Exact conditional tests for cross-classifications: Approximation of attained significance levels

A procedure is proposed for approximating attained significance levels of exact conditional tests. The procedure utilizes a sampling from the null distribution of tables having the same marginal frequencies as the observed table. Application of the approximation through a computer subroutine yields precise approximations for practically any table dimensions and sample size.     

Estimation of covariance structure models with parameters subject to functional restraints

This paper demonstrates the feasibility of using the penalty function method to estimate parameters that are subject to a set of functional constraints in covariance structure analysis. Both types of inequality and equality constraints are studied. The approaches of maximum likelihood and generalized least squares estimation are considered. A modified Scoring algorithm and a modified Gauss-Newton algorithm are implemented to produce the appropriate constrained estimates. The methodology is illustrated by its applications to Heywood cases in confirmatory factor analysis, quasi-Weiner simplex model, and multitrait-multimethod matrix analysis.The author is indebted to several anonymous reviewers for creative suggestions for improvement of this paper. Computer funding is provided by the Computer Services Centre, The Chinese University of Hong Kong.     

基于可达阵的一种Q矩阵标定方法

Q矩阵标定是实施认知诊断评估的前提,已有Q矩阵修正方法并不太适合测验中已知属性向量的题目数较少的情形。根据拓展Q矩阵理论中可达阵R列与简化Q阵列存在布尔“或”关系,在一定认知假设下,率先提出可达阵R与简化Q阵的潜在反应列存在布尔“与”关系,并由此提出基于可达阵的Q矩阵标定方法。研究显示：在已知一个可达阵下,当可达阵项目的猜测或失误参数在.20以下且待标定项目的项目参数约在.30以下时,新方法所得Q矩阵元素返真率基本在.90以上,并且真实Q矩阵与估计Q矩阵下被试分类准确率差异很小;对于含5个属性的独立结构,新方法要求的随机样本的样本量较小;实证研究也印证了模拟研究的结论。新方法只需专家标定少量题目的Q矩阵,即已经标定的Q矩阵对应属性层级结构的可达阵。     

Choice model analysis of the “pick any/n” type of binary data

In the “pick any/n” method, subjects are asked to choose any number of items from a list of n items according to some criterion. This kind of data can be analyzed as a special case of either multiple-choice data or successive categories data where the number of response categories is limited to two. An item response model was proposed for the latter case, which is a combination of an unfolding model and a choice model. The marginal maximum-likelihood estimation method was developed for parameter estimation to avoid incidental parameters, and an expectation-maximization algorithm used for numerical optimization. Two examples of analysis are given to illustrate the proposed method, which we call MAXSC.     

Marginal maximum likelihood estimation for a psychometric model of discontinuous development

Item response theory models posit latent variables to account for regularities in students' performances on test items. Wilson's “Saltus” model extends the ideas of IRT to development that occurs in stages, where expected changes can be discontinuous, show different patterns for different types of items, or even exhibit reversals in probabilities of success on certain tasks. Examples include Piagetian stages of psychological development and Siegler's rule-based learning. This paper derives marginal maximum likelihood (MML) estimation equations for the structural parameters of the Saltus model and suggests a computing approximation based on the EM algorithm. For individual examinees, empirical Bayes probabilities of learning-stage are given, along with proficiency parameter estimates conditional on stage membership. The MML solution is illustrated with simulated data and an example from the domain of mixed number subtraction. The authors' names appear in alphabetical order. We would like to thank Karen Draney for computer programming, Kikumi Tatsuoka for allowing us to use the mixed-number subtraction data, and Eric Bradlow, Chan Dayton, Kikumi Tatsuoka, and four anonymous referees for helpful suggestions. The first author's work was supported by Contract No. N00014-88-K-0304, R&T 4421552, from the Cognitive Sciences Program, Cognitive and Neural Sciences Division, Office of Naval Research, and by the Program Research Planning Council of Educational Testing Service. The second author's work was supported by a National Academy of Education Spencer Fellowship and by a Junior Faculty Research Grant from the Committee on Research, University of California at Berkeley. A copy of the Saltus computer program can be obtained from the second author.     

A maximum likelihood approach to test validation with missing and censored dependent variables

A maximum likelihood approach is described for estimating the validity of a test (x) as a predictor of a criterion variable (y) when there are both missing and censoredy scores present in the data set. The missing data are due to selection on a latent variable (y _s ) which may be conditionally related toy givenx. Thus, the missing data may not be missing random. The censoring process in due to the presence of a floor or ceiling effect. The maximum likelihood estimates are constructed using the EM algorithm. The entire analysis is demonstrated in terms of hypothetical data sets.     

Estimating latent distributions in recurrent choice data

This paper introduces a flexible class of stochastic mixture models for the analysis and interpretation of individual differences in recurrent choice and other types of count data. These choice models are derived by specifying elements of the choice process at the individual level. Probability distributions are introduced to describe variations in the choice process among individuals and to obtain a representation of the aggregate choice behavior. Due to the explicit consideration of random effect sources, the choice models are parsimonious and readily interpretable. An easy to implement EM algorithm is presented for parameter estimation. Two applications illustrate the proposed approach.     

Exploratory longitudinal factor analysis in multiple populations

For multiple populatios, a longtidinal factor analytic model which is entirely exploratory, that is, no explicit identification constraints, is proposed. Factorial collapse and period/practice effects are allowed. An invariant and/or stationary factor pattern is permitted. This model is formulated stochastically. To implement this model a stagewise EM algorithm is developed. Finally a numerical illustration utilizing Nesselroade and Baltes' data is presented.The authors wish to thank Barbara Mellers and Henry Kaiser for their helpful comments and John Nesselroade for providing us the data for our illustration. This research wwa supported in part by a grant (No. AG03164) from the National Institute on Aging to William Meredith. Details of the derivations and a copy of the PROC MATRIX program are available upon request from the first author.