The number of two-way tables satisfying certain additivity axioms

Arbuckle, James; Larimer, James

doi:10.1016/0022-2496(76)90036-5

Cited by 20 publications

(14 citation statements)

References 7 publications

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“…Even if the axioms of transitivity, independence, and double cancellation had been shown to stand up to empirical testing, we ought to be careful in evaluating the status of these necessary, but not sufficient, conditions as "practically sufficient," because it depends on the size of the design. Generally, the larger the design, the smaller the probability that additivity exists when the validity of the three necessary conditions has been established (Arbuckle & Larimer, 1976;McClelland, 1977). Although Arbuckle & Larimer's Monte Carlo studies do not report values for a 6 X 6 design (as was used by Levelt et al), we may take their value for a 7X5 design as a minimum: No less than 68% of all cases in which independence and double cancellation are valid can still be shown to be nonadditive, a proportion that would be even higher for a 6 X 6 matrix of stimuli.…”

Section: Aims Of the Present Articlementioning

confidence: 99%

Are there limits to binaural additivity of loudness?

Gigerenzer¹,

Strube²

1983

Journal of Experimental Psychology: Human Perception and Perfor

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In recent years there has been notable interest in additive models of sensory integration. Binaural additivity has emerged as a main hypothesis in the loudness-scaling literature and has recently been asserted by authors using an axiomatic approach to psychophysics. Restrictions of the range of stimuli used in the majority of former experiments, and inherent weaknesses of the axiomatic study by Levelt, Riemersma, and Bunt (1972) are discussed as providing reasons for the present investigation. A limited binaural additivity (LBA) model is proposed that assumes contralateral binaural inhibition for interaural intensity differences that exceed a critical level. Experimental data are reported for 12 subjects in a loudness-matching task designed to test the axioms of cancellation and of commutativity, both necessary to the existence of strict binaural additivity. In a 2 X 2 design, frequencies of 200 Hz and 2 kHz were used, and mean intensity levels were 20 dB apart. Additivity was found violated in 33 out of 48 possible tests. The LBA model is shown to predict the systematic nonadditivity in the loudness judgments and to conform to results from other studies.

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Section: Aims Of the Present Articlementioning

confidence: 99%

Are there limits to binaural additivity of loudness?

Gigerenzer¹,

Strube²

1983

Journal of Experimental Psychology: Human Perception and Perfor

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“…The result for f (3, 3) is not new since it was noted in Table 2c in Arbuckle 6 Larimer (1976) and is verified in Table 1 in Ullrich 6 Wilson (1993). However, I include a proof of f (3, 3)=3 in Section 5, followed by an outline of a proof of f(4, 4)=4.…”

Section: Results For Finite Structuresmentioning

confidence: 85%

“…It seems extremely hard to determine f (n 1 , n 2 , ..., n N ) precisely except in a few cases. The main reason is that the problem is highly combinatoric, as suggested perhaps by the computations in Arbuckle 6 Larimer (1976), McCelland (1977, and Ullrich 6 Wilson (1993) for the number of ways of ordering cells in finite arrays for N=2 and N=3 that satisfy coordinate independence and more complex cancellation conditions. I have found it a very delicate task, using a combination of linear algebra and combinatorial notions, to construct instances of weak orders or linear orders that satisfy C(2), C(3), ..., C(K&1) but violate C(K) when K appears to be in the neighborhood of f(n 1 , n 2 , ..., n N ).…”

Section: Introductionmentioning

confidence: 98%

Cancellation Conditions for Finite Two-Dimensional Additive Measurement

Fishburn

2001

Journal of Mathematical Psychology

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“…Beyond that, no reliable statistical measure is available from which we could infer the bias and limits of these tests. Some statistical evidence is available (e.g., Arbuckle & Larimer, 1976). These authors urge caution in interpreting double cancellation tests.…”

Section: Resultsmentioning

confidence: 99%

A conjoint measurement approach to color harmony

Pieters

1979

Perception & Psychophysics

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The aim of this study was to find a rule in color harmony that could reveal the mutual influences of the three basic harmonies, those of hue, lightness, and saturation. One hundred and four color pairs were judged for harmony by five subjects five times. The results were analyzed by the conjoint measurement algorithm of . It was concluded that there is an interaction between saturation harmony and hue and lightness harmony together, and an additivity between hue harmony and lightness harmony. This implies the rule: color harmony = saturation harmony", (hue harmony + lightness harmony).In the history of psychology there have been a number of attempts to define a dependent variable in terms of, i.e., as a function of, several independent variables. These independent variables might interact with one another, or might be added together, or both, in order to determine the dependent variable. A salient example is the controversy between Hull and Spence (cited in The aim of this paper is to discover what combination rules are most appropriate for color harmony. The relevant literature is reviewed below, separately for the topics of color harmony and of combination rules. Color HarmonyThe central notion behind a series of experiments on color harmony by Levelt (Berlyne, 1971, p. 179) was to produce evidence for the existence of rules in color harmony by combining the principal color qualities of hue, lightness, I and saturation. Three separate experiments were done, in each of which one quality was varied at a time while the other two were kept constant.In the first experiment, hue was varied in 96 pairs of color chips consisting of one primary hue (red, yellow, green, or blue) plus one of 24 colors which were about equally spaced along the color circle of the Hesselgren system (Hesselgren, 1953). TheThe author expresses his appreciation to Professor W. J. M. Levell, under whose supervision this study was conducted; he is also indebted to Dr. Ch. M. M. de Weert for reading earlier drafts of this paper. Queries and requests for reprints should be addressed to Jules M. Pieters, Department of Psychology, Erasmuslaan 16,6525 GG Nijmegen, The Netherlands. dependent variable was a judgment on a 7-point scale (l, clashing; 4, neutral; 7, harmonizing). The results were equivocal: The response curves for the four primary colors were quite different.The second experiment varied lightness, using the two color pairs judged highest and the two judged lowest, in harmony in the first experiment. Six lightness levels were used for each of the two colors of a pair. The results were similar to those obtained by Cohn (1894): Harmony judgments were an increasing function of the difference between the lightness of the two elements in the color combination. This function was linear on log-log coordinates, with a slope positively related to the harmony of the color pairs.In the third experiment, in which saturation was varied, there were color pairs with five levels of saturation for each of the two chips in a pair. The saturation data r...

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The number of two-way tables satisfying certain additivity axioms

Cited by 20 publications

References 7 publications

Are there limits to binaural additivity of loudness?

Are there limits to binaural additivity of loudness?

Cancellation Conditions for Finite Two-Dimensional Additive Measurement

A conjoint measurement approach to color harmony

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