Testing independence in two-way contingency tables with data subject to misclassification

The misclassification process is represented by a stochastic matrix containing the probabilities that an individual who belongs in one cell is counted as belonging to another (or perhaps the same) cell. These probabilities are supposed known. If misclassification in the row direction is independent of that along the column variable then the size of the usual chi-square test is unchanged. It is shown how to calculate loss of power in this case and also how to calculate the change in size of the test if the errors are not independent. A modified test criterion is suggested when errors are not independent and a numerical example is included.     

Cumulative and local odds-ratio measures of uniformity of association in two-way ordered contingency tables

The cumulative odds-ratio model of uniformity of association in a two-way ordered contingency table is studied, and compared with Goodman's local odds-ratio model. It is shown that the cumulative model can be adapted to the deletion of cells, particularly deletion of diagonal cells in square tables, and an example of the improvement of fit obtained is given, and compared with the fit of the local model. The sensitivity of both models to collapse of categories and to changes in marginal totals is discussed, suggesting criteria for choosing between them. The Appendix describes simple computational algorithms for both models, based on minimizing the likelihood ratio over a range of trial parameters.     

On additivity of duration reproduction functions

In the present paper properties of theoretical response functions in the duration reproduction task (drt) are studied. If the drt response is to be interpreted as the measure of a temporal duration elapsed in the past, the reproduction function (rf) must be additively measuring. The additivity property is defined for bivariate rfs, taking into account the delay between retention and reproduction. The requirement that an rf be additively measuring leads to a system of two functional equations, which are solved by (i) linear functions of time, or (ii) functions of the form yielded by the ‘dual klepsydra model’ of duration reproduction [Wackermann, J., & Ehm, W. (2006). The klepsydra model of internal time representation and time reproduction. Journal of Theoretical Biology, 239, 482-493]. This result supports the outstanding rôle of the ‘klepsydraic reproduction function’ (krf) in models of internal representation and reproduction of temporal durations.     

A method of weighing qualitative preference axioms

A method is developed for determining the absolute and relative strengths of qualitative preference axioms in normative Bayesian decision theory. These strengths are calculated for the three most common qualitative axioms; transitivity, the sure-thing principle, and dominance. The relative strength of the latter two axioms with respect to transitivity is calculated for special cases, and a bound is derived which is applicable to a larger class of decision problems. Possible implications of this theoretical work for decision heuristics are discussed.     

Tests of product and additive scaling axioms

A product axiom for ratio judgments and an additive axiom for difference judgments were tested for six Ss judging brightness. The product axiom was rejected for all six Ss whereas the additive axiom was accepted for two Ss. Power functions did not fit the data well, although somewhat better for difference judgments than ratio judgments. A two-stage model fitted somewhat better, but failed to satisfy one important implication of that model, previously confirmed for lifted weights. The data gave stronger support for a theory of brightness estimation based on an additive axiom for difference judgments than a product axiom based on ratio judgments.     

The principle of additivity and its relation to clinical decision making

Clients (N = 178) with varying degrees of organic indicators were administered a psychological battery including the Memory for Designs Test (MFD; Graham & Kendall, 1960) to assess the extent to which joint presence of organic indicators affected the sensitivity of the MFD in identifying organicity. Joint presence failed to add to the discriminability of the MFD, although individual instruments such as the PIAT (Arithmetic) and WAIS-R (PIQ less than VIQ) clearly discriminated across the MFD.     

The role of cue additivity in salience in taste aversion conditioning

In toxiphobia conditioning certain cues are more easily associated with malaise (i.e., more salient) than others. The present experiments show that cue additivity can account for previously found differences in salience of certain fluids. The relative salience of saccharin, saline and casein hydrolysate was tested in normal rats and in animals rendered anosmic by treatment of their nasal passages with zinc sulphate. The normally more salient casein was reduced in salience to the same level as saccharin and saline in the anosmic animals, while anosmia did not affect the relative salience of saccharin and saline. A second study tested the cue properties of vanilla, a fluid of very low salience. Anosmic animals could not show a conditioned aversion to vanilla. Normal animals readily learned the aversion but could not show it following a later anosmia treatment. Anosmia did not disrupt a saccharin aversion previously conditioned in normal animals. The data show that some of the “tastes” used in toxiphobia conditioning have olfactory components, and some are purely olfactory. Further, olfactory cues can summate with taste cues to form a more salient stimulus.     

Failure of additivity in bisection of length

Subjects were intructed to select one rod to lie halfway in length between two given rods. These bisection instructions imply an additive model in the subjective metric. However, the data were inherently nonadditive; the length of the bisector could be an increasing or a decreasing function of the length of one given rod, depending on the length of the other given rod. A convexity analysis and a nonmetric analysis both showed that no monotone transformation could make the data additive. The bisection problem is used to contrast the axiomatic and functional approaches to measurement theory.     

Measures of kearnels of reducibility axioms and singlets

The present paper is a generalization and further development of the theory of Kernel measures of reducibility axioms formulated in [1], [2], [3] in. the years 1969–1973. In this paper logical connections of Kernel measures with some set-theoretical notions are studied and some suggestions related to these connections are formulated.     

The additivity of loudness across critical bands: A conjoint measurement approach

Five subjects were required in each trial to compare directly two sounds and to indicate which sound was louder. Each of the 64 sounds employed consisted of a combination of one of eight intensity levels of a 2-kHz tone and one of eight intensities of a 5-kHz tone. If, as Fletcher and Munson (1933) argued, loudness is additive for tone combinations in which the frequencies are widely separated, then subjects’ judgments should reflect the summed loudnesses of the 2- and 5-kHz tones in a two-tone combination. Judgments of individual subjects were shown to satisfy the conditions for an additive structure, and individual loudness scales were constructed. These loudness scales varied from subject to subject. Since this paired comparison procedure minimized response biases, the results suggest substantial individual differences in the sensory representation of sound intensity. The relations among sensory scales derived from other structured sensory judgments, such as binaural loudness, are discussed.     

Decision theoretic foundations for axioms of rational preference

Rationality postulates for preferences are developed from two basic decision theoretic principles, namely: (1) the logic of preference is determined by paradigmatic cases in which preferences are choice-guiding, and (2) excessive comparison costs should be avoided. It is shown how the logical requirements on preferences depend on the structure of comparison costs. The preference postulates necessary for choice guidance in a single decision problem are much weaker than completeness and transitivity. Stronger postulates, such as completeness and transitivity, can be derived under the further assumption that the original preference relation should also be capable of guiding choice after any restriction of the original set of alternatives.     

Subjective sampling distributions and the additivity of estimates

Abstract.— Previous studies of sampling distributions have been conducted almost exclusively under the assumption that persons behave in accordance with the "fundamental convention" of probability, i.e. that the sum of all probability estimates will equal 1. When this assumption was tested by asking subjects to give "unrestricted" probability estimates of all possible outcomes of samples from a given population, a general tendency of overestimation made the sum of all probabilities exceed 1 to a considerable extent. The subjective sampling distributions appeared to be unaffected by sample size ( N=5 or 10) and number of outcomes, and were flatter than the corresponding "objective" sampling distributions.     

All roads lead to violations of countable additivity

This paper defends the claim that there is a deep tension between the principle of countable additivity and the one-third solution to the Sleeping Beauty problem. The claim that such a tension exists has recently been challenged by Brian Weatherson, who has attempted to provide a countable additivity-friendly argument for the one-third solution. This attempt is shown to be unsuccessful. And it is argued that the failure of this attempt sheds light on the status of the principle of indifference that underlies the tension between countable additivity and the one-third solution.