3040:
363:, the standard definition of measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (de Boer, 1994/95; Emerson, 2008). For example, the statement "Peter's hallway is 4 m long" expresses a measurement of an hitherto unknown length magnitude (the hallway's length) as the ratio of the unit (the metre in this case) to the length of the hallway. The number 4 is a real number in the strict mathematical sense of this term.
292:) which also contained a schema for deriving higher order cancellation conditions based upon Scott's (1964) work. Using Michell's schema, Ben Richards (Kyngdon & Richards, 2007) discovered that some instances of the triple cancellation axiom are "incoherent" as they contradict the single cancellation axiom. Moreover, he identified many instances of the triple cancellation which are trivially true if double cancellation is supported.
1874:
968:
582:
2929:'s (1927) theory of paired comparisons, multidimensional scaling and Coombs' (1964) theory of unidimensional unfolding. He found support of the cancellation axioms only with Coombs' (1964) theory. However, the statistical techniques employed by Michell (1990) in testing Thurstone's theory and multidimensional scaling did not take into consideration the ordinal constraints imposed by the cancellation axioms (
3005:). Kyngdon found that satisfaction of the cancellation axioms was obtained only through permutation of the matrix in a manner inconsistent with the putative Lexile measures of item difficulty. Kyngdon also tested simulated ability test response data using polynomial conjoint measurement. The data were generated using Humphry's extended frame of reference Rasch model (
2953:
applications biased towards a positive result. With six stimuli, the probability of an interstimulus midpoint order satisfying the double cancellation axioms at random is .5874 (Michell, 1994). This is not an unlikely event. Kyngdon & Richards (2007) employed eight statements and found the interstimulus midpoint orders rejected the double cancellation condition.
174:, 1940), Campbell and the Committee concluded that because psychological attributes were not capable of sustaining concatenation operations, such attributes could not be continuous quantities. Therefore, they could not be measured scientifically. This had important ramifications for psychology, the most significant of these being the creation in 1946 of the
99:. The quantification of psychological attributes such as attitudes, cognitive abilities and utility is therefore logically plausible. This means that the scientific measurement of psychological attributes is possible. That is, like physical quantities, a magnitude of a psychological quantity may possibly be expressed as the product of a
2982:) consisted of completion times and average number of series correct. They found support for the cancellation axioms, however, their study was biased by the small size of the conjoint arrays (3 × 3 is size) and by statistical techniques that did not take into consideration the ordinal restrictions imposed by the cancellation axioms.
201:(1901) anticipated features of the theory of conjoint measurement, it was not until the publication of Luce & Tukey's seminal 1964 paper that the theory received its first complete exposition. Luce & Tukey's presentation was algebraic and is therefore considered more general than Debreu's (1960)
2959:
applied conjoint measurement to item response data to a convict parole questionnaire and to intelligence test data gathered from Danish troops. They found considerable violation of the cancellation axioms in the parole questionnaire data, but not in the intelligence test data. Moreover, they recorded
444:
Length is a quantity for which natural concatenation operations exist. That is, we can combine in a side-by-side fashion lengths of rigid steel rods, for example, such that the additive relations between lengths is readily observed. If we have four 1 m lengths of such rods, we can place them end
217:
proved that via the theory of conjoint measurement, attributes not capable of concatenation could be quantified. N.R. Campbell and the
Ferguson Committee were thus proven wrong. That a given psychological attribute is a continuous quantity is a logically coherent and empirically testable hypothesis.
153:
to estimate the parameters of additive utility functions. Different multi-attribute stimuli are presented to respondents, and different methods are used to measure their preferences about the presented stimuli. The coefficients of the utility function are estimated using alternative regression-based
374:
defined the unit of the
Celsius scale to be 1/100th of the difference in temperature between the freezing and boiling points of water at sea level. A midday temperature measurement of 20 degrees Celsius is simply the difference of the midday temperature and the temperature of the freezing water
287:
Joel
Michell (1988) later identified that the "no test" class of tests of the double cancellation axiom was empty. Any instance of double cancellation is thus either an acceptance or a rejection of the axiom. Michell also wrote at this time a non-technical introduction to the theory of conjoint
2952:
tested the cancellation axioms of upon the interstimulus midpoint orders obtained by the use of Coombs' (1964) theory of unidimensional unfolding. Coombs' theory in all three studies was applied to a set of six statements. These authors found that the axioms were satisfied, however, these were
2147:
possess three levels, the highest order cancellation axiom within Scott's (1964) hierarchy that indirectly tests solvability and
Archimedeaness is double cancellation. With four levels it is triple cancellation (Figure 3). If such tests are satisfied, the construction of standard sequences in
2138:
are not amenable to direct testing in any finite empirical situation. But this does not entail that these axioms cannot be empirically tested at all. Scott's (1964) finite set of cancellation conditions can be used to indirectly test these axioms; the extent of such testing being empirically
456:
Temperature is a quantity for which there is an absence of concatenation operations. We cannot pour a volume of water of temperature 40 °C into another bucket of water at 20 °C and expect to have a volume of water with a temperature of 60 °C. Temperature is therefore an
314:
applications. Karabatsos & Ullrich 2002 demonstrated how this framework could be extended to polynomial conjoint structures. Karabatsos (2005) generalised this work with his multinomial
Dirichlet framework, which enabled the probabilistic testing of many non-stochastic theories of
1725:= 3, then 3! × 3! = 6 × 6 = 36 instances in total of double cancellation. However, all but 6 of these instances are trivially true if single cancellation is true, and if any one of these 6 instances is true, then all of them are true. One such instance is that shown in Figure Two. (
370:. Consider temperature, for example. In the familiar everyday instances, temperature is measured using instruments calibrated in either the Fahrenheit or Celsius scales. What are really being measured with such instruments are the magnitudes of temperature differences. For example,
267:
Shortly after the publication of Krantz, et al., (1971), work focused upon developing an "error theory" for the theory of conjoint measurement. Studies were conducted into the number of conjoint arrays that supported only single cancellation and both single and double cancellation
464:
Psychological attributes, like temperature, are considered to be intensive as no way of concatenating such attributes has been found. But this is not to say that such attributes are not quantifiable. The theory of conjoint measurement provides a theoretical means of doing this.
2630:
142:), however, this has been disputed (e.g., Karabatsos, 2001; Kyngdon, 2008). Order restricted methods for conducting probabilistic tests of the cancellation axioms of conjoint measurement have been developed in the past decade (e.g., Karabatsos, 2001; Davis-Stober, 2009).
2184:
can be expressed as either ratios between magnitudes or ratios between magnitude differences. It is most commonly interpreted as the latter, given that most behavioural scientists consider that their tests and surveys "measure" attributes on so-called "interval scales"
1051:
in which the common terms of two antecedent inequalities cancel out to produce a third inequality. Consider the instance of double cancellation graphically represented by Figure Two. The antecedent inequalities of this particular instance of double cancellation are:
2384:
2118:
whom wrote that "Further, of unequal lines, unequal surfaces, and unequal solids, the greater exceeds the less by such a magnitude as, when added to itself, can be made to exceed any assigned magnitude among those which are comparable with one another "
284:). These studies found that it is highly unlikely that the axioms of the theory of conjoint measurement are satisfied at random, provided that more than three levels of at least one of the component attributes has been identified.
94:
are continuous quantities. Hence the theory of conjoint measurement can be used to quantify attributes in empirical circumstances where it is not possible to combine the levels of the attributes using a side-by-side operation or
2123:, Book I, Assumption 5). Archimedes recognised that for any two magnitudes of a continuous quantity, one being lesser than the other, the lesser could be multiplied by a whole number such that it equalled the greater magnitude.
971:
Figure Two: A Luce–Tukey instance of double cancellation, in which the consequent inequality (broken line arrow) does not contradict the direction of both antecedent inequalities (solid line arrows), so supporting the
2693:
2974:) were defined by the demand placed upon working memory capacity through increasing numbers of working memory place keepers in letter series completion tasks. The rows were defined by levels of motivation (
4367:
295:
The axioms of the theory of conjoint measurement are not stochastic; and given the ordinal constraints placed on data by the cancellation axioms, order restricted inference methodology must be used (
2525:
1881:
The single and double cancellation axioms by themselves are not sufficient to establish continuous quantity. Other conditions must also be introduced to ensure continuity. These are the
844:
causes confusion with statistical concepts of independence, single cancellation is the preferable term. Figure One is a graphical representation of one instance of single cancellation.
2489:
106:
Application of the theory of conjoint measurement in psychology, however, has been limited. It has been argued that this is due to the high level of formal mathematics involved (e.g.,
2246:
248:
providing a schema with which to construct conjoint measurement structures of three or more attributes. Later, the theory of conjoint measurement (in its two variable, polynomial and
2911:, summarized the literature to that date, including the observation that the evaluation of the Thomsen Condition also involves an empirical challenge that they find remedied by the
2078:
2449:
2418:
1424:
2860:
must be tentatively identified as the unit. Hence it would seem that application of conjoint measurement requires some prior descriptive theory of the relevant natural system.
1499:
Double cancellation is satisfied if and only if the consequent inequality does not contradict the antecedent inequalities. For example, if the consequent inequality above was:
1831:
1791:
1553:
1483:
1162:
2721:
1315:
1218:
1103:
166:
established the
Ferguson Committee to investigate the possibility of psychological attributes being measured scientifically. The British physicist and measurement theorist
2517:
1610:
163:
1262:
1356:
417:
3069:
2960:
the supposed "no test" instances of double cancellation. Interpreting these correctly as instances in support of double cancellation (Michell, 1988), the results of
1736:
If single cancellation has been tested upon a set of data first and is established, then only the Luce–Tukey instances of double cancellation need to be tested. For
319:. More recently, Clintin Davis-Stober (2009) developed a frequentist framework for order restricted inference that can also be used to test the cancellation axioms.
4345:
Ullrich, J. R.; Wilson, R. E. (December 1993). "A note on the exact number of two and three way tables satisfying conjoint measurement and additivity axioms".
334:(Kahneman & Tversky, 1979). Prospect theory was a theory of decision making under risk and uncertainty which accounted for choice behaviour such as the
3595:
Karabatsos, G.; Sheu, C. F. (2004). "Bayesian order constrained inference for dichotomous models of unidimensional non-parametric item response theory".
840:, as the ordinal relation between two levels of an attribute is independent of any and all levels of the other attribute. However, given that the term
3009:). He found support of distributive, single and double cancellation consistent with a distributive polynomial conjoint structure in three variables (
3844:
3822:
Kyngdon, A.; Richards, B. (2007). "Attitudes, order and quantity: deterministic and direct probabilistic tests of unidimensional unfolding".
2985:
Kyngdon (2011) used
Karabatsos's (2001) order-restricted inference framework to test a conjoint matrix of reading item response proportions (
2947:
189:
2997:). The levels of reading ability were identified via raw total test score and the levels of reading item difficulty were identified by the
3233:
Davis-Stober, C. P. (February 2009). "Analysis of multinomial models under inequality constraints: applications to measurement theory".
2899:
observed that the evaluation of double cancellation involves considerable redundancy that complicates its empirical testing. Therefore,
3561:
3388:
Berichte Uber die
Verhandlungen der Koeniglich Sachsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physikaliche Klasse
439:
3091:
339:
182:. Stevens' non-scientific theory of measurement is widely held as definitive in psychology and the behavioural sciences generally (
338:. David Krantz wrote the formal proof to prospect theory using the theory of conjoint measurement. In 2002, Kahneman received the
4456:
3957:
McClelland, G. (June 1977). "A note on
Arbuckle and Larimer: the number of two way tables satisfying certain additivity axioms".
4416:
2638:
4446:
241:
3539:
Karabatsos, G. (2001). "The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory".
3926:
Luce, R. D.; Tukey, J. W. (January 1964). "Simultaneous conjoint measurement: a new scale type of fundamental measurement".
3357:
Grayson, D. A. (September 1988). "Two-group classification and latent trait theory: scores with monotone likelihood ratio".
955:, is determined by the satisfaction of the single cancellation axiom, as are all the "left leaning diagonal" relations upon
146:
2625:{\displaystyle \varphi '_{A}=\alpha \varphi _{A}+\beta _{A}{\text{ and }}\varphi '_{X}=\alpha \varphi _{X}+\beta _{X}.\,}
225:(1964), who proposed a hierarchy of cancellation conditions for the indirect testing of the solvability and Archimedean
3166:
Brogden, H. E. (December 1977). "The Rasch model, the law of comparative judgement and additive conjoint measurement".
3052:
4441:
4162:
Stankov, L.; Cregan, A. (1993). "Quantitative and qualitative properties of an intelligence test: series completion".
2769:
The theory of conjoint measurement, however, is not restricted to the quantification of differences. If each level of
2762:. However, it is not clear from the literature as to how a unit could be defined within an additive conjoint context.
96:
3062:
3056:
3048:
1646:
approach infinity, then the number of right leaning diagonal relations is half of the number of total relations upon
847:
Satisfaction of the single cancellation axiom is necessary, but not sufficient, for the quantification of attributes
4184:
3890:
3254:
Debreu, G. (1960). "Topological methods in cardinal utility theory". In Arrow, K.J.; Karlin, S.; Suppes, P. (eds.).
4451:
3999:
Michell, J. (December 1988). "Some problems in testing the double cancellation condition in conjoint measurement".
2379:{\displaystyle (a,x)\succsim (b,y)\iff \varphi _{A}(a)+\varphi _{X}(x)\geqslant \varphi _{A}(b)+\varphi _{X}(y).}
2156:
are possible. Hence these attributes may be dense as per the real numbers or equally spaced as per the integers (
252:-component forms) received a thorough and highly technical treatment with the publication of the first volume of
3475:"Controlling the effect of stimulus context change on attitude statements using Michell's binary tree procedure"
3073:
2970:
applied conjoint measurement to performance on sequence completion tasks. The columns of their conjoint arrays (
2852:. All three must therefore be expressed in the same unit. In such cases, it would appear that a level of either
110:) and that the theory cannot account for the "noisy" data typically discovered in psychological research (e.g.,
2454:
307:
4211:
Stenner, A. J.; Burdick, H.; Sanford, E. E.; Burdick, D. S. (2006). "How accurate are Lexile text measures?".
4185:"Evaluating a model of global psychophysical judgments I: Behavioral properties of summations and productions"
2912:
2805:
consists of densities measured as mass per unit of volume. In such cases, it would appear that one level of
2010:
316:
232:
Work soon focused on extending the theory of conjoint measurement to involve more than just two attributes.
1685:
The number of instances of double cancellation is contingent upon the number of levels identified for both
4436:
3935:
3917:
Luce, R. D.; Suppes, P. (2002). "Representational measurement theory". In
Pashler, H.; Wixted, J. (eds.).
3764:
Kyngdon, A. (2008). "The Rasch model from the perspective of the representational theory of measurement".
3509:
3136:
867:
are ordered. Informally, single cancellation does not sufficiently constrain the order upon the levels of
167:
1364:
179:
3793:
Kyngdon, A. (2011). "Plausible measurement analogies to some psychometric models of test performance".
3722:
Krantz, D. H.; Tversky, A. (1971). "Conjoint measurement analysis of composition rules in psychology".
3106:
Arbuckle, J.; Larimer, J. (1976). "The number of two-way tables satisfying certain additivity axioms".
2904:
1925:
is solved, hence the name of the condition. Solvability essentially is the requirement that each level
4308:
3645:
Krantz, D. H. (July 1964). "Conjoint measurement: the Luce–Tukey axiomatisation and some extensions".
1795:
1755:
4241:
3449:
3281:
3022:
2423:
2392:
2128:
1505:
1435:
1114:
3940:
3514:
3141:
2698:
1270:
1173:
1058:
2494:
1562:
808:
cancel out to leave the same ordinal relationship holding on the remaining elements. For example,
4295:
4108:
4079:
3781:
3612:
3527:
3398:
Michell, J.; Ernst, C. (September 1996). "The axioms of quantity and the theory of measurement".
3374:
3297:
3212:
3183:
2189:). That is, they believe tests do not identify absolute zero levels of psychological attributes.
1964:
be a set of consecutive integers, either finite or infinite, positive or negative. The levels of
1226:
304:
4091:
Scheiblechner, H. (September 1999). "Additive conjoint isotonic probabilistic models (ADISOP)".
3562:"The exchangeable multinomial model as an approach for testing axioms of choice and measurement"
2176:
Satisfaction of the conditions of conjoint measurement means that measurements of the levels of
1323:
976:
Single cancellation does not determine the order of the "right-leaning diagonal" relations upon
384:
3666:
Krantz, D. H. (1968). "A survey of measurement theory". In Danzig, G.B.; Veinott, A.F. (eds.).
2978:), which consisted in different number of times available for completing the test. Their data (
1873:
967:
581:
4257:
4220:
4046:
3877:
3831:
3810:
3752:
3692:
3624:
Karabatsos, G.; Ullrich, J. R. (2002). "Enumerating and testing conjoint measurement models".
3548:
3436:
3415:
3345:
3154:
2135:
322:
Perhaps the most notable (Kyngdon, 2011) use of the theory of conjoint measurement was in the
3309:
Fischer, G. (1995). "Derivations of the Rasch model". In Fischer, G.; Molenaar, I.W. (eds.).
4379:
4354:
4331:
4323:
4287:
4249:
4199:
4171:
4150:
4129:
4100:
4071:
4038:
4008:
3987:
3966:
3945:
3919:
Stevens' handbook of experimental psychology: Vol. 4. Methodology in experimental psychology
3905:
3867:
3859:
3802:
3773:
3731:
3684:
3654:
3633:
3604:
3584:
3576:
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3461:
3407:
3366:
3335:
3327:
3289:
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3204:
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3115:
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axiom, which avoids this redundancy, and found the property supported in binaural loudness.
229:, and David Krantz (1964) who connected the Luce & Tukey work to that of Hölder (1901).
36:
4417:
Kyngdon's R programs for enumerating cancellation tests, testing axioms and prospect theory
4406:
3318:
Gigerenzer, G.; Strube, G. (1983). "Are there limits to binaural additivity of loudness?".
2879:
Several empirical evaluations of the double cancellation have been conducted. Among these,
2750:
are magnitude differences measured relative to some kind of unit difference. Each level of
1865:) and the more stringent test of quantity the application of conjoint measurement becomes.
585:
Figure One: Graphical representation of the single cancellation axiom. It can be seen that
4141:
Sherman, K. (April 1994). "The effect of change in context in Coombs's unfolding theory".
3872:
2926:
327:
323:
3500:
Kahneman, D.; Tversky, A. (1979). "Prospect theory: an analysis of decision under risk".
2813:
must be identified as a tentative unit prior to the application of conjoint measurement.
445:
to end to produce a length of 4 m. Quantities capable of concatenation are known as
375:
divided by the difference of the Celsius unit and the temperature of the freezing water.
4245:
3891:"Theory and tests of the conjoint commutativity axiom for additive conjoint measurement"
3688:
3285:
3195:
Cliff, N. (1992). "Abstract measurement theory and the revolution that never happened".
2766:
proposed a scaling method for conjoint structures but he also did not discuss the unit.
2114:. This condition is a definition of continuity given by the ancient Greek mathematician
1630:
Double cancellation concerns the behaviour of the "right leaning diagonal" relations on
3978:
Michell, J. (June 1994). "Measuring dimensions of belief by unidimensional unfolding".
3863:
3208:
2732:
371:
335:
257:
198:
40:
3743:
Kyngdon, A. (2006). "An empirical study into the theory of unidimensional unfolding".
3637:
4430:
4327:
4299:
4175:
4133:
4112:
4012:
3970:
3949:
3785:
3658:
3616:
3465:
3378:
3340:
3301:
3293:
3216:
3187:
3119:
2884:
4411:
4083:
4029:
Michell, J. (February 2009). "The psychometricians' fallacy: Too clever by half?".
331:
311:
237:
449:
and include mass, time, electrical resistance and plane angle. These are known as
4253:
3427:
Humphry, S. M.; Andrich, D. (2008). "Understanding the unit in the Rasch model".
2735:
in Stevens’ (1946) parlance). The mathematical proof of this result is given in (
2451:
are two other real valued functions satisfying the above expression, there exist
804:
Single cancellation is so-called because a single common factor of two levels of
573:. These relations are presented as axioms in the theory of conjoint measurement.
3150:
1857:, the less probable it is that the cancellation axioms are satisfied at random (
202:
115:
100:
4075:
2989:) where the examinee reading ability comprised the rows of the conjoint array (
2868:
Empirical applications of the theory of conjoint measurement have been sparse (
2785:
is another different quantity whose measurement is expressed as a magnitude of
2131:, in which Euclid presented his theory of continuous quantity and measurement.
2102:
The Archimedean condition argues that there is no infinitely greatest level of
980:. Even though by transitivity and single cancellation it was established that (
4203:
4154:
3909:
3580:
3491:
3474:
3331:
3246:
2115:
222:
44:
4042:
3806:
3777:
3608:
2919:
found conjoint commutativity supported for binaural loudness and brightness.
2887:
of binaural loudness. They found the double cancellation axiom was rejected.
781:, are ordered, then this order holds irrespective of each and every level of
3670:. Providence, Rhode Island: American Mathematical Society. pp. 314–350.
360:
150:
4358:
4261:
4224:
4050:
3991:
3835:
3814:
3756:
3715:
Foundations of Measurement, Vol. I: Additive and polynomial representations
3552:
3440:
3411:
3158:
2993:) and the difficulty of the reading items formed the columns of the array (
170:
was an influential member of the committee. In its Final Report (Ferguson,
3881:
3696:
3419:
3349:
4336:
2095:
which can be found which makes two relevant ordered pairs, the levels of
1946:
1945:— they are either dense like the real numbers or equally spaced like the
276:). Later enumeration studies focused on polynomial conjoint measurement (
221:
Appearing in the next issue of the same journal were important papers by
78:
are known to be quantities. Via specific relations between the levels of
54:
The theory concerns the situation where at least two natural attributes,
32:
3272:
Emerson, W. H. (2008). "On quantity calculus and units of measurement".
4383:
4104:
3531:
3370:
3179:
356:
3386:
Hölder, O. (1901). "Die Axiome der Quantität und die Lehre vom Mass".
1047:
The double cancellation axiom concerns a class of such relations upon
569:
depends upon the behaviour of the relation holding upon the levels of
4291:
4120:
Scott, D. (July 1964). "Measurement models and linear inequalities".
3735:
2998:
2124:
951:). The relation between these latter two ordered pairs, informally a
226:
118:
is a stochastic variant of the theory of conjoint measurement (e.g.,
3523:
3320:
Journal of Experimental Psychology: Human Perception and Performance
3127:
Birnbaum, M. H. (2008). "New paradoxes of risky decision making".
2915:
axiom, which they show to be equivalent to the Thomsen Condition.
1872:
966:
580:
366:
For some other quantities, invariant are ratios between attribute
4271:
Luce's challenge: Quantitative models and statistical methodology
4232:
Stevens, S. S. (1946). "On the theory of scales of measurement".
1849:= 5 then there are 100. The greater the number of levels in both
145:
The theory of conjoint measurement is (different but) related to
3311:
Rasch models: Foundations, recent developments, and applications
3025: – Paradigm for the design, analysis, and scoring of tests
645:
The single cancellation axiom is as follows. The relation upon
3033:
2127:
stated the Archimedean condition as an axiom in Book V of the
1634:
as these are not logically entailed by single cancellation. (
149:, which is a statistical-experiments methodology employed in
4421:
4059:
3843:
Levelt, W. J. M.; Riemersma, J. B.; Bunt, A. A. (May 1972).
3002:
1752:, the number of Luce–Tukey double cancellation instances is
430:
is a real number and is a unit magnitude of the same kind.
4278:
Thurstone, L. L. (1927). "A law of comparative judgement".
2134:
As they involve infinitistic concepts, the solvability and
299:). George Karabatsos and his associates (Karabatsos, 2001;
4412:
Birnbaum's FORTRAN MONANOVA program for testing additivity
4031:
British Journal of Mathematical and Statistical Psychology
3852:
British Journal of Mathematical and Statistical Psychology
3795:
British Journal of Mathematical and Statistical Psychology
3713:
Krantz, D. H.; Luce, R.D; Suppes, P.; Tversky, A. (1971).
3706:
The New Psychometrics: Science, psychology and measurement
2688:{\displaystyle \varphi '_{A},\varphi _{A},\varphi '_{X}\,}
35:. It was independently discovered by the French economist
4022:
An Introduction to the Logic of Psychological Measurement
2891:
conducted a similar investigation and replicated Levelt,
1709:, then the number of instances of double cancellation is
489:
is a continuous quantity, or that both of them are. Let
4407:
Karabatsos' S-Plus programs for testing conjoint axioms
2736:
2157:
1950:
261:
1800:
1760:
836:. Krantz, et al., (1971) originally called this axiom
209:). In the first article of the inaugural issue of the
4309:"A general theory of polynomial conjoint measurement"
2701:
2641:
2528:
2497:
2457:
2426:
2395:
2249:
2013:
1798:
1758:
1658:
are quantities, half of the number of relations upon
1565:
1508:
1438:
1367:
1326:
1273:
1229:
1176:
1117:
1061:
387:
205:
work, the latter being a special case of the former (
39:(1960) and by the American mathematical psychologist
2731:
unique up to affine transformation (i.e. each is an
2172:
Relation to the scientific definition of measurement
1937:. Solvability reveals something about the levels of
517:
represent three independent, identifiable levels of
501:
represent three independent, identifiable levels of
2961:
2956:
2880:
111:
4060:"The Rasch model as additive conjoint measurement"
2715:
2687:
2624:
2511:
2483:
2443:
2412:
2378:
2106:and so hence there is no greatest level of either
2072:
1825:
1785:
1604:
1547:
1477:
1418:
1350:
1309:
1256:
1212:
1156:
1097:
525:, consists of the nine ordered pairs of levels of
411:
351:The classical / standard definition of measurement
164:British Association for the Advancement of Science
2916:
2908:
2900:
1877:Figure Three: An instance of triple cancellation.
1012:) remains undetermined. It could be that either (
378:Formally expressed, a scientific measurement is:
326:proposed by the Israeli – American psychologists
277:
62:, non-interactively relate to a third attribute,
3061:but its sources remain unclear because it lacks
2896:
2888:
4058:Perline, R.; Wright, B. D.; Wainer, H. (1979).
3921:(3rd ed.). New York: Wiley. pp. 1–41.
1858:
1488:Hence double cancellation can only obtain when
1044:) and such ambiguity cannot remain unresolved.
296:
269:
3006:
2930:
773:). What this means is that if any two levels,
300:
256:, which Krantz, Luce, Tversky and philosopher
123:
3452:(1985). "Statistical issues in measurement".
3010:
2967:
1816:
1803:
1776:
1763:
281:
139:
16:General, formal theory of continuous quantity
8:
3668:Mathematics of the Decision Sciences: Part 2
3258:. Stanford University Press. pp. 16–26.
1670:and half are due to additive relations upon
1615:then double cancellation would be violated (
31:) is a general, formal theory of continuous
3256:Mathematical Methods in the Social Sciences
2946:harv error: no target: CITEREFSherman1993 (
2763:
1911:, the fourth exists such that the equation
879:. For example, consider the ordered pairs (
188:harv error: no target: CITEREFMichell1999 (
2287:
2283:
1862:
1841:= 4, then there are 16 such instances. If
855:. It only demonstrates that the levels of
273:
206:
4335:
3939:
3871:
3513:
3490:
3339:
3140:
3092:Learn how and when to remove this message
2712:
2706:
2700:
2684:
2675:
2662:
2646:
2640:
2621:
2612:
2599:
2580:
2571:
2565:
2552:
2533:
2527:
2508:
2502:
2496:
2484:{\displaystyle \alpha >0,\beta _{A}\,}
2480:
2474:
2456:
2440:
2431:
2425:
2409:
2400:
2394:
2358:
2336:
2314:
2292:
2248:
2046:
2021:
2012:
1815:
1802:
1799:
1797:
1775:
1762:
1759:
1757:
1564:
1507:
1437:
1366:
1325:
1272:
1228:
1175:
1116:
1060:
828:) as it is common to both sides, leaving
577:Single cancellation or independence axiom
386:
214:
48:
1429:Cancelling the common terms results in:
797:with respect to each and every level of
557:) (see Figure 1). The quantification of
3889:Luce, R. D.; Steingrimsson, R. (2011).
3263:Embretson, S. E.; Reise, S. P. (2000).
2941:
2940:), Kyngdon (2006), Michell (1994) and (
2937:
2922:
2873:
1726:
1679:
1635:
1616:
289:
183:
127:
119:
4183:Steingrimsson, R; Luce, R. D. (2005).
3265:Item response theory for psychologists
2754:is a difference between the levels of
2073:{\displaystyle (a_{i},x)=(a_{i+1},y).}
903:). If single cancellation holds then (
342:for prospect theory (Birnbaum, 2008).
245:
240:(1967) developed what became known as
233:
3313:. New York: Springer. pp. 15–38.
2869:
2186:
2083:What this basically means is that if
1895:means that for any three elements of
1638:) discovered that when the levels of
1619:) and it could not be concluded that
785:. The same holds for any two levels,
453:quantities in physics and metrology.
135:
131:
107:
7:
3675:Keats, J. A. (1967). "Test theory".
2964:are better than what they believed.
2864:Applications of conjoint measurement
2216:into the real numbers such that for
4164:Learning and Individual Differences
3689:10.1146/annurev.ps.18.020167.001245
2091:, for example, there are levels of
1419:{\displaystyle a+y+b+z>b+x+c+y.}
4347:Journal of Mathematical Psychology
4316:Journal of Mathematical Psychology
4192:Journal of Mathematical Psychology
4122:Journal of Mathematical Psychology
4001:Journal of Mathematical Psychology
3980:Journal of Mathematical Psychology
3959:Journal of Mathematical Psychology
3928:Journal of Mathematical Psychology
3898:Journal of Mathematical Psychology
3864:10.1111/j.2044-8317.1972.tb00477.x
3647:Journal of Mathematical Psychology
3569:Journal of Mathematical Psychology
3400:Journal of Mathematical Psychology
3235:Journal of Mathematical Psychology
3209:10.1111/j.1467-9280.1992.tb00024.x
3108:Journal of Mathematical Psychology
2519:real valued constants satisfying:
2208:, then there exist functions from
1869:Solvability and Archimedean axioms
1807:
1767:
1662:are due to ordinal relations upon
440:Intensive and extensive properties
426:is the magnitude of the quantity,
211:Journal of Mathematical Psychology
14:
4391:van der Ven, A. H. G. S. (1980).
4366:van der Linden, W. (March 1994).
4064:Applied Psychological Measurement
3845:"Binaural additivity of loudness"
3597:Applied Psychological Measurement
2962:Perline, Wright & Wainer 1979
2957:Perline, Wright & Wainer 1979
2903:evaluated instead the equivalent
2881:Levelt, Riemersma & Bunt 1972
2139:determined. For example, if both
1733:instance of double cancellation.
340:Nobel Memorial Prize in Economics
176:operational theory of measurement
112:Perline, Wright & Wainer 1979
4422:R statistical computing software
4143:Australian Journal of Psychology
3560:Karabatsos, G. (February 2005).
3479:Australian Journal of Psychology
3038:
1826:{\displaystyle {\tbinom {m}{3}}}
1786:{\displaystyle {\tbinom {n}{3}}}
473:Consider two natural attributes
434:Extensive and intensive quantity
197:Whilst the German mathematician
2444:{\displaystyle \varphi '_{X}\,}
2413:{\displaystyle \varphi '_{A}\,}
1548:{\displaystyle (a,z)<(c,x),}
1478:{\displaystyle (a,z)>(c,x).}
1157:{\displaystyle (b,z)>(c,y).}
812:cancels out of the inequality (
242:polynomial conjoint measurement
114:). It has been argued that the
4213:Journal of Applied Measurement
3873:11858/00-001M-0000-0013-2CBF-1
3824:Journal of Applied Measurement
3745:Journal of Applied Measurement
3541:Journal of Applied Measurement
3429:Journal of Applied Measurement
2742:This means that the levels of
2716:{\displaystyle \varphi _{X}\,}
2370:
2364:
2348:
2342:
2326:
2320:
2304:
2298:
2284:
2280:
2268:
2262:
2250:
2064:
2039:
2033:
2014:
1596:
1584:
1578:
1566:
1539:
1527:
1521:
1509:
1469:
1457:
1451:
1439:
1310:{\displaystyle (b,z)>(c,y)}
1304:
1292:
1286:
1274:
1213:{\displaystyle (a,y)>(b,x)}
1207:
1195:
1189:
1177:
1148:
1136:
1130:
1118:
1098:{\displaystyle (a,y)>(b,x)}
1092:
1080:
1074:
1062:
481:. It is not known that either
406:
400:
346:Measurement and quantification
21:theory of conjoint measurement
1:
3638:10.1016/S0165-4896(02)00024-0
2917:Luce & Steingrimsson 2011
2909:Luce & Steingrimsson 2011
2901:Steingrimsson & Luce 2005
2848:are lengths so hence must be
996:), the relationship between (
278:Karabatsos & Ullrich 2002
82:, it can be established that
29:additive conjoint measurement
4328:10.1016/0022-2496(67)90039-9
4254:10.1126/science.103.2684.677
4176:10.1016/1041-6080(93)90009-H
4134:10.1016/0022-2496(64)90002-1
4013:10.1016/0022-2496(88)90024-7
3971:10.1016/0022-2496(77)90035-9
3950:10.1016/0022-2496(64)90015-X
3659:10.1016/0022-2496(64)90003-3
3626:Mathematical Social Sciences
3466:10.1016/0165-4896(85)90031-9
3454:Mathematical Social Sciences
3120:10.1016/0022-2496(76)90036-5
2897:Gigerenzer & Strube 1983
2889:Gigerenzer & Strube 1983
2512:{\displaystyle \beta _{X}\,}
1972:if and only if there exists
1605:{\displaystyle (a,z)=(c,x),}
3717:. New York: Academic Press.
3677:Annual Review of Psychology
3151:10.1037/0033-295X.115.2.463
2883:evaluated the axiom to the
2773:is a product of a level of
2206:additive conjoint structure
2168:are continuous quantities.
1859:Arbuckle & Larimer 1976
1257:{\displaystyle a+y>b+x;}
935:). Hence via transitivity (
297:Iverson & Falmagne 1985
270:Arbuckle & Larimer 1976
4473:
4368:"Review of Michell (1990)"
4076:10.1177/014662167900300213
3294:10.1088/0026-1394/45/2/002
3007:Humphry & Andrich 2008
2801:consists of volumes, then
2121:On the Sphere and Cylinder
1351:{\displaystyle b+z>c+y}
437:
412:{\displaystyle Q=r\times }
301:Karabatsos & Sheu 2004
254:Foundations of Measurement
124:Embretson & Reise 2000
66:. It is not required that
4204:10.1016/j.jmp.2005.03.003
4155:10.1080/00049539408259468
3910:10.1016/j.jmp.2011.05.004
3581:10.1016/j.jmp.2004.11.001
3492:10.1080/00049530108255118
3341:21.11116/0000-0000-BC9A-F
3332:10.1037/0096-1523.9.1.126
3247:10.1016/j.jmp.2008.08.003
3011:Krantz & Tversky 1971
2968:Stankov & Cregan 1993
2820:is the sum of a level of
963:Double cancellation axiom
282:Ullrich & Wilson 1993
4043:10.1348/000711007X243582
4024:. Hillsdale NJ: Erlbaum.
3807:10.1348/2044-8317.002004
3778:10.1177/0959354307086924
3609:10.1177/0146621603260678
3047:This article includes a
2832:is the same quantity as
308:Markov chain Monte Carlo
178:by Harvard psychologist
4457:Mathematical psychology
4393:Introduction to Scaling
3766:Theory & Psychology
3076:more precise citations.
3001:Framework for Reading (
2797:consists of masses and
1320:is true if and only if
1223:is true if and only if
749:) is implied for every
689:) is implied for every
653:if and only if for all
317:mathematical psychology
4447:Latent variable models
4359:10.1006/jmps.1993.1037
4269:Stober, C. P. (2009).
3992:10.1006/jmps.1994.1016
3412:10.1006/jmps.1996.0023
3396:(Part 1 translated by
3224:Coombs, C. H. (1964).
2925:applied the theory to
2913:conjoint commutativity
2789:per unit magnitude of
2717:
2689:
2626:
2513:
2485:
2445:
2414:
2380:
2074:
1878:
1827:
1787:
1606:
1549:
1479:
1420:
1352:
1311:
1258:
1214:
1158:
1099:
973:
713:). Similarly, for all
642:
413:
207:Luce & Suppes 2002
168:Norman Robert Campbell
103:and a unit magnitude.
3197:Psychological Science
2718:
2690:
2627:
2514:
2486:
2446:
2415:
2381:
2075:
1992:and for all integers
1958:Archimedean condition
1876:
1828:
1788:
1607:
1550:
1480:
1421:
1353:
1312:
1259:
1215:
1159:
1100:
970:
953:left-leaning diagonal
584:
521:. A third attribute,
414:
215:Luce & Tukey 1964
180:Stanley Smith Stevens
49:Luce & Tukey 1964
4307:Tversky, A. (1967).
4280:Psychological Review
4020:Michell, J. (1990).
3724:Psychological Review
3708:. London: Routledge.
3473:Johnson, T. (2001).
3129:Psychological Review
3023:Item response theory
2723:are measurements of
2699:
2639:
2526:
2495:
2455:
2424:
2393:
2247:
2011:
1796:
1756:
1563:
1506:
1436:
1365:
1324:
1271:
1227:
1174:
1115:
1059:
447:extensive quantities
385:
25:conjoint measurement
4246:1946Sci...103..677S
3286:2008Metro..45..134E
3003:Stenner et al. 2006
2931:van der Linden 1994
2739:, pp. 261–6).
2683:
2654:
2588:
2541:
2439:
2408:
2160:). In other words,
1960:is as follows. Let
1358:, it follows that:
651:single cancellation
158:Historical overview
4442:Statistical theory
4395:. New York: Wiley.
4384:10.1007/BF02294273
4105:10.1007/BF02294297
3704:Kline, P. (1998).
3371:10.1007/BF02294219
3228:. New York: Wiley.
3180:10.1007/BF02295985
3049:list of references
2737:Krantz et al. 1971
2713:
2685:
2671:
2642:
2622:
2576:
2529:
2509:
2481:
2441:
2427:
2410:
2396:
2376:
2158:Krantz et al. 1971
2136:Archimedean axioms
2070:
1951:Krantz et al. 1971
1933:and an element in
1929:has an element in
1879:
1833:. For example, if
1823:
1821:
1783:
1781:
1602:
1545:
1475:
1416:
1348:
1307:
1254:
1210:
1154:
1095:
974:
643:
409:
262:Krantz et al. 1971
162:In the 1930s, the
140:Scheiblechner 1999
4452:Economic theories
4240:(2684): 667–680.
3102:
3101:
3094:
2905:Thomsen condition
2895:(1972) findings.
2816:If each level of
2809:and one level of
2574:
2148:differences upon
1970:standard sequence
1814:
1774:
1717:!. Therefore, if
1555:or alternatively,
147:conjoint analysis
43:and statistician
4464:
4396:
4387:
4362:
4341:
4339:
4313:
4303:
4292:10.1037/h0070288
4274:
4265:
4228:
4207:
4189:
4179:
4158:
4137:
4116:
4087:
4054:
4025:
4016:
3995:
3974:
3953:
3943:
3922:
3913:
3895:
3885:
3875:
3849:
3839:
3818:
3789:
3760:
3739:
3736:10.1037/h0030637
3718:
3709:
3700:
3671:
3662:
3641:
3620:
3591:
3589:
3583:. Archived from
3566:
3556:
3535:
3517:
3496:
3494:
3469:
3444:
3423:
3395:
3382:
3353:
3343:
3314:
3305:
3268:
3259:
3250:
3229:
3226:A Theory of Data
3220:
3191:
3162:
3144:
3123:
3097:
3090:
3086:
3083:
3077:
3072:this article by
3063:inline citations
3042:
3041:
3034:
2951:
2764:van der Ven 1980
2722:
2720:
2719:
2714:
2711:
2710:
2694:
2692:
2691:
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2667:
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2404:
2385:
2383:
2382:
2377:
2363:
2362:
2341:
2340:
2319:
2318:
2297:
2296:
2087:is greater than
2079:
2077:
2076:
2071:
2057:
2056:
2026:
2025:
1832:
1830:
1829:
1824:
1822:
1820:
1819:
1806:
1792:
1790:
1789:
1784:
1782:
1780:
1779:
1766:
1627:are quantities.
1611:
1609:
1608:
1603:
1554:
1552:
1551:
1546:
1496:are quantities.
1484:
1482:
1481:
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1425:
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1357:
1355:
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1349:
1316:
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418:
416:
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410:
310:methodology for
193:
4472:
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4466:
4465:
4463:
4462:
4461:
4427:
4426:
4403:
4390:
4365:
4344:
4311:
4306:
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4231:
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4187:
4182:
4161:
4140:
4119:
4090:
4057:
4028:
4019:
3998:
3977:
3956:
3941:10.1.1.334.5018
3925:
3916:
3893:
3888:
3847:
3842:
3821:
3792:
3763:
3742:
3721:
3712:
3703:
3674:
3665:
3644:
3623:
3594:
3587:
3564:
3559:
3538:
3524:10.2307/1914185
3515:10.1.1.407.1910
3499:
3472:
3450:Falmagne, J. C.
3447:
3426:
3397:
3385:
3356:
3317:
3308:
3271:
3262:
3253:
3232:
3223:
3194:
3165:
3142:10.1.1.144.5661
3126:
3105:
3098:
3087:
3081:
3078:
3067:
3053:related reading
3043:
3039:
3032:
3019:
2945:
2927:L. L. Thurstone
2866:
2840:. For example,
2824:and a level of
2793:. For example,
2777:and a level of
2702:
2697:
2696:
2658:
2637:
2636:
2608:
2595:
2573: and
2561:
2548:
2524:
2523:
2498:
2493:
2492:
2470:
2453:
2452:
2422:
2421:
2391:
2390:
2354:
2332:
2310:
2288:
2245:
2244:
2174:
2042:
2017:
2009:
2008:
1871:
1863:McClelland 1977
1801:
1794:
1793:
1761:
1754:
1753:
1729:) calls this a
1693:. If there are
1561:
1560:
1504:
1503:
1434:
1433:
1363:
1362:
1322:
1321:
1269:
1268:
1225:
1224:
1172:
1171:
1113:
1112:
1057:
1056:
965:
579:
471:
442:
436:
383:
382:
353:
348:
328:Daniel Kahneman
324:prospect theory
274:McClelland 1977
187:
160:
23:(also known as
17:
12:
11:
5:
4470:
4468:
4460:
4459:
4454:
4449:
4444:
4439:
4429:
4428:
4425:
4424:
4419:
4414:
4409:
4402:
4401:External links
4399:
4398:
4397:
4388:
4378:(1): 139–142.
4363:
4342:
4304:
4286:(4): 278–286.
4275:
4266:
4229:
4219:(3): 307–322.
4208:
4198:(4): 290–306.
4180:
4170:(2): 137–169.
4159:
4138:
4128:(2): 233–247.
4117:
4099:(3): 295–316.
4088:
4070:(2): 237–255.
4055:
4026:
4017:
4007:(4): 466–473.
3996:
3986:(2): 224–273.
3975:
3954:
3923:
3914:
3904:(5): 379–389.
3886:
3840:
3819:
3801:(3): 478–497.
3790:
3761:
3751:(4): 369–393.
3740:
3730:(2): 151–169.
3719:
3710:
3701:
3672:
3663:
3653:(2): 248–277.
3642:
3632:(3): 485–504.
3621:
3603:(2): 110–125.
3592:
3590:on 2006-02-06.
3557:
3547:(4): 389–423.
3536:
3508:(2): 263–291.
3497:
3470:
3460:(2): 131–153.
3445:
3435:(3): 249–264.
3424:
3406:(3): 235–252.
3383:
3365:(3): 383–392.
3354:
3326:(1): 126–136.
3315:
3306:
3280:(2): 134–138.
3269:
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3230:
3221:
3203:(3): 186–190.
3192:
3163:
3135:(2): 463–501.
3124:
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3099:
3057:external links
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2733:interval scale
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2295:
2291:
2286:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2173:
2170:
2081:
2080:
2069:
2066:
2063:
2060:
2055:
2052:
2049:
2045:
2041:
2038:
2035:
2032:
2029:
2024:
2020:
2016:
1870:
1867:
1818:
1813:
1810:
1805:
1778:
1773:
1770:
1765:
1613:
1612:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1557:
1556:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1486:
1485:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1441:
1427:
1426:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1318:
1317:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1221:
1220:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1165:
1164:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1106:
1105:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
964:
961:
578:
575:
470:
467:
438:Main article:
435:
432:
420:
419:
408:
405:
402:
399:
396:
393:
390:
372:Anders Celsius
352:
349:
347:
344:
336:Allais Paradox
303:) developed a
258:Patrick Suppes
159:
156:
41:R. Duncan Luce
15:
13:
10:
9:
6:
4:
3:
2:
4469:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4437:Psychometrics
4435:
4434:
4432:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4404:
4400:
4394:
4389:
4385:
4381:
4377:
4373:
4372:Psychometrika
4369:
4364:
4360:
4356:
4352:
4348:
4343:
4338:
4337:2027.42/33362
4333:
4329:
4325:
4321:
4317:
4310:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4276:
4272:
4267:
4263:
4259:
4255:
4251:
4247:
4243:
4239:
4235:
4230:
4226:
4222:
4218:
4214:
4209:
4205:
4201:
4197:
4193:
4186:
4181:
4177:
4173:
4169:
4165:
4160:
4156:
4152:
4148:
4144:
4139:
4135:
4131:
4127:
4123:
4118:
4114:
4110:
4106:
4102:
4098:
4094:
4093:Psychometrika
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4056:
4052:
4048:
4044:
4040:
4036:
4032:
4027:
4023:
4018:
4014:
4010:
4006:
4002:
3997:
3993:
3989:
3985:
3981:
3976:
3972:
3968:
3964:
3960:
3955:
3951:
3947:
3942:
3937:
3933:
3929:
3924:
3920:
3915:
3911:
3907:
3903:
3899:
3892:
3887:
3883:
3879:
3874:
3869:
3865:
3861:
3857:
3853:
3846:
3841:
3837:
3833:
3829:
3825:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3791:
3787:
3783:
3779:
3775:
3771:
3767:
3762:
3758:
3754:
3750:
3746:
3741:
3737:
3733:
3729:
3725:
3720:
3716:
3711:
3707:
3702:
3698:
3694:
3690:
3686:
3682:
3678:
3673:
3669:
3664:
3660:
3656:
3652:
3648:
3643:
3639:
3635:
3631:
3627:
3622:
3618:
3614:
3610:
3606:
3602:
3598:
3593:
3586:
3582:
3578:
3574:
3570:
3563:
3558:
3554:
3550:
3546:
3542:
3537:
3533:
3529:
3525:
3521:
3516:
3511:
3507:
3503:
3498:
3493:
3488:
3484:
3480:
3476:
3471:
3467:
3463:
3459:
3455:
3451:
3448:Iverson, G.;
3446:
3442:
3438:
3434:
3430:
3425:
3421:
3417:
3413:
3409:
3405:
3401:
3393:
3389:
3384:
3380:
3376:
3372:
3368:
3364:
3360:
3359:Psychometrika
3355:
3351:
3347:
3342:
3337:
3333:
3329:
3325:
3321:
3316:
3312:
3307:
3303:
3299:
3295:
3291:
3287:
3283:
3279:
3275:
3270:
3266:
3261:
3257:
3252:
3248:
3244:
3240:
3236:
3231:
3227:
3222:
3218:
3214:
3210:
3206:
3202:
3198:
3193:
3189:
3185:
3181:
3177:
3173:
3169:
3168:Psychometrika
3164:
3160:
3156:
3152:
3148:
3143:
3138:
3134:
3130:
3125:
3121:
3117:
3113:
3109:
3104:
3103:
3096:
3093:
3085:
3075:
3071:
3065:
3064:
3058:
3054:
3050:
3045:
3036:
3035:
3029:
3024:
3021:
3020:
3016:
3014:
3012:
3008:
3004:
3000:
2996:
2992:
2988:
2983:
2981:
2977:
2973:
2969:
2965:
2963:
2958:
2954:
2949:
2943:
2939:
2934:
2932:
2928:
2924:
2920:
2918:
2914:
2910:
2906:
2902:
2898:
2894:
2890:
2886:
2885:psychophysics
2882:
2877:
2875:
2871:
2863:
2861:
2859:
2855:
2851:
2847:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2814:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2767:
2765:
2761:
2757:
2753:
2749:
2745:
2740:
2738:
2734:
2730:
2726:
2707:
2703:
2680:
2676:
2672:
2668:
2663:
2659:
2655:
2651:
2647:
2643:
2618:
2613:
2609:
2605:
2600:
2596:
2592:
2589:
2585:
2581:
2577:
2566:
2562:
2558:
2553:
2549:
2545:
2542:
2538:
2534:
2530:
2522:
2521:
2520:
2503:
2499:
2475:
2471:
2467:
2464:
2461:
2458:
2436:
2432:
2428:
2405:
2401:
2397:
2373:
2367:
2359:
2355:
2351:
2345:
2337:
2333:
2329:
2323:
2315:
2311:
2307:
2301:
2293:
2289:
2277:
2274:
2271:
2265:
2259:
2256:
2253:
2243:
2242:
2241:
2239:
2235:
2231:
2227:
2223:
2219:
2215:
2211:
2207:
2203:
2199:
2195:
2192:Formally, if
2190:
2188:
2183:
2179:
2171:
2169:
2167:
2163:
2159:
2155:
2151:
2146:
2142:
2137:
2132:
2130:
2126:
2122:
2117:
2113:
2109:
2105:
2100:
2098:
2094:
2090:
2086:
2067:
2061:
2058:
2053:
2050:
2047:
2043:
2036:
2030:
2027:
2022:
2018:
2007:
2006:
2005:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1975:
1971:
1967:
1963:
1959:
1954:
1952:
1948:
1944:
1940:
1936:
1932:
1928:
1924:
1921:
1917:
1914:
1910:
1906:
1902:
1898:
1894:
1890:
1888:
1884:
1875:
1868:
1866:
1864:
1860:
1856:
1852:
1848:
1844:
1840:
1836:
1811:
1808:
1771:
1768:
1751:
1747:
1743:
1739:
1734:
1732:
1728:
1724:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1683:
1681:
1677:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1628:
1626:
1622:
1618:
1599:
1593:
1590:
1587:
1581:
1575:
1572:
1569:
1559:
1558:
1542:
1536:
1533:
1530:
1524:
1518:
1515:
1512:
1502:
1501:
1500:
1497:
1495:
1491:
1472:
1466:
1463:
1460:
1454:
1448:
1445:
1442:
1432:
1431:
1430:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1361:
1360:
1359:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1301:
1298:
1295:
1289:
1283:
1280:
1277:
1267:
1266:
1265:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1204:
1201:
1198:
1192:
1186:
1183:
1180:
1170:
1169:
1168:
1151:
1145:
1142:
1139:
1133:
1127:
1124:
1121:
1111:
1110:
1109:
1089:
1086:
1083:
1077:
1071:
1068:
1065:
1055:
1054:
1053:
1050:
1045:
1043:
1039:
1035:
1031:
1027:
1023:
1019:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
969:
962:
960:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
862:
858:
854:
850:
845:
843:
839:
835:
831:
827:
823:
819:
815:
811:
807:
802:
800:
796:
792:
788:
784:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
668:
664:
660:
656:
652:
648:
640:
636:
632:
628:
624:
620:
616:
612:
608:
604:
600:
596:
592:
588:
583:
576:
574:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
480:
476:
468:
466:
462:
460:
454:
452:
448:
441:
433:
431:
429:
425:
403:
397:
394:
391:
388:
381:
380:
379:
376:
373:
369:
364:
362:
358:
350:
345:
343:
341:
337:
333:
329:
325:
320:
318:
313:
309:
306:
302:
298:
293:
291:
288:measurement (
285:
283:
279:
275:
271:
265:
263:
259:
255:
251:
247:
243:
239:
235:
230:
228:
224:
219:
216:
212:
208:
204:
200:
195:
191:
185:
181:
177:
173:
169:
165:
157:
155:
152:
148:
143:
141:
137:
133:
129:
125:
121:
117:
113:
109:
104:
102:
98:
97:concatenation
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
52:
50:
46:
42:
38:
37:Gérard Debreu
34:
30:
26:
22:
4392:
4375:
4371:
4353:(4): 624–8.
4350:
4346:
4319:
4315:
4283:
4279:
4270:
4237:
4233:
4216:
4212:
4195:
4191:
4167:
4163:
4149:(1): 41–47.
4146:
4142:
4125:
4121:
4096:
4092:
4067:
4063:
4037:(1): 41–55.
4034:
4030:
4021:
4004:
4000:
3983:
3979:
3965:(3): 292–5.
3962:
3958:
3931:
3927:
3918:
3901:
3897:
3858:(1): 51–68.
3855:
3851:
3827:
3823:
3798:
3794:
3769:
3765:
3748:
3744:
3727:
3723:
3714:
3705:
3680:
3676:
3667:
3650:
3646:
3629:
3625:
3600:
3596:
3585:the original
3575:(1): 51–69.
3572:
3568:
3544:
3540:
3505:
3502:Econometrica
3501:
3482:
3478:
3457:
3453:
3432:
3428:
3403:
3399:
3391:
3387:
3362:
3358:
3323:
3319:
3310:
3277:
3273:
3264:
3255:
3238:
3234:
3225:
3200:
3196:
3174:(4): 631–4.
3171:
3167:
3132:
3128:
3111:
3107:
3088:
3079:
3068:Please help
3060:
2994:
2990:
2986:
2984:
2979:
2975:
2971:
2966:
2955:
2942:Sherman 1993
2938:Johnson 2001
2935:
2923:Michell 1990
2921:
2892:
2878:
2874:Michell 2009
2867:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2825:
2821:
2817:
2815:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2778:
2774:
2770:
2768:
2759:
2755:
2751:
2747:
2743:
2741:
2728:
2724:
2634:
2388:
2237:
2233:
2229:
2225:
2221:
2217:
2213:
2209:
2205:
2201:
2197:
2193:
2191:
2181:
2177:
2175:
2165:
2161:
2153:
2149:
2144:
2140:
2133:
2120:
2111:
2107:
2103:
2101:
2096:
2092:
2088:
2084:
2082:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1965:
1961:
1957:
1955:
1942:
1938:
1934:
1930:
1926:
1922:
1919:
1915:
1912:
1908:
1904:
1900:
1896:
1892:
1891:
1889:conditions.
1886:
1882:
1880:
1854:
1850:
1846:
1842:
1838:
1834:
1749:
1745:
1741:
1737:
1735:
1730:
1727:Michell 1988
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1684:
1680:Michell 2009
1675:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1636:Michell 2009
1631:
1629:
1624:
1620:
1617:Michell 1988
1614:
1498:
1493:
1489:
1487:
1428:
1319:
1222:
1167:Given that:
1166:
1107:
1048:
1046:
1041:
1037:
1033:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
981:
977:
975:
956:
952:
948:
944:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
884:
880:
876:
872:
871:to quantify
868:
864:
860:
856:
852:
848:
846:
842:independence
841:
838:independence
837:
833:
829:
825:
821:
817:
813:
809:
805:
803:
798:
794:
790:
786:
782:
778:
774:
770:
766:
762:
758:
754:
750:
746:
742:
738:
734:
730:
726:
722:
718:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
644:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
586:
570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
533:. That is, (
530:
526:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
472:
463:
458:
455:
450:
446:
443:
427:
423:
421:
377:
367:
365:
354:
332:Amos Tversky
321:
312:psychometric
294:
290:Michell 1990
286:
266:
253:
249:
238:Amos Tversky
231:
220:
210:
196:
184:Michell 1999
175:
171:
161:
144:
128:Fischer 1995
120:Brogden 1977
105:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
53:
28:
24:
20:
18:
3934:(1): 1–27.
3830:(1): 1–34.
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3082:August 2011
3074:introducing
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697:such that (
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234:Krantz 1968
203:topological
199:Otto Hölder
116:Rasch model
101:real number
4431:Categories
3772:: 89–109.
3274:Metrologia
3267:. Erlbaum.
3114:: 89–100.
3030:References
2870:Cliff 1992
2187:Kline 1998
2116:Archimedes
1740:levels of
1731:Luce–Tukey
1697:levels of
649:satisfies
505:; and let
461:quantity.
223:Dana Scott
136:Kline 1998
132:Keats 1967
108:Cliff 1992
45:John Tukey
4300:144782881
4113:120080826
3936:CiteSeerX
3786:143679173
3617:122303701
3510:CiteSeerX
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