Correctness of Kruskal's algorithms for monotone regression with ties

Kruskal has proposed two modifications of monotone regression that can be applied if there are ties in nonmetric scaling data. In this note we prove Kruskal's conjecture that his algorithms give the optimal least squares solution of these modified monotone regression problems. We also propose another (third) approach for dealing with ties.Comments by Dr. J. B. Kruskal have been most helpful.     

Test theory without an answer key

A general model is presented for homogeneous, dichotomous items when the answer key is not known a priori. The model is structurally related to the two-class latent structure model with the roles of respondents and items interchanged. For very small sets of respondents, iterative maximum likelihood estimates of the parameters can be obtained by existing methods. For other situations, new estimation methods are developed and assessed with Monte Carlo data. The answer key can be accurately reconstructed with relatively small sets of respondents. The model is useful when a researcher wants to study objectively the knowledge possessed by members of a culturally coherent group that the researcher is not a member of.This research was supported by NSF Grant No. SES-8320173 to the authors. We gratefully acknowledge comments and suggestions from John Boyd, Tarow Indow, and Kathy Maher as well as the editor and several anonymous referees.     

Upward versus Downward Anchoring in Frequency Judgments of Social Facts

Frequency estimation of social facts in two methods of judgment elicitation was investigated. In the “narrow-range” condition, subjects answered questions in the format: “Out of 100 incidents, how many belong to category X?” In the “wide-range” condition, the frequency for the same event was assessed with respect to “Out of 10,000”. Judged frequencies in the wide-range condition were divided by 100, and were compared with the corresponding judgments in the narrow-range condition. Such comparisons were made for low-frequency and high-frequency events. Previous research has shown that, for low-frequency events, judged frequencies are proportionally greater in the narrow-range than in the wide-range condition. These results reflect cognitive processes of implicit anchoring, whereby judged frequencies lie close to small numbers within the response ranges provided. I call this process “downward anchoring,” and predicted that this tendency would be replicated in the present study. Moreover, I predicted that assessments about high-frequency events would evoke similar cognitive processes operating in the opposite direction. By such “upward anchoring,” judged frequencies would lie close to relatively larger numbers within the given response ranges. Consequently, I predicted that judged frequencies for high-frequency events would be proportionally greater in the wide-range condition than in the narrow-range condition. These predictions were confirmed.     

Interpreting canonical correlation analysis through biplots of structure correlations and weights

This paper extends the biplot technique to canonical correlation analysis and redundancy analysis. The plot of structure correlations is shown to the optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biplots may be interpreted.     

The asymptotic posterior normality of the latent trait in an IRT model

It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.This research was partially supported by Office of Naval Research Cognitive and Neural Sciences Grant N0014-J-90-1940, 442-1548, National Science Foundation Mathematics Grant NSF-DMS-91-01436, and the National Center for Supercomputing Applications. We wish to thank Kumar Joag-dev and Zhiliang Ying for enlightening suggestions concerning the proof of the basic result.The authors wish to thank Kumar Joag-Dev, Brian Junker, Bert Green, Paul Holland, Robert Mislevy, and especially Zhiliang Ying for their useful comments and discussions.     

Correlation and prediction in ordinal test theory

Based on the test theory model for ordinal measurements proposed by Schulman and Haden, the present paper considers correlations between tests, attenuation, regressions involving true and observed scores, and prediction of test reliability.The population correlation between tests is shown to be related to the expected sample correlation for samples of sizen ₁ andn ₂. Errors of estimation, measurement and prediction are found to be similar to their counterparts in interval test theory, while attenuation is identical to its counterpart. The bias in estimating population reliability from sample data is compared for Kendall's tau and Spearman's rho.The author wishes to thank the referees for their helpful comments on an earlier draft of this paper, and in particular, for the suggested alternative methods of establishing some of the presented results.     

Estimation and visualization of confusability matrices from adaptive measurement data

We present a simple but effective method based on Luce’s choice axiom [Luce, R.D. (1959). Individual choice behavior: A theoretical analysis. New York: John Wiley & Sons] for consistent estimation of the pairwise confusabilities of items in a multiple-choice recognition task with arbitrarily chosen choice-sets. The method combines the exact (non-asymptotic) Bayesian way of assessing uncertainty with the unbiasedness emphasized in the classical frequentist approach.We apply the method to data collected using an adaptive computer game designed for prevention of reading disability. A player’s estimated confusability of phonemes (or more accurately, phoneme-grapheme connections) and larger units of language is visualized in an easily understood way with color cues and explicit indication of the accuracy of the estimates. Visualization of learning-related changes in the player’s performance is considered.The empirical validity of the choice axiom is evaluated using the game data itself. The axiom appears to hold reasonably well although a small systematic violation is observable for the smallest choice-set sizes.     

Monaural Loudness Adaptation for Middle-Intensity Middle-Frequency Signals: The Importance of Measurement Technique

Using the Simple Adaptation technique (SA) and the Ipsilateral Comparison Paradigm (ICP), the authors studied monaural loudness adaptation to a middle-intensity [60 dB(A)] tone at signal frequencies of 250, 1000, and 4000 Hz in the left and right ears. Adaptation effects were absent when the SA procedure was used. However, they were observed uniformly across all frequency values with the ICP, a result that challenges the assertion in the literature, on the basis of SA measures, that loudness adaptation for middle-intensity signals occurs only at frequencies above 4000 Hz. The ICP features periodic intensity modulations (±10 dB relative to the base signal) to accommodate listeners' needs for referents by which they can gauge subtle changes in the loudness of the adapting tone, a key component that is missing in the SA method. Adaptation effects in this investigation were similar in both ears, supporting the equal susceptibility assumption common in loudness adaptation studies.     

单病种平均住院日影响因素分析

为探讨单病种平均住院日的影响因素,采用多元逐步回归分析,对住院构成前5位病种的平均住院日影响因素进行分析。结果显示,有10个影响因素进入回归方程,不同病种平均住院日影响因素略有不同,是否转科对平均住院日影响最大。在医院管理过程中,需不断加强医院流程管理与改造,实施临床路径,合理制定医院平均住院日指标。