A note on the computation of the second-order derivatives of the elementary symmetric functions in the rasch model

Jansen (1984) gave explicit formulas for the computation of the second-order derivatives of the elementary symmetric functions. But they are only applicable to those pairs of items which have unequal parameters. It is shown here that for equal parameters similar explicit formulas do exist, too, facilitating the application of the Newton-Raphson procedure to estimate the parameters in the Rasch model and related models according to the conditional maximum likelihood principle.The author wishes to thank the reviewers for their helpful comments on an earlier draft of this paper.     

Maximum likelihood estimates of item parameters using the logistic function

The logistic function is proposed as an alternative to the integrated normal function when estimating parameters of test items. The logistic curve is described; an iterative method for finding maximum likelihood estimates of its parameters is given, and an example of its use is presented.     

A new model of sensorimotor coupling in the development of speech

We present a computational model that learns a coupling between motor parameters and their sensory consequences in vocal production during a babbling phase. Based on the coupling, preferred motor parameters and prototypically perceived sounds develop concurrently. Exposure to an ambient language modifies perception to coincide with the sounds from the language. The model develops motor mirror neurons that are active when an external sound is perceived. An extension to visual mirror neurons for oral gestures is suggested.     

Identifiability and statistical properties of two-stage learning with no successes in the initial stage

A model is considered for learning where there is an initial stage with only errors, an intermediate state with errors and correct responses, and an absorbing state with only correct responses. A model with observable states is constructed, and sufficient identifiable parameters are given for the original model. Distribution statistics and other properties of data are derived as functions of the identifiable parameters. Informal estimates and sufficient statistics are given for the identifiable parameters. A number of testable hypotheses about the theoretical parameters are described. Some identifying restrictions for the model are given, along with the estimates for the unrestricted free parameters which follow. An application is made to data from a pairedassociate learning experiment.This research was sponsored by NSF Grant GB2791 and by PHS Grant MH12717. Part of the work was done while the author held a visiting faculty appointment at Stanford University.     

Tactile roughness of grooved surfaces: A model and the effect of friction

Previous experimental data on the roughness of grooved aluminum tiles provide a data base against which to test theories of roughness perception. A model based on the static deformation of the skin touching the stimulus tile is developed, and 11 parameters of the deformation are individually compared with the experimental data. All parameters were tested first in an approximate way, and then the better parameters were recalculated in a more exact manner. Three parameters, the depth to which the finger penetrates the groove, the cross-sectional area of the finger within the groove, and the cross-sectional area of the deviation of the skin from its resting position, all predict the roughness well as a function of finger force and groove width. The last of the three predicts roughness best as a function of land width, and is tentatively preferred as “the stimulus for roughness.” All predictions from the static model indicate that variation of the coefficient of friction between skin and tile should have little or no effect. This counterintuitive prediction was confirmed by an experiment.     

Estimating and Testing the Sources of Evoked Potentials in the Brain

The source of an event related brain potential (ERP) is estimated from multivariate measurements of this ERP on the head under several mathematical and physical constraints on the parameters of the source model. We will discuss statistical aspects of standard methods to estimate these parameters and their confidence intervals. In addition new principled tests of the goodness of fit as well as of differences between estimated sources are proposed and compared with customary approaches in psychophysiology. Several factors which influence the statistical analysis, in particular the number of measurement leads, will be discussed.     

Planning and preparing expected and unexpected movements: reexamining the relationships of arm, direction, and extent of movement

Using variations of the movement precue method, this study sought to define the operational characteristics of motoric decisions that govern the planning and preparation of arm, direction, and extent of movement. Experiment 1 examined how these parameters are programmed when the precue method does not confound motoric and nonmotoric decision processes. Experiment 2 examined how an already planned and prepared response is modified (reprogrammed) when an unexpected response must be executed in its place. The collective results of both experiments demonstrated that (a) these parameters were planned and prepared in a specific order; knowledge about direction was necessary for the programming of arm or extent; (b) arm and extent were reprogrammed independently from direction but changes in direction caused all parameters to be reprogrammed, and (c) programming and reprogramming processes operated in a parallel mode when two or more parameters were involved. The results also showed that these parameters were organized within a hierarchical structure. The present findings were discussed in relation to those reported in previous precue studies and existing models of response programming.     

Why We Walk the Way We Do

By using inverse dynamics and forceplate recordings, this study established the principle of oscillating systems and the influence of gravity and body parameters on the programming of the gait parameters, step frequency and length. Calculation of the ratio of the amplitude of the center of mass (CM) and the center of foot pressure (CP) oscillations yielded an equation and established a biomechanical constant, the natural body frequency (NBF). NBF appears to be an absolute invariant parameter, specific to human standing posture and gait in terrestrial gravity, which influences the relative positions of CM and CP and whose value separates the frequency bands of standing posture from those for gait. This equation was tested by using the experimental paradigm of stepping in place and then used in calculating the magnitude of CM oscillations during gait. The biomechanical analysis of the experimental observations allows one to establish the relationships between body parameters and gravity and the central programming of locomotor parameters.     

Response coding and visuomotor transformation in the Simon task: the role of action goals

Manual responses can be defined by differing response parameters. Any of them may generate a Simon effect. For all those response parameters, the same implementation of the Simon effect (in terms of subserving mechanism) is assumed. In 3 experiments, subjects had to respond with either fingers or sticks. Temporal properties of the Simon effect changed with response parameters relevant in a task. The Simon effect for manual responses decayed. For stick responses, in which the action goal differed from the anatomical mapping of the acting hand, a sustained Simon effect was observed. However, if the action goal for stick responses was not instrumental for selecting the correct response, the Simon effect decayed. The findings are consistent with the notion of different mechanisms involved in generating a Simon effect.     

Why we walk the way we do

By using inverse dynamics and forceplate recordings, this study established the principle of oscillating systems and the influence of gravity and body parameters on the programming of the gait parameters, step frequency and length. Calculation of the ratio of the amplitude of the center of mass (CM) and the center of foot pressure (CP) oscillations yielded an equation and established a biomechanical constant, the natural body frequency (NBF). NBF appears to be an absolute invariant parameter, specific to human standing posture and gait in terrestrial gravity, which influences the relative positions of CM and CP and whose value separates the frequency bands of standing posture from those for gait. This equation was tested by using the experimental paradigm of stepping in place and then used in calculating the magnitude of CM oscillations during gait. The biomechanical analysis of the experimental observations allows one to establish the relationships between body parameters and gravity and the central programming of locomotor parameters.     

Automatic measurement of aspects of speech reflecting motor coordination

An experimental setting and a method of measurement are described in which five parameters are recorded to reflect speech motor activity. From a psychological or clinical point of view, these parameters are a valuable supplement to traditional motor parameters. In the first experiment, it is shown that this method allows for an objective measure of these parameters. The second experiment proves their high short-term stability. This method requires an IBM-compatible PC, an AD-converter card, software, and microphones. Since a computer is used to evaluate the parameters, a simple registration of these parameters is possible. Applications in the area of speech disorders and short-term changes of speech are discussed.     

A Comparison of Using the Fixed Common-Precalibrated Parameter Method and the Matched Characteristic Curve Method for Linking Multiple-Test Items

A linking design typically consists of a data collection procedure together with an item linking procedure that places item parameters calibrated from multiple test forms onto a common scale. This study considered 2 potentially useful item response theory linking designs. The first one is characterized by selecting a single set of common items across all multiple test forms, the precalibrated item parameters of which are kept fixed while the unknown parameters of the other items are being estimated. This linking design will be referred to as the fixed common-precalibrated item parameter design. However, data collected under this design could also be analyzed by the characteristic curve method, which constituted an alternative linking procedure. In this study, the relative merits of the 2 linking designs were examined with respect to their robustness against 3 manipulated conditions-namely, when the common items have imprecise estimates, when there is a noticeable difference in the average item difficulty between the common and the noncommon items, and when the examinees are heterogeneous in terms of their abilities. A parameter recovery study was conducted to achieve this purpose. The results indicated that both linking designs were capable of producing accurate linking of items and equivalent estimation of ability parameters under the 3 conditions. When the 2 designs were actually utilized in the development of an item bank, it was found that both linking designs produced quite consistent solutions despite minor differences on some item and ability estimates. Condition under which a linking design is preferred over the other is also provided in the Discussion section of this article.     

Shrinkage estimation of the three-parameter logistic model

The three-parameter logistic model is widely used to model the responses to a proficiency test when the examinees can guess the correct response, as is the case for multiple-choice items. However, the weak identifiability of the parameters of the model results in large variability of the estimates and in convergence difficulties in the numerical maximization of the likelihood function. To overcome these issues, in this paper we explore various shrinkage estimation methods, following two main approaches. First, a ridge-type penalty on the guessing parameters is introduced in the likelihood function. The tuning parameter is then selected through various approaches: cross-validation, information criteria or using an empirical Bayes method. The second approach explored is based on the methodology developed to reduce the bias of the maximum likelihood estimator through an adjusted score equation. The performance of the methods is investigated through simulation studies and a real data example.     

Beliefs about the social orientations of others: A parametric test of the triangle,false consensus,and cone hypotheses

This study tests a number of hypotheses proposed in the literature concerning the relationship between an actor's social orientation and her beliefs about the social orientations of others. In contrast to the existing literature, this study employs a parametric approach with an innovative methodology. First, the social orientation parameters of actors are estimated: the weights respondents add to (1) the outcomes of Alter and to (2) the absolute difference between the outcomes for Ego and Alter. Then, the mean and the variance of the distribution of beliefs about the social orientation parameters of others are estimated, conditional on the actor's social orientation parameters. The results show that (1) there is a positive association between an actor's social orientation and her belief about the mean of the social orientations of others and (2) those who have approximately zero social orientation parameter values (individualists) expect the variation of others' social orientations to be lower than those with smaller (competitors) or larger (cooperators/egalitarians) social orientation parameter values. These results support the cone model, which models the “false” consensus effect where the “false” consensus is highest for individualists.     

Evaluation of six multinomial models of conscious and unconscious processes with the recall-recognition paradigm

Six multinomial processing-tree models (W. H. Batchelder & D. M. Riefer, 1999), which include parameters representing conscious and unconscious memory processes, were tested using the recall-recognition paradigm. Data from 2 experiments were fit equally well by 3 of the 6 models. One model recognition was an extension of the generate-recognize model (L. L. Jacoby, 1998), and another was an extension of the non-high-threshold model (D. M. McBride & B. A. Dosher, 1999). The 3rd model was the source evaluation model (D. M. McBride & B. A. Dosher, 1999). Values of the parameters of 2 of these 3 models, excepting the non-high-threshold model, responded to experimental manipulations in accordance with the meaning of the parameters. The equivalence of models with regard to goodness-of-fit tests is discussed as is how experiments can be designed to demonstrate the superiority of one model over another. The potential usefulness of these models in the study of amnesia is considered.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

Aging and the restructuring of precued movements

A precue paradigm was used to examine the time it takes to restructure a planned motor response. Two groups of subjects, a young group and an elderly group, performed an aiming task in which 75% of the trials involved no change of movement parameters. On remaining trials, subjects had to change one or more of the movement parameters. Elderly subjects had slower reaction times (RTs), movement times, and made more errors in both conditions. Elderly subjects had proportionally longer RTs overall, independent of restructuring a movement plan. Preparation of arm and direction also exhibited a proportional increase in RT. However, differential aging effects were found for preparation of extent. Elderly subjects were slower preparing short movements compared with long movements, whereas young subjects showed the opposite trend. These results suggest that with advancing age, operations concerned with movement-plan restructuring for arm and direction undergo change in processing rate, whereas operations for extent undergo more extensive alteration.     

Estimating the parameters of the latent population distribution

Under consideration is a test battery of binary items. The responses ofn individuals are assumed to follow a Rasch model. It is further assumed that the latent individual parameters are distributed within a given population in accordance with a normal distribution. Methods are then considered for estimating the mean and variance of this latent population distribution. Also considered are methods for checking whether a normal population distribution fits the data. The developed methods are applied to data from an achievement test and from an attitude test.