Matrix correlation

A correlational measure for ann byp matrixX and ann byq matrixY assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to transformationsXL andYM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which causeXL andYM to ben bys, wheres may be any number between 1 and min (p, q). An efficient algorithm is described for maximizing the correlation betweenXL andYM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.This research was supported by grant APA 0320 from the Natural Sciences and Engineering Research Council of Canada. The authors wish to acknowledge valuable discussion of this problem with Jan de Leeuw, University of Leiden.     

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.     

Comparison of ranks of cross-product and covariance solutions in component analysis

SupposeD is a data matrix forN persons andn variables, and is the matrix obtained fromD by expressing the variables in deviation-score form. It is shown that ifD has rankr, will always have rank (r−1) ifr=N<n, otherwise it will generally have rankr. If has ranks,D will always have ranks ifs=n, but ifs<n it will generally have rank (s+1). Thus two cases can arise, Case A in whichD has rank one greater than , and Case B in whichD has rank equal to . Implications of this distinction for analysis of cross products versus analysis of covariances are briefly indicated.     

The assessment of knowledge and learning in competence spaces: The gain–loss model for dependent skills

The gain–loss model (GaLoM) is a formal model for assessing knowledge and learning. In its original formulation, the GaLoM assumes independence among the skills. Such an assumption is not reasonable in several domains, in which some preliminary knowledge is the foundation for other knowledge. This paper presents an extension of the GaLoM to the case in which the skills are not independent, and the dependence relation among them is described by a well‐graded competence space. The probability of mastering skill s at the pretest is conditional on the presence of all skills on which s depends. The probabilities of gaining or losing skill s when moving from pretest to posttest are conditional on the mastery of s at the pretest, and on the presence at the posttest of all skills on which s depends. Two formulations of the model are presented, in which the learning path is allowed to change from pretest to posttest or not. A simulation study shows that models based on the true competence space obtain a better fit than models based on false competence spaces, and are also characterized by a higher assessment accuracy. An empirical application shows that models based on pedagogically sound assumptions about the dependencies among the skills obtain a better fit than models assuming independence among the skills.     

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.     

Effects of response type on pigeons' sensitivity to variation in reinforcer amount and reinforcer delay

Twelve pigeons, divided into two groups, responded on concurrent nonindependent variable-interval schedules to obtain access to grain by either pecking keys or pressing treadles. Either the amount of grain or the delay to the receipt of grain was varied in separate conditions to determine the sensitivity of relative responding to variation in reinforcer amount (s_A), the sensitivity to variation in reinforcer delay (s_D), and s_A/s_D, a measure related to self-control. There were no significant differences between the two groups in the values of s_A, s_D, and s_A/s_D. These results suggest that the values of s_A, s_D, and s_A/s_D for pigeons may be similar across these two types of responses.     

EM algorithms for ML factor analysis

The details of EM algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models. The algorithm is essentially the same for both cases and involves only simple least squares regression operations; the largest matrix inversion required is for aq ×q symmetric matrix whereq is the matrix of factors. The example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other; such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.     

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.     

On a property of BCK-identities

A BCK-algebra is an algebra in which the terms are generated by a set of variables, 1, and an arrow. We mean by aBCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two termss andt, if neithers=1 nort=1 is a BCK-identity, ands=t is a BCK-identity, then the rightmost variables of the two terms are identical.This theorem was conjectured firstly in [5], and then in [3]. As a corollary of this theorem, we derive that the BCK-algebras do not form a variety, which was originally proved algebraically by Wroski ([4]).To prove the main theorem, we use a Gentzen-type logical system for the BCK-algebras, introduced by Komori, which consists of the identity axiom, the right and the left introduction rules of the implication, the exchange rule, the weakening rule and the cut. As noted in [2], the cut-elimination theorem holds for this system.Presented byJan Zygmunt     

Significance Testing with No Alternative Hypothesis: A Measure of Surprise

A pure significance test would check the agreement of a statistical model with the observed data even when no alternative model was available. The paper proposes the use of a modified p-value to make such a test. The model will be rejected if something surprising is observed (relative to what else might have been observed). It is shown that the relation between this measure of surprise (the s-value) and the surprise indices of Weaver and Good is similar to the relationship between a p-value, a corresponding odds-ratio, and a logit or log-odds statistic. The s-value is always larger than the corresponding p-value, and is not uniformly distributed. Difficulties with the whole approach are discussed.     

Full-information item bi-factor analysis

A plausibles-factor solution for many types of psychological and educational tests is one that exhibits a general factor ands − 1 group or method related factors. The bi-factor solution results from the constraint that each item has a nonzero loading on the primary dimension and at most one of thes − 1 group factors. This paper derives a bi-factor item-response model for binary response data. In marginal maximum likelihood estimation of item parameters, the bi-factor restriction leads to a major simplification of likelihood equations and (a) permits analysis of models with large numbers of group factors; (b) permits conditional dependence within identified subsets of items; and (c) provides more parsimonious factor solutions than an unrestricted full-information item factor analysis in some cases. Supported by the Cognitive Science Program, Office of Naval Research, Under grant #N00014-89-J-1104. We would like to thank Darrell Bock for several helpful suggestions.     

Quantification Theory in *9 of Principia Mathematica

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown that with a minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition.     

A covariance analysis of multiple paired comparisons

In the method of multiple paired comparisons the dominance of objectj over objectk is observed uponp attributes. The present paper develops a covariance analysis for these paired comparisons in terms of a linear model which includes scale, bias, and interaction effects, along withs covariants upon which the comparisons are presumably dependent. The covariance model gives rise to adjusted parameter estimates and hypothesis tests for the residual pairwise layout from which the effects of thes covariants have been removed. These estimation and testing procedures are illustrated with an analysis of political judgment data, and their relevance to the general problem of residual scaling is discussed.This research was supported by National Institute of Mental Health grants MH 12972 and MH 15506. The author would like to express his appreciation to Jerry Solomon and Wei-Ching Chang of the Oregon Research Institute for the data analysis reported at the end of the paper. Computing assistance was obtained from the Health Sciences Computing Facility, UCLA, sponsored by N.I.H. Grant FR-3.     

The role of probability of success estimates in the sunk cost effect

The sunk cost effect is manifested in a tendency to continue an endeavor once an investment has been made. Arkes and Blumer (1985) showed that a sunk cost increases one's estimated probability that the endeavor will succeed [p(s)]. Is this p(s) increase a cause of the sunk cost effect, a consequence of the effect, or both? In Experiment 1 participants read a scenario in which a sunk cost was or was not present. Half of each group read what the precise p(s) of the project would be, thereby discouraging p(s) inflation. Nevertheless these participants manifested the sunk cost effect, suggesting p(s) inflation is not necessary for the effect to occur. In Experiment 2 participants gave p(s) estimates before or after the investment decision. The latter group manifested higher p(s), suggesting that the inflated estimate is a consequence of the decision to invest. Copyright © 2000 John Wiley & Sons, Ltd.     

Matching of numerical symbols with number of responses by pigeons

Pigeons were trained to peck a certain number of times on a key that displayed one of several possible numerical symbols. The particular symbol displayed indicated the number of times that the key had to be pecked. The pigeons signalled the completion of the requirement by operating a separate key. They received a food reward for correct response sequences and time-out penalties for incorrect response sequences. In the first experiment nine pigeons learned to allocate 1, 2, 3 or 4 pecks to the corresponding numerosity symbols s ₁, s ₂, s ₃ and s ₄ with levels of accuracy well above chance. The second experiment explored the maximum set of numerosities that the pigeons were capable of handling concurrently. Six of the pigeons coped with an s ₁–s ₅ task and four pigeons even managed an s ₁–s ₆ task with performances that were significantly above chance. Analysis of response times suggested that the pigeons were mainly relying on a number-based rather than on a time-based strategy. Received: 11 October 1999 / Accepted after revision: 27 January 2000     

Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

Distribution of the midgap density of states and their capture cross-sections in hydrogenated amorphous silicon deduced from space-charge-limited conduction in the dark and under illumination

Space-charge-limited current measurements in the dark and under illumination offer the means of obtaining the density of states N(E) and their capture cross-sections s(E) in a range of energy E above the equilibrium Fermi level in hydrogenated amorphous silicon (a-Si:H) n⁺–i–n⁺ structures. N(E) and s(E) have been measured in the energy region from 0.65 to 0.5?eV below the conduction-band edge E _c. The decrease on s(E) observed as the energy E increases agrees with recent observations and is attributed to the coexistence of two types of defect having different capture cross-sections. This is confirmed by light soaking of a-Si:H which makes the defects with high s(E) predominant.     

Health-related Internet habits and health anxiety in university students

Health-related Internet use has grown rapidly, yet little research has considered how health anxious individuals use the Internet for this purpose. Our aim was to examine the relationships between health anxiety and the extent of, reasons for, and consequences of health-related Internet usage in university students (n = 255). Responses on a purpose-made Internet use questionnaire were correlated with health anxiety scores; multiple regression analyses controlling for depression and anxiety were also conducted. Health anxiety positively correlated with (all ps < .01): frequency of health-related searching (r_s = .163), proportion of health-related information sought (r_s = .200), time spent online for health purposes (r_s = .166), and number of searches for both illness (r_s = .453) and wellness (r_s = .208) information. Health anxiety further positively correlated with advantages perceived in health-related Internet use (r_s = .183), heightened tension (r_s = .364) and relief (r_s = .174) post-search, and perceived doctor disadvantages (r_s = .306), yet a greater likelihood to visit a doctor post-search (r_s = .217). Health anxiety also correlated with six measures of possible addiction to using the Internet for health purposes (r_s range = .171 to .366, all ps < .01). Some (including several potentially dysfunctional) aspects of health-related Internet use correlate with health anxiety. Research evaluating the possible role of Internet use in the development and maintenance of health anxiety is warranted.     

Correlation Weights in Multiple Regression

A general theory on the use of correlation weights in linear prediction has yet to be proposed. In this paper we take initial steps in developing such a theory by describing the conditions under which correlation weights perform well in population regression models. Using OLS weights as a comparison, we define cases in which the two weighting systems yield maximally correlated composites and when they yield minimally similar weights. We then derive the least squares weights (for any set of predictors) that yield the largest drop in R ² (the coefficient of determination) when switching to correlation weights. Our findings suggest that two characteristics of a model/data combination are especially important in determining the effectiveness of correlation weights: (1) the condition number of the predictor correlation matrix, R _xx , and (2) the orientation of the correlation weights to the latent vectors of R _xx .