Confidence regions for multidimensional scaling analysis

Techniques are developed for surrounding each of the points in a multidimensional scaling solution with a region which will contain the population point with some level of confidence. Bayesian credibility regions are also discussed. A general theorem is proven which describes the asymptotic distribution of maximum likelihood estimates subject to identifiability constraints. This theorem is applied to a number of models to display asymptotic variance-covariance matrices for coordinate estimates under different rotational constraints. A technique is described for displaying Bayesian conditional credibility regions for any sample size.The research reported here was supported by grant number APA 320 to the author by the National Research Council of Canada.     

Multidimensional unfolding analyses of ranking data for groups

One probabilistic version of Coombs' unfolding model called the MMUR (Marginalization model for the Multidimensional Unfolding analysis of Ranking data) is extended to treat ranking data for groups. One favorable feature of the model is that it can both take into consideration individual differences without estimating the subject parameters and capture the differences between the groups in a systematic manner. Another advantage lies in the fact that one can see the group differences in the geometrical point configuration, since the model shows how the ideal points of the groups differ from each other in space. Four applications are provided which demonstrate that the model is useful for clarifying systematic differences in this type of data.     

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.     

Covariance structure analysis of ordinal ipsative data

Data are ipsative if they are subject to a constant-sum constraint for each individual. In the present study, ordinal ipsative data (OID) are defined as the ordinal rankings across a vector of variables. It is assumed that OID are the manifestations of their underlying nonipsative vector y, which are difficult to observe directly. A two-stage estimation procedure is suggested for the analysis of structural equation models with OID. In the first stage, the partition maximum likelihood (PML) method and the generalized least squares (GLS) method are proposed for estimating the means and the covariance matrix of A_cy, where A_c is a known contrast matrix. Based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, the generalized least squares method is used to estimate the structural parameters in the second stage. A goodness-of-fit statistic is given for testing the hypothesized covariance structure. Simulation results show that the proposed method works properly when a sufficiently large sample is available.This research was supported by National Institute on Drug Abuse Grants DA01070 and DA10017. The authors are indebted to Dr. Lee Cooper, Dr. Eric Holman, Dr. Thomas Wickens for their valuable suggestions on this study, and Dr. Fanny Cheung for allowing us to use her CPAI data set in this article. The authors would also like to acknowledge the helpful comments from the editor and the two anonymous reviewers.     

On the sampling interpretation of confidence intervals and hypothesis tests in the context of conditional maximum likelihood estimation

In the context ofconditional maximum likelihood (CML) estimation, confidence intervals can be interpreted in three different ways, depending on the sampling distribution under which these confidence intervals contain the true parameter value with a certain probability. These sampling distributions are (a) the distribution of the data given theincidental parameters, (b) the marginal distribution of the data (i.e., with the incidental parameters integrated out), and (c) the conditional distribution of the data given the sufficient statistics for the incidental parameters. Results on the asymptotic distribution of CML estimates under sampling scheme (c) can be used to construct asymptotic confidence intervals using only the CML estimates. This is not possible for the results on the asymptotic distribution under sampling schemes (a) and (b). However, it is shown that theconditional asymptotic confidence intervals are also valid under the other two sampling schemes. I am indebted to Theo Eggen, Norman Verhelst and one of Psychometrika's reviewers for their helpful comments.     

标准参照测验中的信度估计公式

标准参照测验是与常模参照测验不同的一种测验,在标准参照测验中,一个人在测验上的分数不是和他人相比较而是和某个已经设定的标准作比较。如果测验是从某功课论域中随机抽样构造而成,则使用者希望知道考生在这份测验上的观测分数与其在该功课论域上的分数（假如已知）的接近程度;如果使用者想根据测验分数对考生作掌握分类,则他们关心这个推断与假设考生论域分数已知时所作推断一致程度有多高。本文对这两个问题的信度估计进行探讨,得到几个有用的估计公式。     

A parametric procedure for ultrametric tree estimation from conditional rank order proximity data

The psychometric and classification literatures have illustrated the fact that a wide class of discrete or network models (e.g., hierarchical or ultrametric trees) for the analysis of ordinal proximity data are plagued by potential degenerate solutions if estimated using traditional nonmetric procedures (i.e., procedures which optimize a STRESS-based criteria of fit and whose solutions are invariant under a monotone transformation of the input data). This paper proposes a new parametric, maximum likelihood based procedure for estimating ultrametric trees for the analysis of conditional rank order proximity data. We present the technical aspects of the model and the estimation algorithm. Some preliminary Monte Carlo results are discussed. A consumer psychology application is provided examining the similarity of fifteen types of snack/breakfast items. Finally, some directions for future research are provided.     

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.     

Ability estimation for conventional tests

Five different ability estimators—maximum likelihood [MLE ()], weighted likelihood [WLE ()], Bayesian modal [BME ()], expected a posteriori [EAP ()] and the standardized number-right score [Z ()]—were used as scores for conventional, multiple-choice tests. The bias, standard error and reliability of the five ability estimators were evaluated using Monte Carlo estimates of the unknown conditional means and variances of the estimators. The results indicated that ability estimates based on BME (), EAP () or WLE () were reasonably unbiased for the range of abilities corresponding to the difficulty of a test, and that their standard errors were relatively small. Also, they were as reliable as the old standby—the number-right score.