Deviations from Prescribed Prompting Procedures: Implications for Treatment Integrity

The acquisition of new skills may be hindered when teaching procedures vary from previously validated approaches or contain errors. In the present study, we compared the acquisition and maintenance of response chains taught using a perfectly implemented system of least prompts and a multiple verbal prompts procedure (i.e., addition of multiple verbal prompts and failure to follow through with more intrusive prompts). Four children, aged 6–9, participated in the study. An adapted alternating treatments design was used to compare the effectiveness and efficiency of learning during the system of least prompts and the alternative system of least prompts. Results were consistent with those obtained in previous studies in that the perfectly implemented and alternative prompting procedures were effective in teaching new skills for all participants. However, the perfectly implemented treatment required fewer trials to mastery for 4 of the 5 evaluations. Response chains taught under the multiple verbal prompts condition had poorer maintenance for 2 of the 5 evaluations. The results of the current study suggest that deviations from empirically identified teaching procedures may reduce the speed with which new skills are acquired.     

Structural equation models with continuous and polytomous variables

A two-stage procedure is developed for analyzing structural equation models with continuous and polytomous variables. At the first stage, the maximum likelihood estimates of the thresholds, polychoric covariances and variances, and polyserial covariances are simultaneously obtained with the help of an appropriate transformation that significantly simplifies the computation. An asymptotic covariance matrix of the estiates is also computed. At the second stage, the parameters in the structural covariance model are obtained via the generalized least squares approach. Basic statistical properties of the estimates are derived and some illustrative examples and a small simulation study are reported.This research was supported in part by a research grant DA01070 from the U. S. Public Health Service. We are indebted to several referees and the editor for very valuable comments and suggestions for improvement of this paper. The computing assistance of King-Hong Leung and Man-Lai Tang is also gratefully acknowledged.     

Estimating an unobserved component of a serial response time model

A method is developed for estimating the response time distribution of an unobserved component in a two-component serial model, assuming the components are stochastically independent. The estimate of the component's density function is constrained only to be unimodal and non-negative. Numerical examples suggest that the method can yield reasonably accurate estimates with sample sizes of 300 and, in some cases, with sample sizes as small as 100.The author wishes to thank David Kohfeld, Jim Ramsay, Jim Townsend and two anonymous referees for a number of useful and stimulating comments on an earlier version of this paper.     

Mapclus: A mathematical programming approach to fitting the adclus model

We present a new algorithm, MAPCLUS (MAthematicalProgrammingCLUStering), for fitting the Shepard-Arabie ADCLUS (forADditiveCLUStering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS (INdividualDifferencesCLUStering).We are indebted to Scott A. Boorman, W. K. Estes, J. A. Hartigan, Lawrence J. Hubert, Carol L. Krumhansl, Joseph B. Kruskal, Sandra Pruzansky, Roger N. Shepard, Edward J. Shoben, Sigfrid D. Soli, and Amos Tversky for helpful discussions of this work, as well as the anonymous referees for their suggestions and corrections on an earlier version of this paper. We are also grateful to Pamela Baker and Dan C. Knutson for technical assistance. The research reported here was supported in part by LEAA Grant 78-NI-AX-0142 and NSF Grants SOC76-24512 and SOC76-24394.     

A generalization of Takane's algorithm for dedicom

An algorithm is described for fitting the DEDICOM model for the analysis of asymmetric data matrices. This algorithm generalizes an algorithm suggested by Takane in that it uses a damping parameter in the iterative process. Takane's algorithm does not always converge monotonically. Based on the generalized algorithm, a modification of Takane's algorithm is suggested such that this modified algorithm converges monotonically. It is suggested to choose as starting configurations for the algorithm those configurations that yield closed-form solutions in some special cases. Finally, a sufficient condition is described for monotonic convergence of Takane's original algorithm.Financial Support by the Netherlands organization for scientific research (NWO) is gratefully acknowledged. The authors are obliged to Richard Harshman.     

Intuitive prediction: ecological validity versus representativeness

Insufficiently regressive intuitive predictions have been attributed to mistaken reliance on the representativeness heuristic. In contrast, we suggest that intuitive predictions stem from a conceptualization of ‘goodness of prediction’ that differs from the accepted statistical definition in terms of error minimization, namely, ecological validity—that is, representation of the substantive characteristics of the predicted variable Y and its distribution as well as of the relationship between Y and the predictor X—rather than minimization of prediction errors. Simultaneous satisfaction of the above representation requirements is achieved by multivalued prediction: The prediction of different Y′ values for the same X value, resulting in conditional distributions Y′|X for at least some X values. Empirical results supporting this hypothesis are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

等级数据的测量不变性检验及影响因素模拟研究

研究介绍了针对等级数据的模型建构（LRV,潜在反应变量模型）和参数估计（WLSMV）方法,以及在此基础上的测量不变性检验（DIFFTEST）方法,同时采用蒙特卡洛模拟研究方法,考察样本总量大小、组间样本量对比情况、阈值差异程度、量表长度等因素,对DIFTEST进行针对等级数据的测量不变性检验效果的影响情况,以及WLSMV估计方法下的参数复原情况。研究结果发现WLSMV估计方法参数的复原效果很好;DIFFTEST的一类错误概率达到可接受水平,在大样本情况下、组间样本量基本相等、阈值差异程度较大时,DIFFTEST检测力较好。在控制测量不变性遭受破坏的项目个数情况下,随着测验长度的增加,DIFFTEST的检测力下降。     

An Introduction to Kristof’s Theorem for Solving Least-Square Optimization Problems Without Calculus

Kristof’s Theorem (Kristof, 1970 Kristof, W. (1970). A theorem on the trace of certain matrix products and some applications. Journal of Mathematical Psychology, 7(3), 515–530. https://doi.org/10.1016/0022-2496(70)90037-4[Crossref], [Web of Science ®] , [Google Scholar]) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof’s Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof’s (1964 Kristof, W. (1964). Die beste orthogonale Transformation zur gegenseitigen Uberfiihrung zweier Faktorenmatrizen. Diagnostica, 10, 87–90. [Google Scholar], 1970 Kristof, W. (1970). A theorem on the trace of certain matrix products and some applications. Journal of Mathematical Psychology, 7(3), 515–530. https://doi.org/10.1016/0022-2496(70)90037-4[Crossref], [Web of Science ®] , [Google Scholar]) writings. In this article, I describe the underlying logic of Kristof’s Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem’s proof. I then show how Kristof’s Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017 R Core Team (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from https://www.R-project.org/ [Google Scholar]) code to perform the calculations described in the text.