Nearest neighbor analysis of point processes: Applications to multidimensional scaling

A new approach for evaluating spatial statistical models based on the (random) number 0 ≤ N(i, n) ≤ n of points whose nearest neighbor is i in an ensemble of n + 1 points is discussed. The second moment of N(i, n) offers a measure of the centrality of the ensemble. The asymptotic distribution of N(i, n) and the expected degree of centrality for several spatial and nonspatial point processes is described. The use of centrality as a diagnostic statistic for multidimensional scaling is explored.     

Analysis of the Talmudic Argumentum A Fortiori Inference Rule (Kal Vachomer) using Matrix Abduction

We motivate and introduce a new method of abduction, Matrix Abduction, and apply it to modelling the use of non-deductive inferences in the Talmud such as Analogy and the rule of Argumentum A Fortiori. Given a matrix \({\mathbb {A}}\) with entries in {0, 1}, we allow for one or more blank squares in the matrix, say a _i,j =?. The method allows us to decide whether to declare a _i,j = 0 or a _i,j = 1 or a _i,j =? undecided. This algorithmic method is then applied to modelling several legal and practical reasoning situations including the Talmudic rule of Kal-Vachomer. We add an Appendix showing that this new rule of Matrix Abduction, arising from the Talmud, can also be applied to the analysis of paradoxes in voting and judgement aggregation. In fact we have here a general method for executing non-deductive inferences.     

A quasi-nonmetric method for multidimensional scaling VIA an extended euclidean model

An Extended Two-Way Euclidean Multidimensional Scaling (MDS) model which assumes both common and specific dimensions is described and contrasted with the standard (Two-Way) MDS model. In this Extended Two-Way Euclidean model then stimuli (or other objects) are assumed to be characterized by coordinates onR common dimensions. In addition each stimulus is assumed to have a dimension (or dimensions) specific to it alone. The overall distance between objecti and objectj then is defined as the square root of the ordinary squared Euclidean distance plus terms denoting the specificity of each object. The specificity,s _j , can be thought of as the sum of squares of coordinates on those dimensions specific to objecti, all of which have nonzero coordinatesonly for objecti. (In practice, we may think of there being just one such specific dimension for each object, as this situation is mathematically indistinguishable from the case in which there are more than one.)We further assume that _ij =F(d _ij ) +e _ij where _ij is the proximity value (e.g., similarity or dissimilarity) of objectsi andj,d _ij is the extended Euclidean distance defined above, whilee _ij is an error term assumed i.i.d.N(0, ²).F is assumed either a linear function (in the metric case) or a monotone spline of specified form (in the quasi-nonmetric case). A numerical procedure alternating a modified Newton-Raphson algorithm with an algorithm for fitting an optimal monotone spline (or linear function) is used to secure maximum likelihood estimates of the paramstatistics) can be used to test hypotheses about the number of common dimensions, and/or the existence of specific (in addition toR common) dimensions.This approach is illustrated with applications to both artificial data and real data on judged similarity of nations.     

On linear classifications under varying choice probabilities

The paper deals with certain problems connected with the assumption that choice probabilities p_s(x, y) depend on the subject s. A set of postulates is given, which implies the existence of sequences of “classification standards”, i.e., sequences {z_j} such that whenever we have 0 < p_s0(x, z_i) < 1 for some s₀ and i, then p_s(z_i+k, x) = p_s(x, z_i?k) = 1 for all s, and k ≥ 1. Elements of any such sequence {z_j} can serve as boundaries between successive categories of classification based on the following rule: Assign x to jth category if you feel it is “to the right” of z_j and “to the left” of z_j+1. Under the condition stated above this rule is unambiguous, and the resulting classification has the property that every element is assigned to one of the two neighboring categories, regardless who performs the classification.Next, the postulates are enriched so as to imply the existence of “tightest” among such sequences {z_j}, hence leading to a classification with largest number of categories.     

Is nevtral NEUTRAL? Visual similarity effects in the early phases of written-word recognition

For simplicity, contemporary models of written-word recognition and reading have unspecified feature/letter levels—they predict that the visually similar substituted-letter nonword PEQPLE is as effective at activating the word PEOPLE as the visually dissimilar substituted-letter nonword PEYPLE. Previous empirical evidence on the effects of visual similarly across letters during written-word recognition is scarce and nonconclusive. To examine whether visual similarity across letters plays a role early in word processing, we conducted two masked priming lexical decision experiments (stimulus-onset asynchrony = 50 ms). The substituted-letter primes were visually very similar to the target letters (u/v in Experiment 1 and i/j in Experiment 2; e.g., nevtral–NEUTRAL). For comparison purposes, we included an identity prime condition (neutral–NEUTRAL) and a dissimilar-letter prime condition (neztral-NEUTRAL). Results showed that the similar-letter prime condition produced faster word identification times than the dissimilar-letter prime condition. We discuss how models of written-word recognition should be amended to capture visual similarity effects across letters.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

Segregation by color and stereoscopic depth in three-dimensional visual space

Two experiments were conducted to investigate how color and stereoscopic depth information are used to segregate objects for visual search in three-dimensional (3-D) visual space. Eight observers were asked to indicate the alphanumeric category (letter or digit) of the target which had its unique color and unique depth plane. In Experiment 1, distractors sharing a common depth plane or a common color appeared in spatial contiguity in thexy plane. The results suggest that visual search for the target involves examination of kernels formed by homogeneous items sharing the same color and depth. In Experiment 2, thexy contiguity of distractors sharing a common color or a common depth plane was varied. The results showed that when target-distractor distinction becomes more difficult on one dimension, the other dimension becomes more important in performing visual search, as indicated by a larger effect on search time. This suggests that observers can make optimal use of the information available. Finally, color had a larger effect on search time than did stereoscopic depth. Overall, the results support models of visual processing which maintain that perceptual segregation and selective attention are determined by similarity among objects in 3-D visual space on both spatial and nonspatial stimulus dimensions.     

Models for measurement,precision, and the nondichotomization of graded responses

It is common in educational, psychological, and social measurement in general, to collect data in the form of graded responses and then to combine adjacent categories. It has been argued that because the division of the continuum into categories is arbitrary, any model used for analyzing graded responses should accommodate such action. Specifically, Jansen and Roskam (1986) enunciate ajoining assumption which specifies that if two categoriesj andk are combined to form categoryh, then the probability of a response inh should equal the sum of the probabilities of responses inj andk. As a result, they question the use of the Rasch model for graded responses which explicitly prohibits the combining of categories after the data are collected except in more or less degenerate cases. However, the Rasch model is derived from requirements of invariance of comparisons of entities with respect to different instruments, which might include different partitions of the continuum, and is consistent with fundamental measurement. Therefore, there is a strong case that the mathematical implication of the Rasch model should be studied further in order to understand how and why it conflicts with the joining assumption. This paper pursues the mathematics of the Rasch model and establishes, through a special case when the sizes of the categories are equal and when the model is expressed in the multiplicative metric, that its probability distribution reflects the precision with which the data are collected, and that if a pair of categories is collapsed after the data are collected, it no longer reflects the original precision. As a consequence, and not because of a qualitative change in the variable, the joining assumption is destroyed when categories are combined. Implications of the choice between a model which satisfies the joining assumption or one which reflects on the precision of the data collection considered are discussed.     

How one gambles if one must: Effects of differing return rates on multistage betting decisions

In the multistage betting game (MBG), a decision maker (DM) is provided with some capital x which he is required to bet over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_{i = 1}^mp_i = 1). If y_i is bet on alternative i (y_i ≥ 0, Σ_{i = 1}^my_i = x) and alternative i obtains, the DM's capital for the next stage is y_ir_i, (r_i > 0). The MBG lasts until either the DM loses his capital or N stages elapse, whichever comes first. Each of six subjects participated in six sessions consisting of several hundred 3-alternative MBG stages. A within-subject design assigned negative expected value (EV) bets to the first three sessions and positive EV bets to three more sessions. Significant effects were found due to return rate, capital size, homogeneous runs of either wins or losses, and individual differences. Four maximization of expected utility and two minimization of risk models were presented and tested. A modified logarithmic utility model is proposed, which provides the best fit to the data. The implications of the results and directions for further research are briefly discussed.     

Stochastic processes and the guttman simplex

The problem of whether a precise connection exists between stochastic processes of the type considered in mathematical learning theory and the Guttman simplex is investigated. The approach is to consider a class of models that characterize the sequential properties of discrete data, and to derive a set of conditions which a model must satisfy in order to generate inter-trial correlationsr _ij with the ‘perfect simplex’ property:r _ik =r _ij r _jk for all trialsi<j<k. It is shown that the Chapman-Kolomogorov Equations provide a necessary and sufficient condition for this property to hold. It follows that a process which is Markovian in the errors and successes will have this property. It is also shown that, if certain reliability assumptions are introduced, then the all-or-none model has the simplex property if the appropriate correlation coefficients are corrected for attenuation. This research was supported by a grant from the United States Public Health Service, NIMH 07722.     

Nested models of relative judgment: Applications to a similarity averaging model

A theoretical analysis of the Eisler and Ekman (1959) model of similarity judgments for unidimensional continua is presented, based on a general model of relative judgment. This general model assumes that judgments are mediated by perceived relations of pairs of stimuli, that there exists a transformation of the judgmental response that is a function of the sensory ratio of the two stimuli, and that response bias operates in a multiplicative manner. Three structural conditions are presented, each imposing constraints on the structure of observed judgments. The structural conditions define threenested models of relative judgment, with the second a weakened version of the first, and the third a weakened version of the second. The special virtue of the general model is that it is applicable to a variety of judgmental tasks (e.g., ratio estimation, similarity, pair comparison), the key being derivation of theresponse transformation conforming to the structural conditions. The structural conditions thus constitute necessary conditions for several different judgmental models. The theory was first applied with success to ratio estimation judgments (Fagot, 1978), and this paper applies the general model to the Eisler and Ekman similarity “averaging” model. Empirical tests were carried out on published data for pitch, darkness, visual area, and heaviness judgments. Although the strong form of the model presented by Eisler and Ekman was rejected, weakened versions were generally supported by the data. These results were similar to those obtained for ratio estimation (Fagot, 1978), and are interpreted to be very promising for the general model of relative judgment.     

The multidimensional randomized response design: Estimating different aspects of the same sensitive behavior

The conventional randomized response design is unidimensional in the sense that it measures a single dimension of a sensitive attribute, like its prevalence, frequency, magnitude, or duration. This paper introduces a multidimensional design characterized by categorical questions that each measure a different aspect of the same sensitive attribute. The benefits of the multidimensional design are (i) a substantial gain in power and efficiency, and the potential to (ii) evaluate the goodness-of-fit of the model, and (iii) test hypotheses about evasive response biases in case of a misfit. The method is illustrated for a two-dimensional design measuring both the prevalence and the magnitude of social security fraud.     

A comparison of two response time models applied to perceptual matching

Two models, a Poisson race model and a diffusion model, are fit to data from a perceptual matching task. In each model, information about the similarity or the difference between two stimuli accumulates toward thresholds for either response. Stimulus variables are assumed to influence the rate at which information accumulates, and response variables are assumed to influence the level of the response thresholds. Three experiments were conducted to assess the performance of each model. In Experiment 1, observers performed under different response deadlines; in Experiment 2, response bias was manipulated by changing the relative frequency ofsame anddifferent stimuli. In Experiment 3, stimulus pairs were presented at three eccentricities: foveal, parafoveal, and peripheral. We examined whether the race and diffusion models could fit the response time and accuracy data through changes only in response parameters (for Experiments 1 and 2) or stimulus parameters (for Experiment 3). Comparisons between the two models suggest that the race model, which has not been studied extensively, can account for perceptual matching data at least as well as the diffusion model. Furthermore, without the constraints on the parameters provided by the experimental conditions, the diffusion and the race models are indistinguishable. This finding emphasizes the importance of fitting models across several conditions and imposing logical psychological constraints on the parameters of models.     

A note concerning the creager—Valentine paper

TheU _i statistic, discussed by Creager and Valentine in a recent paper, is defined as a standard error of estimate for the regression of a composite, consisting ofn components, on then–1 components, exclusive of thejth component. It is shown that this standard error equals the standard error of estimate for the prediction of thejth component from the othern–1 components. This equality is used to suggest an efficient procedure to compute all primary uniquenessesU _i .     

Detection errors onthe andand: Evidence for reading units larger than the word

In five experiments, subjects read lO0-word passages and circled instances of a given target letter, letter group, or word. In each case subjects made a disproportionate number of detection errors on the common function wordsthe andand. The predominance of errors on these two words was reduced for passages in which the words were placed in an inappropriate syntactic context and for passages in which word-group identification was disturbed by the use of mixed typecases or a list, rather than a paragraph, format. These effects for the wordand were not found for the control wordant. These results were taken as evidence that familiar word sequences may be read in units larger than the word, probably short syntactic phrases or word frames. A tentative model of the reading process consistent with these results is proposed.     

A Statistical Hypothesis Testing Method for the Rank Ordering of the Priorities of the Alternatives in the Analytic Hierarchy Process

In analytic hierarchy process (AHP), a ratio scale (π₁, π₂, ⋯, π_t) for the priorities of the alternatives {T₁, T₂, ⋯, T_t} is used for a decision problem in which π_i/π_j is used to quantify the ratio of the priority of T_i to that of T_j. In a group decision‐making setup, the subjective estimates of π_i/π_j are obtained as entries of a pairwise comparison matrix for each member of the group. On the basis of these pairwise comparison matrices, one of the topics of interest in some situation is the total rank ordering of the priorities of the alternatives. In this article, a statistical method is proposed for testing a specific total rank ordering of the priorities of the alternatives. The method developed is then illustrated using numerical examples. Copyright © 2014 John Wiley & Sons, Ltd.     

Cultural threat and perceived realistic group conflict as dual predictors of prejudice

Recent research has focused on how perceived intergroup similarity influences stereotyping and prejudice. Very little is known, however, regarding how the quality or type of similarity influences intergroup relations. Presented is a methodology that allows one to manipulate the quality of perceived intergroup similarity. This methodology is used to test contrasting predictions about how perceptions of intergroup similarity on self-stereotyped interpersonal and work-related traits predict attitudes towards immigrants. Predictions were derived from cultural threat and perceived realistic group conflict theories. Some participants were asked to rate how similar they perceived their in-group was to Mexican immigrants, whereas others were asked to evaluate how the groups differed on the given traits. Control participants evaluated themselves on the given traits. Participants were presented with either interpersonal traits or work related traits as stimuli. The main dependent measures were a perceived realistic conflict scale, a prejudice scale, and a stereotyping scale. All three scales used Mexican immigrants as the target category. When interpersonal traits were made salient, contrast comparisons led to more negative attitudes towards immigrants, supporting a cultural threat hypothesis. When work-related traits were made salient, similarity comparisons led to more prejudice and more negative attitudes towards immigrants, supporting a perceived realistic conflict hypothesis. Thus, a perceived threat to either the cultural norm or economic well being led to more negative attitudes towards immigrants. Results are discussed for their relevance to models of intergroup relations.     

Fairness in Technological Design

This paper addresses an important multi-disciplinary issue of current interest, that is, the implications of technological design for fairness. A visual, graphical methodology centered on the Taylor-Russell diagram is proposed to address this issue. The Taylor-Russell diagram helps to identify and explore ways in which predictions built into designs can pit the interests of different constituencies against one another. The configuration of the design represents a trade-off between the interests of the communities involved. Whether or not the trade-off is appropriate constitutes a problem of fairness or distributive justice. The breadth of this methodology is supported by a diversity of examples analyzed. These include a surveillance system, an automotive safety system, a civic information system, and the international food distribution system. These examples provide models for application of the methodology to the analysis of designs in further areas of concern. Limitations of the methodology are also discussed. While it helps to identify and clarify issues of fairness in technology design, the methodology does not provide a general theory of fairness, nor can it provide fair solutions to such issues without appeal to further principles or concepts.