A comparison of two procedures for calculating discriminant function coefficients

Alternative procedures for calculating discriminant function coefficients have been illustrated in reported research. One method proceeds from data which has been expressed in deviation score units, whereas the other method implies that the data has been expressed in standard units. It is shown algebraically that both procedures yield identical discriminant function coefficients, and therefore, the method involving less mathematical manipulation and computation is preferable.     

Development and validation of discriminant functions for the Strong Interest Inventory

This study compared the relative accuracy of (a) single Occupational Scale (OS) scores on the Strong Interest Inventory (SII) and (b) multiple-predictor scoring functions for discriminating members of nine occupations from people-in-general. The functions were constructed using discriminant function analysis with 4797 adults drawn from criterion samples for the 1985 SII. The analyses included Basic Interest Scale (BIS) scores only or combinations of OSs and BISs. Two subsequent analyses, one of which was cross-validation using 1302 young adults with little work experience, also were conducted. Results indicated that BIS-only functions yielded hit rates comparable to single OSs, and functions that combined OSs and BISs generally produced slight improvements in accuracy. These findings are potentially relevant for future SII scale development decisions.     

A comparison of methods for calculating a severe discrepancy

Given the substantial rise in the number of students identified as learning-disabled, increasing attention has centered on methods for determining a severe discrepancy between ability and achievement. Using scores from 86 learning disabilities referrals, we compared four such methods (a z-score discrepancy, an estimated true score discrepancy, an unadjusted regression procedure, and an adjusted regression procedure). Each student was evaluated with the WISC-R, PIAT, and K-ABC. A high degree of agreement was found between z-score and estimated true score difference approaches. Less agreement was found between the unadjusted regression procedure and the other methods. It was concluded that the four methods cannot be used interchangeably in the calculation of severe discrepancies. Of the four methods that were analyzed, the unadjusted regression procedure selected the smallest percentage of students.     

Adaptive psychophysical methods for nonmonotonic psychometric functions

Many psychophysical tasks in current use render nonmonotonic psychometric functions; these include the oddball task, the temporal generalization task, the binary synchrony judgment task, and other forms of the same–different task. Other tasks allow for ternary responses and render three psychometric functions, one of which is also nonmonotonic, like the ternary synchrony judgment task or the unforced choice task. In all of these cases, data are usually collected with the inefficient method of constant stimuli (MOCS), because extant adaptive methods are only applicable when the psychometric function is monotonic. This article develops stimulus placement criteria for adaptive methods designed for use with nonmonotonic psychometric functions or with ternary tasks. The methods are transformations of conventional up–down rules. Simulations under three alternative psychophysical tasks prove the validity of these methods, their superiority to MOCS, and the accuracy with which they recover direct estimates of the parameters determining the psychometric functions, as well as estimates of derived quantities such as the point of subjective equality or the difference limen. Practical recommendations and worked-out examples are provided to illustrate how to use these adaptive methods in empirical research.     

Alternate methods and software for calculating interobserver agreement for continuous observation data

Portable electronic data collection devices permit investigators to collect large amounts of observational data in a form ready for computer analysis. These devices are particularly efficient for gathering continuous data on multiple behavior categories. We expect that the increasing availability of these devices will lead to greater use of continuous data collection methods in observational research. This paper addresses the difficulties encountered when calculating traditional interobserver agreement statistics for continuous, multiple-code scoring. Two alternative strategies are described that yield interobserver agreement values based on the exact time of behavior code entries by the primary and secondary observers.Work on this paper was supported in part by NICHD Grants P01HD15051 and R01HD17650 and Office of Special Education and Rehabilitation Services Grant G008302980.     

The role of dichotic listening performance and tasks of executive functions in reading impairment: a discriminant function analysis.

This study focused on the role of dichotic listening performance for the identification of reading impaired subtypes. Dichotic listening (DL), using verbal stimuli, has shown to be a valid measure of language lateralization. Usually, lateralization is estimated from the proportion of right ear over left ear accuracy during a free recall test procedure. However, it has been suggested that a more accurate estimate of laterality can be obtained by using a directed attention procedure. A sample of 43 reading disabled children of whom 18 showed signs of impaired language comprehension skills and 25 without language comprehension impairments, were compared to 20 age, and gender, matched controls on dichotic listening performance in both an unbiased free recall task and in a directed attention task using consonant-vowel syllables as dichotic stimuli. A laterality index was calculated for left and right ear stimuli reported during both the free recall condition and the attended ear-scores for the two directed attention conditions. Although both DL procedures yielded significant group differences, with a lower laterality score for the reading disabled compared to the controls, there was no main effect of DL-procedure or group x procedure interaction. Taken alone, DL performance could correctly classify 42% of the reading impaired samples, but together with other measures of executive functions, discriminant function analyses yielded 90.74% accuracy in classifying reading impaired children. The result indicates that DL together with tests of executive functions are valuable tools for assessment of reading impaired subjects.     

Robust canonical discriminant analysis

A method for robust canonical discriminant analysis via two robust objective loss functions is discussed. These functions are useful to reduce the influence of outliers in the data. Majorization is used at several stages of the minimization procedure to obtain a monotonically convergent algorithm. An advantage of the proposed method is that it allows for optimal scaling of the variables. In a simulation study it is shown that under the presence of outliers the robust functions outperform the ordinary least squares function, both when the underlying structure is linear in the variables as when it is nonlinear. Furthermore, the method is illustrated with empirical data.The research of the first author was supported by the Netherlands Organization of Scientific Research (NWO grant 560-267-029).     

Approximate truth

Conclusion The technical results presented here on continuity and approximate implication are obviously incomplete. In particular, a syntactic characterization of approximate implication is highly desirable. Nevertheless, I believe the results above do show that the theory has considerable promise for application to the areas mentioned at the top of the paper.Formulation and defense of realist interpretations of science, for example, require approximate truth because we hardly ever have evidence that a particular scientific theory corresponds perfectly with a portion of the real world. Realists need to assert, then, that evidence for a theory is evidence for its approximate truth, not its truth (see [3] and [18]). Approximate truth is, however, a vague notion, and specification of quantity terms and of a sense of approximation are needed to make precise applications of it. Suitability of both vocabulary and sense of approximation depend on the subject matter, and their selection is a partly empirical matter that raises complex issues. In light of the number of common inferences which are not continuous, realists also need to be concerned about indiscriminate use of deductive logic to derive consequences from approximately true theories. These issues will be considered further in a future paper.Approximate truth also has potential application in areas of artificial intelligence that require inference from inaccurate data. In the qualitative physical theories of de Kleer and Brown [6], for example, qualitative values are derived by partitioning the real numbers into regions. Inferences leading from inside to outside a region must be identified and avoided, and approximate implication and continuity may prove useful in doing this. More generally, growing use of predicate logic as a programming language invites application of the theory of approximate truth as a symbolic substitute for numerical evaluation of computation errors. This too will be the subject of a future paper.Thanks to R. Boyd, A. Garfinkel, H. Hertz, P. Solomon, P. Suppes, S. Weissman, and anonymous referees for advice and criticism.     

Ideal point discriminant analysis

A new method of multiple discriminant analysis was developed that allows a mixture of continuous and discrete predictors. The method can be justified under a wide class of distributional assumptions on the predictor variables. The method can also handle three different sampling situations, conditional, joint and separate. In this method both subjects (cases or any other sampling units) and criterion groups are represented as points in a multidimensional euclidean space. The probability of a particular subject belonging to a particular criterion group is stated as a decreasing function of the distance between the corresponding points. A maximum likelihood estimation procedure was developed and implemented in the form of a FORTRAN program. Detailed analyses of two real data sets were reported to demonstrate various advantages of the proposed method. These advantages mostly derive from model evaluation capabilities based on the Akaike Information Criterion (AIC).The work reported in this paper has been supported by Grant A6394 from the Natural Sciences and Engineering Research Council of Canada and by a leave grant from the Social Sciences and Humanities Research Council of Canada to the first author. Portions of this study were conducted while the first author was at the Institute of Statistical Mathematics in Tokyo on leave from McGill University. He would like to express his gratitude to members of the Institute for their hospitality. Thanks are also due to T. Komazawa at the Institute for letting us use his data, to W. J. Krzanowski at the University of Reading for providing us with Armitage, McPherson, and Copas' data, and to Don Ramirez, Jim Ramsay and Stan Sclove for their helpful comments on an earlier draft of this paper.     

人类的近似数量系统

近似数量系统(Approximate Number System, ANS)指个体在不需要依赖于计算和数量符号的情况下, 对一组数量进行近似表征的系统。通过总结近十年来研究者们在ANS的遗传和神经基础、干预训练等方面取得的新进展, 指出未来应综合运用各种认知神经科学研究手段立足于ANS的基因和脑生理基础研究, 进一步揭示ANS的本质和内在发生发展机制, 并将有关研究发现运用到教育教学中, 对数学困难儿童进行干预, 以提高其数学能力和适应社会的能力。     

A comparison of six methods to estimate thresholds from psychometric functions

There are many ways in which to estimate thresholds from psychometric functions. However, almost nothing is known about the relationships between these estimates. In the present experiment, Monte Carlo techniques were used to compare psychometric thresholds obtained using six methods. Three psychometric functions were simulated using Naka-Rushton and Weibull functions and a probit/logit function combination. Thresholds were estimated using probit, logit, and normit analyses and least-squares regressions of untransformed orz-score and logit-transformed probabilities versus stimulus strength. Histograms were derived from 100 thresholds using each of the six methods for various sampling strategies of each psychometric function. Thresholds from probit, logit, and normit analyses were remarkably similar. Thresholds fromz-score- and logit-transformed regressions were more variable, and linear regression produced biased threshold estimates under some circumstances. Considering the similarity of thresholds, the speed of computation, and the ease of implementation, logit and normit analyses provide effective alternatives to the current “gold standard”—probit analysis—for the estimation of psychometric thresholds.     

The discriminant function and its use in psychology

R. A. Fisher's method of determining a set of weights for the linear combination of the scores in a battery of tests is explained and illustrated. The relationship of this method to combination by multiple regression methods is shown.     

Hierarchical Approximate Bayesian Computation

Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper, we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function.     

Aggression in African-American boys: a discriminant analysis.

The main purposes of this study were to determine whether familial variables found to be related to the development of aggressive behavior in Caucasian boys may also be related to similar aggressive patterns in inner-city African-American boys and to assess the relative importance of these variables for classifying the subject population into aggressive and nonaggressive groups. Discriminant analyses were performed using 83 African-American boys, with a mean age of 13.9 years, who were classified as institutionalized aggressive, noninstitutionalized aggressive, and noninstitutionalized nonaggressive subjects. Although the same father and mother variables were used in the analyses, the father variables were not related to group membership. Furthermore, the results showed that both groups of aggressive boys reported more aggression within as well as outside their family home settings than their nonaggressive counterparts. Socioeconomic-related factors such as employment status of parent(s), size of family, and number of parents in the household were not useful predictors of aggressive and nonaggressive group membership.