. According to (5), C,,,, is a linear function of the values tj,, and so the variance in this estimate is determined immediately var(C

) = c:‘=‘m’ m,r

e(m

D2

where D is the determinant in the denominator of (5) and the Dj are the minors of the coefficients II/, in the numerator determinant. The size of var(C,,,) depends largely on the sizes of the determinants D and Di. To illustrate, suppose that the physical scale values cp, are equally spaced so that

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‘pi+, = cp, + 6 for some small increment 6. Then D2 will be of the order of 6’(‘+‘) while the coefficients (Di)* will be of the order of 6”- ‘jr, and their ratio will be of the order d-2r. Roughly, the expected error of the rth derivative estimate will be the rth power of that of the first derivative estimate, and the latter will in turn be inversely proportional to the sizes of the intervals between the physical scale values on which the estimate is based. The foregoing may partially explain why we are reluctant to draw conclusions about the values of higher derivatives based on inexact measurements of small differences, though there are obviously many more issues of both a conceptual and a mathematical nature to be explored in this connection. In spite of this, however, we feel that systematic qualitatioe properties of higher derivative estimates (e.g., that they are uniformly positive or uniformly negative) may be significant, even when their numerical values are of little significance because they are subject to too much error. 5.3. Psychophysical

Invariants

Obviously the results of the pilot experiment reported on in Section 4 are merely suggestive, and there are many directions for further experimental study that might be pursued. One of these has to do with the properties of psychophysical functions that are invariant under changes of scaling variables, subjects, sensory modalities, and so on. In fact the search for qualitative invariants is a large part of the motivation for the present study, given the lack of invariance of such quantitative properties as Power Law exponent under changes in scaling variables, even where power functions give a good fit to experimental data. There are a number of experimental studies that relate to this and to Convexity in particular that may be cited in this connection. Direction of Convexity appears to be invariant in many contexts. For example, Marks ( 1974) reported Power Law exponents varying from 0.13 to 0.85 in 40 studies of loudness using 13 different scaling methods, all of which are consistent with Downwards Convexity but which are otherwise quite non-invariant. Downwards Convexity also appears to be invariant across subjects in the case of brightness scaling (Fagot & Stewart, 1968; Marks and Stevens, 1966), while Upwards Convexity is invariant across subjects for handgrip (Stevens and Mack, 1959), muscular effort (Bernyer, 1962) and saltness and sweetness (Ekman & Akesson, 1965). On the other hand the study of magnitude estimates of heaviness by Marks and Cain (1972) yielded power function exponents ranging from 0.71 to 1.48 (with a mean of 1.11) among subjects, which shows that Direction of Convexity is not invariant over subjects in the case of heaviness. The findings cited above suggest many intriguing questions of a qualitative nature. Can the invariance in the Direction of Convexity in the cases of brightness, loudness, handgrip, saltness, and sweetness (as against that of heaviness) be accounted for in neural-sensory terms? It is worth noting in this connection that Stevens (1970) argued that sensory scales directly reflect some aspect of neural activity, and he described power functions obtained from direct ratio scaling of

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these modalities in terms of “transducer function” which are of the expander type if the exponent is greater than 1 (Upwards Convexity) and of the compressor type if the exponent is less than 1 (Downwards Convexity). Given a physiological basis for these functions, and the plausible assumption that all subjects have the same kind of function--expander or compressor-for the same sense modality, we would have an immediate explanation for the invariance across subjects of Direction of Convexity for the given modalities. This would be a direct explanation of a qualitative property, Convexity, and not of anything specifically quantitative, since the explanation says nothing at all about the magnitudes of the Power Law exponents. Even more: the explanation itself is plausibly “invariant” in the sense that it does not really depend on the assumption that the psychophysical function must be a Power Law. 5.4. Invariance

Considerations;

Lute’s

Principles

qf Theor?) Construction

The qualitative laws we have been considering and many of the psychophysical functions that satisfy them do not conform in many of their applications to important invariance-theoretic principles of theory construction formulated by Lute (1959). We will start by stating the principles, and then we will comment on the significance of failures to conform to them. Lute’s principles can be stated in terms of physical scale transformations t and psychological scale transformations u in transformation groups T and U, as follows. If (A, S, R) is a psychophysical system we will call the system (A, S’, R’) such that S’(x)= t(S(x)) and R’(x)= u(R’(x)) for all .Y in A the t, u-transform of (A, S, R). Given that t and u are in T and U, respectively, (A, S’, R’) may also be said to be a t, U- or a T, u- or a T, U-transform of (A, S, R). Especially important are the so-called changes of scale or permissible transformations that result when units or other arbitrary scale factors are altered on either the physical or the psychological scale. In these cases the systems (A, S, R) and (A, S’, R’) can be regarded as “different descriptions of the same phenomena.“’ However, we shall not confine attention exclusively to permissible transformations since physical and psychological scales are often subjected for analytical purposes to much wider ranges of transformations. That is often the case with log-log transforms. Any function f is a psychophysical function for (A, S, R) if and only if the function f’(v)

= u(f(t-‘(cp)))3

(12)

is a psychophysical function for the t, u-transform of (A, S, R), where t -’ is the group-theoretic inverse of t. The function f' defined by (12) can be called the ’ These change-of-scale transforms are not to be confused with isomorphisms or “similitude transforms,” g, mapping (A, S, R) into (A’, S’, R’) such that g maps A onto A’, and s’(g(x)) = S(s) and R’(g(.x))= R(x) for all x in A. These are the fundamental tools of recent work on the theory of meaningfulness (cf. Narens, 1985; Falmagne & Narens, 1983; but see also the Appendix of the earlier work of Adams et nl., 1964). When A and A’ are different, (A, R, S) and
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t, u-transform off, and also a t, U- or a T, u- or a T, U-transform of it. A law L may be said to be consistent with transformations in T and U if all T, U-transforms of functions in f (L) are in f (L), and a psychophysical function f is invariant relative to these transformations if for every t in T there is a t, U-transform off which is identical with f Consistency is a closure condition on f(L), while invariance is a condition on its individual members. One way of stating Lute’s principles is to say that psychophysical laws should be consistent with permissible transformations and psychophysical functions should be invariant relative to them. The latter is what fails for many monotone and convex functions, but before turning to that a word should be said about the significance of the principles. We believe that it is a mistake to regard Lute’s Principles as ones which psychophysical laws and functions must satisfy, at least if statements of the laws are to be meaninaful (cf. Falmagne & Narens, 1983; Lute & Narens, 1983; Narens, 1985; among others, as well as Adams, Fagot & Robinson, 1965). We suggest instead that for a law to be inconsistent with transformations in groups T and U means that the property of satisfying it is not always preserved by the transformations, and for a psychophysical function not to be invariant with respect to them may be an indication that it is “inadequately” formulated. These can be illustrated as follows. The function f(q)= -oscp'+ 3cp- 1.5 is convex downwards and strictly monotone increasing in the interval (0, 3), but its log-log transform f ‘(cp) = ln(f(eV)) = ln( -0.5e’V + 3eV - 1.5) has a point of inflection at cp= 1. This means that convexity is inconsistent with logarithmic transformations on both scales, but it does not mean that convex functions may never be subjected to log-log transforms or that applying them to them is never appropriate for any purpose. Rather, these transforms are not appropriate to the study of convexity, since convexity properties are not always preserved by them. As to invariance, consider the log function with a non-zero threshold parameter, f(q) = ln(cp - I), which is a special case of Eq. (2~). Lute’s results entail that f is not invariant relative to ratio transformations on the physical scale and affine transformations on the psychological scale. This does not mean that f could not be a psychophysical function for any particular system (A, S, R), since it could well be that given the particular scales S(x) and R(x) of this system, R(x) would equal ln(S(s) - 1) for all x in A. Nevertheless the expression ln(q - 1) might not adequately represent the dependency of psychological scale values on physical ones. For instance, if the factor -1 in ln(cp - 1) represents the physical scale value of a threshold stimulus x0, then the proper form of the psychophysical law should not be R(x) = ln(S(x) - 1) but rather R(x, x0) = ln(S(x) - S(x,)), which is invariant relative to ratio transformations on the physical scale and affine ones on the psychological scale. It is a matter for further study whether something like this always holds when invariance fails, but even if that were so it would not prove that

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qualitative laws like monotonicity and convexity were wrong or that any particular psychophysical function satisfying them was wrong. At best it would show thatf(L) did not adequately represent the content of the law. Of course this is very vague, and we hope to return to it in another paper. We will end this one with brief observations on mathematical problems concerning the transformation groups T and U with which monotonicity and convexity are consistent, and relative to which psychophysical functions satisfying these laws are invariant. First, consistency. Perhaps surprisingly, while increasing and decreasing monotonicity are consistent with increasing monotone transformations on both scales, convexity is consistent only with much more restrictive transformations. Upwards Convexity alone is preserved by downwardly convex transformations on the physical scale and upwardly convex ones on the psychological scale, and Downwards Convexity is preserved by upwardly convex transformations on the physical scale and downwardly convex ones on the psychological scale. It follows immediately that the only transformations that preserve both weak Upwards and Downwards Convexity are affine. That is because the affine transformations are the only ones that always preserve linearity, and linear functions are both upwardly and downwardly convex in the weak sense. Somewhat more complex arguments which will not be given here show that afline transformations are also the only ones that preserve both kinds of strong convexity as well as convexity simpliciter; i.e., these are the only ones which carry convex functions into convex functions, though possibly carrying upwardly convex ones into downwardly convex ones and vice versa. The analogue of the problem just commented on relating to consistency is that of determining the transformation groups T and U relative to which psychophysical functions satisfying certain laws are invariant. This is a kind of converse of the problem considered by Lute, which was to determine the psychophysical functions that are invariant relative to given transformation groups T and U. Lute reduced that problem to determining the classes of solutions to functional equations associated with these groups, but it appears that answering the converse question leads to possibly more difficult problems which we can only state in closing. If we takeS(L) and T to be givens we may ask what classes U of psychological scale transformations satisfy the condition that for all f in F(L) and t in T there exists u in U such that u(f(tr’(cp)))=f(cp) for all cp in the range off, and in particular what the smallest class U, satisfying this condition is. Obviously U, is the class of all functions j”( t(f- ‘( cp))) for f in f( L) and t in T. Clearly if T contains only the identity transformation t(q) = cp thenf(t(f ‘(9))) = cp and U, will contain only the identity transform. If f(L) contains the identity function then U, must be a superset of T, because f(t(f-‘(cp))) = t(q) in this case. If f(L) is a group (which holds in the cases of monotonicity and upwards or downwards convexity separately, but not of upwards and downwards Convexity) and moreover it is consistent with T, then f(t(f‘(cp))) must itself belong to ,f(L); i.e., TC U, s f(L). Thus, if f(L) E T, as is the case when both are ratio functions or both are afline, then U, must equal both. The difficult problem is to characterize U, in other cases, and we will end by just noting that U. is not always identical with T. In particular,

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if T is the set of ratio transformations and f(L) is the set of monotone increasing functions then t(q) = acp belongs to T andf(cp) = c?’ belongs tof(L). It follows that j-(@-‘(q))) = (pa belongs to UO. But unless 01= 1 this is not a ratio transformation.

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