Avoiding degeneracy in multidimensional unfolding by penalizing on the coefficient of variation

Multidimensional unfolding methods suffer from the degeneracy problem in almost all circumstances. Most degeneracies are easily recognized: the solutions are perfect but trivial, characterized by approximately equal distances between points from different sets. A definition of an absolutely degenerate solution is proposed, which makes clear that these solutions only occur when an intercept is present in the transformation function. Many solutions for the degeneracy problem have been proposed and tested, but with little success so far. In this paper, we offer a substantial modification of an approach initiated bythat introduced a normalization factor based on thevariance in the usual least squares loss function. Heiser unpublishedthesis, (1981) and showed that the normalization factor proposed by Kruskal and Carroll was not strong enough to avoid degeneracies. The factor proposed in the present paper, based on the coefficient of variation, discourages or penalizes nonmetric transformations of the proximities with small variation, so that the procedure steers away from solutions with small variation in the interpoint distances. An algorithm is described for minimizing the re-adjusted loss function, based on iterative majorization. The results of a simulation study are discussed, in which the optimal range of the penalty parameters is determined. Two empirical data sets are analyzed by our method, clearly showing the benefits of the proposed loss function.The authors would like to thank the editor, an associate editor, and three reviewers for their valuable comments and suggestions to improve the quality of this work.This revised article was published online in August 2005 with the PDF paginated correctly.     

Magnitude estimation of short electrocutaneous pulses

Summary A comparison was made of electrocutaneous magnitude estimation data across two experiments with contextual differences not involving stimulus parameters, such as number and range of stimuli and relative position of the standard in the stimulus range. The data were fitted by 2-parameter linear, log-linear and power functions. When the data are fitted by either linear or log-linear equations, both intercept and slope parameters are significantly affected by the different contextual factors. When the data are fitted by a power function, however, only the intercept is altered; the slope remains invariant despite contextual changes introduced in the second experiment.Although the empirically derived psychophysical power law has been applied to magnitude estimation data for all other sensory modalities, its application to electrocutaneous stimuli has been less successful.     

Bias in longitudinal data analysis with missing data using typical linear mixed‐effects modelling and pattern‐mixture approach: An analytical illustration

We analytically derive the fixed‐effects estimates in unconditional linear growth curve models by typical linear mixed‐effects modelling (TLME) and by a pattern‐mixture (PM) approach with random‐slope‐dependent two‐missing‐pattern missing not at random (MNAR) longitudinal data. Results showed that when the missingness mechanism is random‐slope‐dependent MNAR, TLME estimates of both the mean intercept and mean slope are biased because of incorrect weights used in the estimation. More specifically, the estimate of the mean slope is biased towards the mean slope for completers, whereas the estimate of the mean intercept is biased towards the opposite direction as compared to the estimate of the mean slope. We also discuss why the PM approach can provide unbiased fixed‐effects estimates for random‐coefficients‐dependent MNAR data but does not work well for missing at random or outcome‐dependent MNAR data. A small simulation study was conducted to illustrate the results and to compare results from TLME and PM. Results from an empirical data analysis showed that the conceptual finding can be generalized to other real conditions even when some assumptions for the analytical derivation cannot be met. Implications from the analytical and empirical results were discussed and sensitivity analysis was suggested for longitudinal data analysis with missing data.     

Latent growth curve analysis with dichotomous items: Comparing four approaches

A Monte Carlo study was used to compare four approaches to growth curve analysis of subjects assessed repeatedly with the same set of dichotomous items: A two‐step procedure first estimating latent trait measures using MULTILOG and then using a hierarchical linear model to examine the changing trajectories with the estimated abilities as the outcome variable; a structural equation model using modified weighted least squares (WLSMV) estimation; and two approaches in the framework of multilevel item response models, including a hierarchical generalized linear model using Laplace estimation, and Bayesian analysis using Markov chain Monte Carlo (MCMC). These four methods have similar power in detecting the average linear slope across time. MCMC and Laplace estimates perform relatively better on the bias of the average linear slope and corresponding standard error, as well as the item location parameters. For the variance of the random intercept, and the covariance between the random intercept and slope, all estimates are biased in most conditions. For the random slope variance, only Laplace estimates are unbiased when there are eight time points.     

Multilevel SEM with random slopes in discrete data using the pairwise maximum likelihood

Pairwise maximum likelihood (PML) estimation is a promising method for multilevel models with discrete responses. Multilevel models take into account that units within a cluster tend to be more alike than units from different clusters. The pairwise likelihood is then obtained as the product of bivariate likelihoods for all within-cluster pairs of units and items. In this study, we investigate the PML estimation method with computationally intensive multilevel random intercept and random slope structural equation models (SEM) in discrete data. In pursuing this, we first reconsidered the general ‘wide format’ (WF) approach for SEM models and then extend the WF approach with random slopes. In a small simulation study we the determine accuracy and efficiency of the PML estimation method by varying the sample size (250, 500, 1000, 2000), response scales (two-point, four-point), and data-generating model (mediation model with three random slopes, factor model with one and two random slopes). Overall, results show that the PML estimation method is capable of estimating computationally intensive random intercept and random slopes multilevel models in the SEM framework with discrete data and many (six or more) latent variables with satisfactory accuracy and efficiency. However, the condition with 250 clusters combined with a two-point response scale shows more bias.     

Separating bias and sensitivity in judgments of associative memory

Ratings of the degree of association between words are linearly related to normed associative strengths, but the intercept is high, and the slope is shallow (the judgments of associative memory [JAM] function). Two experiments included manipulations intended to decrease the intercept and increase the slope. Discrimination training on many pairs of words and constraining ratings to sum to a constant both reduced the intercept but failed to change the slope. The intercept of the JAM function appears to contain a bias component that can be manipulated independently of the slope, which reflects sensitivity to associative strengths.     

Why 'Sounds Are Judged Longer Than Lights': Application of a Model of the Internal Clock in Humans

Three experiments, using temporal generalization and verbal estimation methods, studied judgements of duration of auditory (500-Hz tone) and visual (14-cm blue square) stimuli. With both methods, auditory stimuli were judged longer, and less variable, than visual ones. The verbal estimation experiments used stimuli from 77 to 1183 msec in length, and the slope of the function relating mean estimate to real length differed between modalities (but the intercept did not), consistent with the idea that a pacemaker generating duration representations ran faster for auditory than for visual stimuli. The different variability of auditory and visual stimuli was attributed to differential variability in the operation of a switch of a pacemaker-accumulator clock, and experimental datasuggested that such switch effects were separable from changes in pacemaker speed. Overall, the work showed how a clock model consistent with scalar timing theory, the leading account of animal timing, can address an issue derived from the classical literature on human time perception.     

Trajectories of teacher-student warmth and conflict at the transition to middle school: Effects on academic engagement and achievement

Using piece-wise longitudinal trajectory analysis, this study investigated trajectories of teacher-reported warmth and conflict in their relationships with students 4 years prior to and 3 years following the transition to middle school in a sample of 550 academically at-risk and ethnically diverse adolescents. At the transition to middle school, teacher reports of warmth showed a significant drop (shift in intercept), above age-related declines. Both warmth and conflict declined across the middle school years. Structural equation modeling (SEM) tested effects of the shifts in intercept and the post-transition slopes on reading and math achievement, teacher-rated engagement, and student-reported school belonging 3 years post-transition, above pre-transition levels of the outcome. For warmth, a drop in intercept predicted lower math scores and engagement, and a more positive slope predicted higher engagement. For conflict, an increase in intercept and a negative slope predicted lower engagement. Implications of findings for reducing normative declines in academic engagement in middle school are discussed.     

Magnitude estimates for electrical pulses: evidence for two neural mechanisms

The hypothesis that there are two neural mechanisms for electrocutaneous stimulation--one that is sensitive to low current and is adaptive to repeated stimulation and another that is responsive to high current and is less adaptive--was tested in a control and four main experiments. In the main experiments, magnitude estimates obtained for single electrical pulses (of 2-msec duration) were described by a simple power function for each combination of high- and low-current levels and 10 trial blocks. The results were: (1) The slope of the power function for low current was steeper than was that for high current; (2) for low current, the intercept of the power function decreased with increasing block, whereas for high current, it remained constant over blocks; (3) this decrease of the intercept for low current disappeared when judgmental blocks were separated by a rest period of 8 min; (4) the modulus did not affect the slope; (5) for a large modulus combined with low current, the intercept decreased rapidly over trial blocks, whereas for a small modulus combined with high current, the intercept increased over trial blocks. The first four findings support the two-mechanism hypothesis, but the last one may also be interpretable in terms of the regression to absolute scale values.     

Differential prediction and the use of multiple predictors: the omitted variables problem

Moderated regression is widely used to examine differential prediction by race or gender. When using multiple predictors in a selection system, guidance as to whether differential prediction analysis should be conducted on each predictor individually, or on the set of predictors in combination, is lacking. Analyzing predictors individually creates the possibility of an omitted variable problem. Army Project A data were used to examine differential prediction by race with the use of personality measures for 79 predictor-criterion combinations. Traditional analysis indicated predictive bias by intercept in 45 instances and by slope in 7 instances; the inclusion of an Armed Services Vocational Aptitude Battery general factor as an additional predictor changed the conclusion in 32 cases for the intercept and in 3 cases for the slope.     

The wandering ideal point model: A probabilistic multidimensional unfolding model for paired comparisons data

In analogy with the wandering vector model, a probabilistic multidimensional unfolding model is proposed for representing paired comparisons data. It is shown that contrary to other stochastic multidimensional unfolding models, the present model does not imply strong stochastic transitivity, only moderate stochastic transitivity. A maximum likelihood parameter estimation procedure is developed and an illustrative application is presented.     

GGUMLINK: A computer program to link parameter estimates of the generalized graded unfolding model from item response theory

The generalized graded unfolding model (GGUM) is an item response theory (IRT) model that implements symmetric, nonmonotonic, single-peaked item characteristic curves. The GGUM is appropriate for measuring individual differences for a variety of psychological constructs, especially attitudes. Like other IRT models, the location and scale (i.e., the metric) of parameter estimates from the GGUM are data dependent. Therefore, parameter estimates from alternative calibrations will generally not be comparable, even when responses to the same items are analyzed. GGUMLINK is a computer program developed to reexpress parameter estimates from two separate GGUM calibrations in a common metric. In this way, the results from separate calibrations of model parameters can be compared. GGUMLINK can secure a common metric by using one of five methods that have recently been generalized to the GGUM. The GGUMLINK executable program is available free and may be downloaded from http://www.education.umd.edu/EDMS/tutorials/index.html.     

A nonmetric variety of linear factor analysis

The numbers in each column of ann ×m matrix of multivariate data are interpreted as giving the measured values of alln of the objects studied on one ofm different variables. Except for random error, the rank order of the numbers in such a column is assumed to be determined by a linear rule of combination of latent quantities characterizing each row object with respect to a small number of underlying factors. An approximation to the linear structure assumed to underlie the ordinal properties of the data is obtained by iterative adjustment to minimize an index of over-all departure from monotonicity. The method is “nonmetric” in that the obtained structure in invariant under monotone transformations of the data within each column. Except in certain degenerate cases, the structure is nevertheless determined essentially up to an affine transformation. Tests show (a) that, when the assumed monotone relationships are strictly linear, the recovered structure tends closely to approximate that obtained by standard (metric) factor analysis but (b) that, when these relationships are severely nonlinear, the nonmetric method avoids the inherent tendency of the metric method to yield additional, spurious factors. From the practical standpoint, however, the usefulness of the nonmetric method is limited by its greater computational cost, vulnerability to degeneracy, and sensitivity to error variance.     

What Can Bilingual Children Tell Us About the Developmental Relationship Between Vocabulary and Grammar?

Strong correlations between vocabulary and grammar are well attested in language development in monolingual and bilingual children. What is less clear is whether there is any directionality in the relationship between the two constructs, whether it is predictive over time, and the extent to which it is affected by language input. In the present study, we analyzed data from 100 bilingual children with English as an additional language who were tested on measures of vocabulary breadth and depth, morphology, and syntax at three time points at 6-month intervals from the age of 5;8. We used bivariate growth models to test the directionality of the relationship between vocabulary breadth and depth, and measures of morphology and syntax; testing bilingual children allowed us to use measures of English input as covariates in the analyses. All the models showed a correlation between vocabulary and grammar, but no correlation between their growth slopes, suggesting that vocabulary and grammar grow independently. Three of the four bivariate models showed a significant correlation between the intercept of grammar skills and the slope of vocabulary growth. Length of exposure to English predicted the intercept of vocabulary breadth and grammar, suggesting that children exposed to English earlier had larger vocabularies and better morpho-syntactic skills. Current English input predicted the intercept of both measures of vocabulary as well as the slope for vocabulary depth, the only measure for which there was a significant relationship between intercept and slope, suggesting a Matthew effect for this dimension of vocabulary. All materials, data, and code are available at https://osf.io/x3wht/ . Research highlights

• Vocabulary breadth and morphological and syntactic skills increased linearly for all participants, without any difference between lower and higher achieving children.
• Vocabulary depth grew more over time for those children with deeper vocabulary knowledge and higher levels of current English input at the start of the study.
• All of the bivariate growth models showed a correlation between vocabulary and grammar, but failed to show any correlation between their growth.
• Significant relationships between the intercept of grammar and the growth of vocabulary showed steeper lexical growth in children with better grammar skills.
• Length of exposure to English had an effect on morphological and syntactic skills, while only current English input had an effect on vocabulary depth.

     

协变量相关对时变效应模型参数估计的影响

时变效应模型被广泛应用于密集追踪研究中,研究者往往会同时纳入2个或以上协变量。然而,协变量相关对其参数估计的影响较少被研究者关注。本研究在不同类型协变量的情境下,采用蒙特卡洛模拟,探讨协变量相关对时变效应模型参数估计的影响,结果表明：（1）在两种协变量类型的情境下,协变量相关都会影响时变效应模型斜率函数β_1和斜率函数β_2参数估计的准确性;（2）两种协变量类型的情境下,协变量相关和样本量的交互作用都会影响时变效应模型斜率函数β_1和斜率函数β_2参数估计的准确性;（3）两种协变量类型的情境下,样本量、观测数据缺失率主要通过主效应影响时变效应模型参数估计的准确性。     

Resting-state aperiodic neural dynamics predict individual differences in visuomotor performance and learning

An emerging body of work has demonstrated that resting-state non-oscillatory, or aperiodic, 1/f neural activity is a functional and behaviorally relevant marker of cognitive function capacity. In the motor domain, previous work has only applied 1/f analyses to investigations of motor coordination and performance measures. The value of aperiodic resting-state neural dynamics as a marker of individual visuomotor performance capacity remains unknown. Accordingly, the aim of this work was to investigate if individual 1/f intercept and slope parameters of aperiodic resting-state neural activity predict reaction time and perceptual sensitivity in an immersive virtual reality marksmanship task. The marksmanship task required speeded selection of target stimuli and avoidance of selecting non-target stimuli. Motor and perceptual demands were incrementally increased across task blocks and participants performed the task across three training sessions spanning one week. When motor demands were high, steeper individual 1/f slope predicted shorter reaction time. This relationship did not change with practice. Increased 1/f intercept and a steeper 1/f slope were associated with higher perceptual sensitivity, measured as d’. However, this association was only observed under the highest levels of perceptual demand and only in the initial exposure to these conditions. Individuals with a lower 1/f intercept and a shallower 1/f slope demonstrated the greatest gains in perceptual sensitivity from task practice. These findings demonstrate that individual differences in motor and perceptual performance can be accounted for with resting-state aperiodic neural dynamics. The 1/f aperiodic parameters are most informative in predicting visuomotor performance under complex and demanding task conditions. In addition to predicting capacity for high visuomotor performance with a novel task, 1/f aperiodic parameters might also be useful in predicting which individuals might derive the most improvements from practice.     

Multidimensional Fechnerian Scaling: Probability-Distance Hypothesis

The probability-distance hypothesis states that the probability with which one stimulus is discriminated from another is a function of some subjective distance between these stimuli. The analysis of this hypothesis within the framework of multidimensional Fechnerian scaling yields the following results. If the hypothetical subjective metric is internal (which means, roughly, that the distance between two stimuli equals the infimum of the lengths of all paths connecting them), then the underlying assumptions of Fechnerian scaling are satisfied and the metric in question coincides with the Fechnerian metric. Under the probability-distance hypothesis, the Fechnerian metric exists (i.e., the underlying assumptions of Fechnerian scaling are satisfied) if and only if the hypothetical subjective metric is internalizable, which means, roughly, that by a certain transformation it can be made to coincide in the small with an internal metric; and then this internal metric is the Fechnerian metric. The specialization of these results to unidimensional stimulus continua is closely related to the so-called Fechner problem proposed in 1960's as a substitute for Fechner's original theory.     

Recall of enacted and nonenacted instructions compared: forgetting functions

Summary Three experiments are reported in which under one condition subjects were given instructions to perform a series of acts (SPTs) and to remember the acts performed, and under a second condition were asked to remember the same verbal instructions without being required to perform any acts. Different groups of subjects were tested at different retention intervals. The forgetting curves for the two conditions differ with respect to intercept, but are similar in slope. The similarity in slope for the two conditions is present irrespective of differences in intercept. Several methodological issues are discussed as well as the nature of the relationship between recall of SPTs and verbal instructions.     

Least squares metric,unidimensional unfolding

The partial derivatives of the squared error loss function for the metric unfolding problem have a unique geometry which can be exploited to produce unfolding methods with very desirable properties. This paper details a simple unidimensional unfolding method which uses the geometry of the partial derivatives to find conditional global minima; i.e., one set of points is held fixed and the global minimum is found for the other set. The two sets are then interchanged. The procedure is very robust. It converges to a minimum very quickly from a random or non-random starting configuration and is particularly useful for the analysis of large data sets with missing entries.This paper benefits from many conversations with and suggestions from Howard Rosenthal.