Estimating the polychoric correlation from misclassified data

Many variables that are used in social and behavioural science research are ordinal categorical or polytomous variables. When more than one polytomous variable is involved in an analysis, observations are classified in a contingency table, and a commonly used statistic for describing the association between two variables is the polychoric correlation. This paper investigates the estimation of the polychoric correlation when the data set consists of misclassified observations. Two approaches for estimating the polychoric correlation have been developed. One assumes that the probabilities in relation to misclassification are known, and the other uses a double sampling scheme to obtain information on misclassification. A parameter estimation procedure is developed, and statistical properties for the estimates are discussed. The practicability and applicability of the proposed approaches are illustrated by analysing data sets that are based on real and generated data. Excel programmes with visual basic for application (VBA) have been developed to compute the estimate of the polychoric correlation and its standard error. The use of the structural equation modelling programme Mx to find parameter estimates in the double sampling scheme is discussed.     

The Relation of Severity and Comorbidity to Treatment Outcome with Cognitive Behavioral Therapy for Childhood Anxiety Disorders

The present study investigated the impact of comorbidity over and above the impact of symptom severity on treatment outcome of Cognitive Behavioral Therapy for children with anxiety disorders. Children (aged 8–12, n = 124) diagnosed with an anxiety disorder were treated with a short-term CBT protocol. Severity was assessed with a composite measure of parent-reported behavior problems. Two approaches to comorbidity were examined; “total comorbidity” which differentiated anxiety disordered children with (n = 69) or without (n = 55) a co-occurring disorder and “non-anxiety comorbidity’ which differentiated anxious children with (n = 22) or without a non-anxiety comorbid disorder (n = 102). Treatment outcome was assessed in terms of Recovery, represented by post-treatment diagnostic status, and Reliable Change, a score reflecting changes in pre- to post-treatment symptom levels. Severity contributed to the prediction of (no) Recovery and (more) Reliable Change in parent-reported internalizing and externalizing symptoms and self-reported depressive symptoms. Total and non-anxiety comorbidity added to the prediction of diagnostic recovery. Non-anxiety comorbidity added to the prediction of Reliable Change in parent reported measures by acting as a suppressor variable. Non-anxiety comorbidity operated as a strong predictor that explained all of the variance associated with severity for self-reported depressive symptoms. The results support the need for further research on mechanisms by which treatment gains in children with higher symptom severity and non-anxiety comorbidity can be achieved.     

Joint Attention and Attachment in Toddlers with Autism

Joint attention is often referred to as a triadic relation between self, other and object. Young children with autism show deficiencies in the use of joint attention behaviors. Individual differences may be expected, and they may be determined by the children’s cognitive development or the characteristics of the relationship of the child with the caregiver. Although most joint attention skills develop under the age of three, most studies of joint attention in children with autism involved children older than 3 years of age, due to difficulties in diagnosing autism under this age. In this study we investigated joint attention behaviors of 78 young children (mean age 25.7 months, SD 6.1) with autism spectrum disorders (n = 20), other developmental delays (n = 18), and typically developing children (n = 40). Following the pertinent literature and confirmed by factor analysis, two types of joint attention behaviors were distinguished, Basic Joint Attention (BJA) and Associated Joint Attention (AJA). We found that cognitive delays and autistic symptoms—but not attachment insecurity or disorganization—were related to less joint attention. Already at the age of 2 years, children with more autistic symptoms show less joint attention, even after controlling for developmental level.     

Cohen’s Linearly Weighted Kappa is a Weighted Average of 2×2 Kappas

An agreement table with n∈ℕ_≥3 ordered categories can be collapsed into n−1 distinct 2×2 tables by combining adjacent categories. Vanbelle and Albert (Stat. Methodol. 6:157–163, 2009c) showed that the components of Cohen’s weighted kappa with linear weights can be obtained from these n−1 collapsed 2×2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2×2 tables, where the weights are the denominators of the 2×2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.     

A Person Fit Test For Irt Models For Polytomous Items

A person fit test based on the Lagrange multiplier test is presented for three item response theory models for polytomous items: the generalized partial credit model, the sequential model, and the graded response model. The test can also be used in the framework of multidimensional ability parameters. It is shown that the Lagrange multiplier statistic can take both the effects of estimation of the item parameters and the estimation of the person parameters into account. The Lagrange multiplier statistic has an asymptotic χ²-distribution. The Type I error rate and power are investigated using simulation studies. Results show that test statistics that ignore the effects of estimation of the persons’ ability parameters have decreased Type I error rates and power. Incorporating a correction to account for the effects of the estimation of the persons’ ability parameters results in acceptable Type I error rates and power characteristics; incorporating a correction for the estimation of the item parameters has very little additional effect. It is investigated to what extent the three models give comparable results, both in the simulation studies and in an example using data from the NEO Personality Inventory-Revised.     

Inferential reasoning by exclusion in pigeons, dogs, and humans

The ability to reason by exclusion (which is defined as the selection of the correct alternative by logically excluding other potential alternatives; Call in Anim Cogn 9:393–403 2006) is well established in humans. Several studies have found it to be present in some nonhuman species as well, whereas it seems to be somewhat limited or even absent in others. As inconsistent methodology might have contributed to the revealed inter-species differences, we examined reasoning by exclusion in pigeons (n = 6), dogs (n = 6), students (n = 6), and children (n = 8) under almost equal experimental conditions. After being trained in a computer-controlled two-choice procedure to discriminate between four positive (S+) and four negative (S−) photographs, the subjects were tested with displays consisting of one S− and one of four novel stimuli (S′). One pigeon, half of the dogs and almost all humans preferred S′ over S−, thereby choosing either by novelty, or by avoiding S− without acquiring any knowledge about S′, or by inferring positive class membership of S′ by excluding S−. To decide among these strategies the subjects that showed a preference for S′ were then tested with displays consisting of one of the S′ and one of four novel stimuli (S′′). Although the pigeon preferentially chose the S′′ and by novelty, dogs and humans maintained their preference for S′, thereby showing evidence of reasoning by exclusion. Taken together, the results of the present study suggest that none of the pigeons, but half of the dogs and almost all humans inferred positive class membership of S′ by logically excluding S−. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: II: A search procedure to locate the greatest lower bound

Finding the greatest lower bound for the reliability of the total score on a test comprisingn non-homogenous items with dispersion matrix Σ _x is equivalent to maximizing the trace of a diagonal matrix Σ _E with elements θ _I , subject to Σ _E and Σ _T =Σ _x − Σ _E being non-negative definite. The casesn=2 andn=3 are solved explicity. A computer search in the space of the θ _i is developed for the general case. When Guttman's λ₄ (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ _i makes the rank of Σ _T less thann − 1. Numerical examples of various bounds are given. Present affiliation of the first author: St. Hild's College of Education, Durham City, England.     

Inhibition of action rules

Inhibition is often proposed as an important executive-control process, but its existence is difficult to establish empirically. Recently, Mayr and Keele (2000) reported that it takes longer to switch to a recently disengaged task set (i.e.,n−2 task-set repetitions) than to a less recently disengaged task set. Thissetalternation cost may indicate inhibition of the disengaged task set. The primary goal of the present study was to test the inhibition account of set-alternation costs against an important alternative account based on automatic episodic retrieval. The episodic retrieval account predicts reaction time facilitation instead of costs for the special case ofcomplete n−2 repetitions of both task sets and all stimulus— response aspects. A new task-switching paradigm, in which action rules varied on a trial-by-trial basis while the relevant stimulus dimension remained invariant, allowed the implementation of a high proportion of completen−2 repetitions. Consistent with the inhibition view, set-alternation costs were obtained even for these constellations.     

On the asymptotic distribution of Pearson’s X 2 in cross-validation samples

In categorical data analysis, two-sample cross-validation is used not only for model selection but also to obtain a realistic impression of the overall predictive effectiveness of the model. The latter is of particular importance in the case of highly parametrized models capable of capturing every idiosyncracy of the calibrating sample. We show that for maximum likelihood estimators or other asymptotically efficient estimators Pearson's X ² is not asymptotically chi-square in the two-sample cross-validation framework due to extra variability induced by using different samples for estimation and goodness-of-fit testing. We propose an alternative test statistic, X _xval ², obtained as a modification of X ² which is asymptotically chi-square with C−1 degrees of freedom in cross-validation samples. Stochastically, X _xval ² ≤ X ². Furthermore, the use of X ² instead of X _xval ² with a χ _C −1² reference distribution may provide an unduly poor impression of fit of the model in the cross-validation sample. This paper is dedicated to the memory of Michael V. Levine. Requests for reprints should be sent to Albert Maydeu-Olivares, Faculty of Psychology, University of Barcelona, P. Valle de Hebrón, 171, 0835 Barcelona, Spain.     

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 .     

Knowledge and Error in Early Chinese Thought

Drawing primarily on the Mòzǐ and Xúnzǐ, the article proposes an account of how knowledge and error are understood in classical Chinese epistemology and applies it to explain the absence of a skeptical argument from illusion in early Chinese thought. Arguments from illusion are associated with a representational conception of mind and knowledge, which allows the possibility of a comprehensive or persistent gap between appearance and reality. By contrast, early Chinese thinkers understand mind and knowledge primarily in terms of competence or ability, not representation. Cognitive error amounts to a form of incompetence. Error is not explained as a failure to accurately represent the mind-independent reality due to misleading or illusory appearances. Instead, it can be explained metaphorically by appeal to part-whole relations: cognitive error typically occurs when agents incompetently respond to only part of their situation, rather than the whole.     

Health Related Quality of Life After Gastric Bypass Surgery

The majority of studies which have evaluated health related quality of life are limited in the duration of follow up. The objective of this study was to prospectively conduct an evaluation using a repeated cross sectional analysis of separate patient cohorts who were up to four years after gastric bypass surgery. Adult Roux-en-Y gastric bypass patients were recruited to the study. All patients were requested to complete a general health status questionnaire, the Short Form-36 (SF-36), before surgery or at their post operative out patient follow up visits. Patient weight was documented at each follow up visit. A cross sectional analysis was performed to evaluate SF-36 scores in each annual cohort. Data are reported as mean +/− S.D. Three-hundred-eight patients completed at least one SF-36 assessment [Initial assessment at the time of surgery, time 1, n = 245, 1y n = 149, 2y n = 70, 3y n = 59, 4y n = 61]. The SF-36 scores were greater (p < 0.05) in each of the separate post surgery cohorts for physical functioning, role limitations due to physical health, social functioning, pain, vitality, general health and the physical component summary (PCS) scores. While not comparing changes in scores within individuals over time, these data suggest early improvement especially in the physical dimension of health related quality of life. In this analysis, this finding was also observed in each of the separate cohorts up to 4 years after gastric bypass surgery.     

Some Results on Mean Square Error for Factor Score Prediction

For the confirmatory factor model a series of inequalities is given with respect to the mean square error (MSE) of three main factor score predictors. The eigenvalues of these MSE matrices are a monotonic function of the eigenvalues of the matrix Γ _p = Φ ^1/2 Λ _p ′Ψ _p ⁻¹ Λ _p Φ ^1/2. This matrix increases with the number of observable variables p. A necessary and sufficient condition for mean square convergence of predictors is divergence of the smallest eigenvalue of Γ _p or, equivalently, divergence of signal-to-noise (Schneeweiss & Mathes, 1995). The same condition is necessary and sufficient for convergence to zero of the positive definite MSE differences of factor predictors, convergence to zero of the distance between factor predictors, and convergence to the unit value of the relative efficiencies of predictors. Various illustrations and examples of the convergence are given as well as explicit recommendations on the problem of choosing between the three main factor score predictors. The author is obliged to Maarten Speekenbrink and Peter van Rijn for their assistance with plotting the figures. In addition, I am obliged to the referees for their stimulating remarks.     

N-backer: an auditory n-back task with automatic scoring of spoken responses

The n-back task is commonly used to load working memory (WM) in dual-task and neuroimaging experiments. However, it typically involves visual presentation and buttonpress responses, making it unsuitable for combination with primary tasks that involve vision and action, such as sequential object use and other tasks of daily living. The N-backer software presented here will automatically present and score auditory–verbal n-back sequences utilising the standard speech synthesis and recognition facilities that come with Microsoft Windows. Data are presented from an experiment in which 12 student participants carried out three tasks from the Naturalistic Action Test (NAT) while their attention was divided between the primary task and a continuous auditory–verbal 2-back secondary task. The participants’ 2-back performance was scored in two ways: by hand, from video recordings, and automatically, using the software, allowing us to evaluate the accuracy of N-backer. There was an extremely high correlation between these scores (.933). The videos were also used to obtain a comprehensive error score for the NAT. Significantly more errors were made in the more complex NAT tasks when participants were 2-backing, as compared with when they were not, showing that the auditory–verbal n-back task can be used to disrupt sequential object use. This dual-task method may simulate the attentional deficits of patients with brain injury, providing insights into the difficulties they face in tasks of daily living.     

Investigating the Performance of Alternate Regression Weights by Studying All Possible Criteria in Regression Models with a Fixed Set of Predictors

We describe methods for assessing all possible criteria (i.e., dependent variables) and subsets of criteria for regression models with a fixed set of predictors, x (where x is an n×1 vector of independent variables). Our methods build upon the geometry of regression coefficients (hereafter called regression weights) in n-dimensional space. For a full-rank predictor correlation matrix, R _xx, of order n, and for regression models with constant R ² (coefficient of determination), the OLS weight vectors for all possible criteria terminate on the surface of an n-dimensional ellipsoid. The population performance of alternate regression weights—such as equal weights, correlation weights, or rounded weights—can be modeled as a function of the Cartesian coordinates of the ellipsoid. These geometrical notions can be easily extended to assess the sampling performance of alternate regression weights in models with either fixed or random predictors and for models with any value of R ². To illustrate these ideas, we describe algorithms and R (R Development Core Team, 2009) code for: (1) generating points that are uniformly distributed on the surface of an n-dimensional ellipsoid, (2) populating the set of regression (weight) vectors that define an elliptical arc in ℝ ⁿ , and (3) populating the set of regression vectors that have constant cosine with a target vector in ℝ ⁿ . Each algorithm is illustrated with real data. The examples demonstrate the usefulness of studying all possible criteria when evaluating alternate regression weights in regression models with a fixed set of predictors.     

ADHD and Behavioral Inhibition: A Re-examination of the Stop-signal Task

The current study investigates two recently identified threats to the construct validity of behavioral inhibition as a core deficit of attention-deficit/hyperactivity disorder (ADHD) based on the stop-signal task: calculation of mean reaction time from go-trials presented adjacent to intermittent stop-trials, and non-reporting of the stop-signal delay metric. Children with ADHD (n = 12) and typically developing (TD) children (n = 11) were administered the standard stop-signal task and three variant stop-signal conditions. These included a no-tone condition administered without the presentation of an auditory tone; an ignore-tone condition that presented a neutral (i.e., not associated with stopping) auditory tone; and a second ignore-tone condition that presented a neutral auditory tone after the tone had been previously paired with stopping. Children with ADHD exhibited significantly slower and more variable reaction times to go-stimuli, and slower stop-signal reaction times relative to TD controls. Stop-signal delay was not significantly different between groups, and both groups’ go-trial reaction times slowed following meaningful tones. Collectively, these findings corroborate recent meta-analyses and indicate that previous findings of stop-signal performance deficits in ADHD reflect slower and more variable responding to visually presented stimuli and concurrent processing of a second stimulus, rather than deficits of motor behavioral inhibition.     

Task-set inhibition in chunked task sequences

Exploring the hierarchical control relationship between different levels of representation and processing is important for understanding how the mind controls itself. In the present study, the relationship between chunking (a sequence-level process) and task-set inhibition (a task-level process) in the performance of task sequences was investigated to evaluate the hypothesis that within-chunk facilitation reduces then−2 repetition cost (slower performance forn−2 task repetitions than forn−2 task switches) attributed to task-set inhibition. An experiment is reported in which subjects were induced to chunk sequences such thatn−2 repetitions occurred within or between chunks. Direct evidence of chunking was obtained, andn−2 repetition cost was smaller whenn−2 repetitions occurred within chunks than between chunks. These findings are consistent with an elaborated hypothesis that attributes the reduction inn−2 repetition cost to priming of task goals rather than direct modulation of task-set inhibition.     

Single-letter recognition as a function of exposure duration

The time course of visual letter recognition was investigated in a single-stimulus identification experiment. On each trial, a randomly chosen stimulus letter was presented at 1 of 12 equiprobable positions that were equally spaced around the circumference of an imaginary circle centered on fixation. Exposure duration was varied from 10 to 200 ms, and the letter was followed by a pattern mask. The subject's task was to report the identity of the stimulus letter but refrain from guessing. For the briefest exposures, correct reports never occurred. For longer exposures, the function relating the probability p of recognizing the letter to the duration t of the stimulus exposure was well approximated by an exponential distribution function: p(t) = 1 − exp[−v·(t−t ₀)], where v is the rate of processing and t ₀ is the minimum effective exposure duration. The generality of this finding may be limited to cases in which stimuli are highly discriminable and response criteria are conservative. Extensions to Poisson counter or random walk models are considered for cases in which stimuli are confusable. Received: 1 July 1997 / Accepted: 2 July 1998     

All Finitely Axiomatizable Tense Logics of Linear Time Flows Are CoNP-complete

We prove that all finitely axiomatizable tense logics with temporal operators for ‘always in the future’ and ‘always in the past’ and determined by linear fows time are coNP-complete. It follows, for example, that all tense logics containing a density axiom of the form ■ⁿ⁺¹_F p → ⁿ_F p, for some n ≥ 0, are coNP-complete. Additionally, we prove coNP-completeness of all ∩-irreducible tense logics. As these classes of tense logics contain many Kripke incomplete bimodal logics, we obtain many natural examples of Kripke incomplete normal bimodal logics which are nevertheless coNP-complete.