Sequential dependencies and regression in psychophysical judgments

A tendency for judgments of stimulus magnitude to be biased in the direction of the value of the immediately preceding stimulus is found in magnitude estimations of loudness. This produces a bias in the empirical psychophysical function that results in underestimation of the exponent of the unbiased function presumed to relate number and stimulus intensity, N = aSⁿ. The biased judgment can be represented as a power product of focal and preceding stimulus intensity, N_ij= aS ^m S_j ^b. A bias-free estimate of the correct exponent, n, can be obtained from the relation n = m + b.     

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.     

Ratio scale measurement of the perceived lengths of lines.

In Experiments 1 and 2, magnitude and category estimates of perceived line length difference were used to rank order 36 pairs of lines with respect to the psychological length difference between the lines forming a pair. In Experiment 3 magnitude estimates of line length ratios were used to order the same 36 pairs with respect to the psychological length ratio between the lines comprising a pair. A nonmetric analysis of the rank-order of the difference and ratio judgments was used to construct a ratio scale of perceived line length. Perceived line length was found to be a power function of physical line length with an exponent of .46. In Experiments 4 and 5, magnitude and category estimates of line length similarity were obtained. A nonmetric analysis of these data indicated that similarity judgments were monotonic inverses of judgments of psychological ratios.     

The PIG in sPrInG: Evidence on letter grouping from the reading of buried words

We introduce a novel procedure for investigating factors that determine selective attention to letters in words. Participants were presented with words (in Experiments 1 and 3) and nonwords (in Experiment 2) that contained a buried word whose letters differed in color relative to the other letters present (e.g.,pig, in spring). The strings were presented in single case or mixed case, keeping the letters of the buried words in one case (SpRiNg). The time in which the whole stimulus was named was shorter for same-case than for mixed-case strings (forspring: spring < SpRiNg). In contrast, the time in which buried words were named was shorter in mixed- than in same-case strings (forpig: spring > SpRiNg). Across items, the effects of case mixing were negatively correlated across the two tasks. The positive effect of case mixing for buried words also occurred irrespective of whether the whole string was a word or a nonword, and there were contributions from similarity of both letter size and case. The results suggest that case mixing can facilitate selective attention to letters, which is otherwise disrupted by size- and case-based grouping across letter strings. The study provides evidence for letter grouping using size and case information.     

A deduction theorem schema for deductive systems of propositional logics

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) _s q.     

A direct test of the unequal-variance signal detection model of recognition memory

Analyses of the receiver operating characteristic (ROC) almost invariably suggest that, on a recognition memory test, the standard deviation of memory strengths associated with the lures (σ_lure) is smaller than that of the targets (σ_target). Often, σ_lure/σ_target ≈ 0.80. However, that conclusion is based on a model that assumes that the memory strength distributions are Gaussian in form. In two experiments, we investigated this issue in a more direct way by asking subjects to simply rate the memory strengths of targets and lures using a 20-point or a 99-point strength scale. The results showed that the standard deviation of the ratings made to the targets (s _target) was, indeed, larger than the standard deviation of the ratings made to the lures (s _lure). Moreover, across subjects, the ratios _lure/s _target correlated highly with the estimate of σ_lure/σtarget obtained from ROC analysis, and both estimates were, on average, approximately equal to 0.80.     

Reasons and Entailment

What is the relation between entailment and reasons for belief? In this paper, I discuss several answers to this question, and I argue that these answers all face problems. I then propose the following answer: for all propositions p ₁,…,p _n and q, if the conjunction of p ₁,…, and p _n entails q, then there is a reason against a person’s both believing that p ₁,…, and that p _n and believing the negation of q. I argue that this answer avoids the problems that the other answers to this question face, and that it does not face any other problems either. I end by showing what the relation between deductive logic, reasons for belief and reasoning is if this answer is correct.     

Stimulus Presentation Ratios And The Outcomes For Correct Responses In Signal-detection Procedures

Three pigeons were trained to discriminate between two line orientations (S₁ and S₂). A left-key peck was correct when S₁ was presented, and a right key-peck was correct when S₂ was presented. In all procedures, correct responses were occasionally reinforced with food paired with the presentation of the magazine light. Incorrect responses produced a blackout. Six detection procedures were used. In the first, the signal presentation ratio was varied across conditions and the reinforcer ratio was allowed to covary. In the second, the signal presentation ratio was held constant at 1:1 and the reinforcer ratio varied across conditions. In the third, the signal presentation ratio was varied across conditions and the reinforcer ratio was held constant at 1:1. In these three procedures, correct responses that were not scheduled for reinforcement were followed by blackout. The remaining three procedures repeated those described above with one procedural change: Nonreinforced but correct trials were followed by the presentation of the magazine light. Birds showed systematic preferences for the key associated with the stimulus presented or reinforced most often. There was no change in the birds' performance over changes in the feedback for nonreinforced but correct responses.     

How accurate is size and distance perception for very far terrestrial objects? Function and causality

This study investigated absolute estimation of size and distance for natural and artificial objects at viewing distances of 1.1–15.3 km (Experiments 1 and 2) and 0.4–5.0 m (Experiment 3). The main results were that, regardless of distance range, size and distance estimates (S′ andD′) were related to objective size and distance (S andD), respectively, by a power function with an exponent of unity, but great individual differences in exponent were obtained for the far objects. The ratioS′/D′ was reasonably represented byS′/D′ =Kθ ⁿ andS′/D′ = tan(aθ +b), rather thanS′/D′ = tan θ, where θ is the visual angle. Partial correlations were obtained to examine whether (1) apparent size is determined by taking apparent distance into account or (2) both apparent size and apparent distance are determined directly by external stimuli. The combined data for the far objects and the data for the close objects showed that there were high correlations betweenS andS′ and betweenD andD′ and a low correlation betweenD′ andS′. The data of Experiment 2 showed that bothD′ andS′ were highly correlated withS D, and θ, and there was a high positive correlation betweenD′ andS′. It was suggested that the direct-perception model is valid under some situations, but the taking-into-account model is not supported in any set of data.     

The cognitive implications of asymmetric color generalization in honeybees

Generalization occurs when a conditioned response formed to one stimulus is also elicited by other stimuli which have not been used in the course of conditioning. Here, we studied color generalization in honeybees Apis mellifera trained to two rewarded colors, S₁+ and S₂+. After training, bees were tested with non-rewarded novel stimuli, which lay between the trained stimuli in a honeybee color space (Int) or outside the range defined by the trained stimuli (E ₁ and E ₂). We analyzed whether bees interpolated their choice to Int and/or extrapolated it to E ₁ and E ₂. We compared the performances of the group trained with S₁+ and S₂+ to those of control groups trained only with S₁+ or S₂+. Bees trained with S₁+ and S₂+ responded similarly and highly to all test stimuli. These results do not allow discerning between generalization models based on the presence of interpolation and/or extrapolation. Nevertheless, bee’s performance was consistent with a linear summation of the two generalization gradients generated by S₁+ and S₂+, respectively. These gradients were asymmetric because control bees responded to the test stimuli as if these belonged to different similarity classes in spite of the fact that they had similar perceptual distances separating them. Stimuli treated as similar were located in the same half of the color spaces, whereas stimuli treated as different were located in opposite halves. Our results suggest that color categories could exist in honeybees and may underlie the performance of the control groups. Under this assumption, color categories would be also present in simpler nervous systems, and would not require factors such as language to be expressed.     

On finite linear intermediate predicate logics

An intermediate predicate logicS ₊ ⁿ (n>0) is introduced and investigated. First, a sequent calculusGS _n is introduced, which is shown to be equivalent toS ₊ ⁿ and for which the cut elimination theorem holds. In § 2, it will be shown thatS ₊ ⁿ is characterized by the class of all linear Kripke frames of the heightn.To the memory of the late Professor Iwao Nishimura     

SCALES FOR SUBJECTIVE DISTANCE

By means of the method of ratio estimation, scale values were obtained for subjective distance. In three experiments different stimulus ranges of the objective distances were used. It was found: ( 1 ) Subjective distance is a power function of the objective distance. ( 2 ) The exponent of the function varies with the stimulus range. With increasing stimulus range the exponent has a tendency to decrease. It is conceivable that the change of the exponent may be explained by an adaptation of the subjective range to the stimulus range.     

From marine ecology to crime analysis: Improving the detection of serial sexual offences using a taxonomic similarity measure

Jaccard has been the choice similarity metric in ecology and forensic psychology for comparison of sites or offences, by species or behaviour. This paper applies a more powerful hierarchical measure—taxonomic similarity (Δ_s), recently developed in marine ecology—to the task of behaviourally linking serial crime. Forensic case linkage attempts to identify behaviourally similar offences committed by the same unknown perpetrator (called linked offences). Δ_s considers progressively higher‐level taxa, such that two sites show some similarity even without shared species. We apply this index by analysing 55 specific offence behaviours classified hierarchically. The behaviours are taken from 16 sexual offences by seven juveniles where each offender committed two or more offences. We demonstrate that both Jaccard and Δ_s show linked offences to be significantly more similar than unlinked offences. With up to 20% of the specific behaviours removed in simulations, Δ_s is equally or more effective at distinguishing linked offences than where Jaccard uses a full data set. Moreover, Δ_s retains significant difference between linked and unlinked pairs, with up to 50% of the specific behaviours removed. As police decision‐making often depends upon incomplete data, Δ_s has clear advantages and its application may extend to other crime types. Copyright © 2007 John Wiley & Sons, Ltd.     

Transcending Lindemann's predictions for the melting temperatures of metals and for the long-wavelength limit of their liquid structure factors

Lindemann's melting criterion remains useful. However, one prediction it makes for liquid metals (our focus here) is that the long-wavelength limit of the structure factor S(q) at freezing, S _{T m} (0), where T _m is the melting temperature, is a universal constant. For 34 metals we have calculated S _{T m} (0) from input data, which is essentially the measured T _m and the surface thickness L, defined near freezing as the product of isothermal compressibility and surface tension. To complete the characterization of S _{T m} (0) we fit to one metal, chosen as Rb, for which S _{T m} (0) is well established experimentally. For a wide variety of metals considered, S _{T m} (0) is then found to vary by a factor of 10.     

Average exponent from magnitude production

When magnitude production is used to obtain a psychophysical power function for a group ofSs, the group exponent is shown to be the harmonic mean of individual-S exponents under the conditions usually employed in empirical research. The conditions are (a) the same numbers are presented to eachS, (b) responses to each number are pooled overSs by taking geometric means, and (c) the function is fit by the method of least squares in which regression is of the logarithm ofSs' responses upon the logarithm of numbers presented byE.This paper was supported in part by grant no. APA-151 from the National Research Council of Canada. The author wishes to thank William W. Rozeboom for his helpful comments on this paper.     

Some necessary conditions for common-factor analysis

LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU ² such thatG = R – U ² is Gramian and of minimum rankr. Lets ₁ be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr s ₁. Two further lower bounds tor are also established that are better thans ₁. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.This research was made possible in part by an uncommitted grant-in-aid from the Behavioral Sciences Division of the Ford Foundation.     

Changes in absolute detection threshold and in subjective intensity of suprathreshold stimuli during olfactory adaptation and recovery

Olfactory adaptation and recovery to methyl isobutyl ketone at a concentration 10 times the absolute detection threshold (I_{t o}) was intensively studied in two human Ss. A combined psychophysical procedure was used that allowed comparisons of changes in threshold (I_t) with changes in the subjective intensity of suprathreshold stimuli. Information was also obtained on the effect of the adapting stimulus on the psychophysical power function for this odorant. A threshold detection procedure was used to estimate changes in I_t; an unstructured magnitude-estimation procedure was used to monitor changes in the subjective intensity of suprathreshold stimuli and the psychophysical power function. The data provided additional information on the behavioral course of olfactory adaptation and recovery and suggested that this combined method can be used profitably for further investigations of this kind. Complementary to the work of Cain and Engen (1969), the results suggested an increase in the exponent of the power function with increasing adaptation.     

Maximum likelihood estimation of the joint covariance matrix for sections of tests given to distinct samples with application to test equating

Consider an old testX consisting ofs sections and two new testsY andZ similar toX consisting ofp andq sections respectively. All subjects are given testX plus two variable sections from either testY orZ. Different pairings of variable sections are given to each subsample of subjects. We present a method of estimating the covariance matrix of the combined test (X ₁, ...,X _s ,Y ₁, ...,Y _p ,Z ₁, ...,Z _q ) and describe an application of these estimation techniques to linear, observed-score, test equating.The author is indebted to Paul W. Holland and Donald B. Rubin for their encouragement and many helpful comments and suggestions that contributed significantly to the development of this paper.This research was supported by the Program Statistics Research Project of the ETS Research Statistics Group.     

The Impact of Fallible Item Parameter Estimates on Latent Trait Recovery

In this paper we propose an upward correction to the standard error (SE) estimation of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} , the maximum likelihood (ML) estimate of the latent trait in item response theory (IRT). More specifically, the upward correction is provided for the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} when item parameter estimates obtained from an independent pretest sample are used in IRT scoring. When item parameter estimates are employed, the resulting latent trait estimate is called pseudo maximum likelihood (PML) estimate. Traditionally, the SE of [^(q)]_ML\hat{\theta}_{\mathrm{ML}} is obtained on the basis of test information only, as if the item parameters are known. The upward correction takes into account the error that is carried over from the estimation of item parameters, in addition to the error in latent trait recovery itself. Our simulation study shows that both types of SE estimates are very good when θ is in the middle range of the latent trait distribution, but the upward-corrected SEs are more accurate than the traditional ones when θ takes more extreme values.     

Knowledge despite falsehood

I examine the claim, made by some authors, that we sometimes acquire knowledge from falsehood. I focus on two representative cases in which a subject S infers a proposition q from a false proposition p. If S knows that q, I argue, S's false belief that p is not essential to S's cognition. S's knowledge is instead due to S's belief that p′, a proposition in the neighbourhood of p that S (dispositionally) believes (and knows). S thus knows despite her false belief. The widely accepted and plausible principle that inferential knowledge requires known premises is unscathed.