Application of the bootstrap methods in factor analysis

A Monte Carlo experiment is conducted to investigate the performance of the bootstrap methods in normal theory maximum likelihood factor analysis both when the distributional assumption is satisfied and unsatisfied. The parameters and their functions of interest include unrotated loadings, analytically rotated loadings, and unique variances. The results reveal that (a) bootstrap bias estimation performs sometimes poorly for factor loadings and nonstandardized unique variances; (b) bootstrap variance estimation performs well even when the distributional assumption is violated; (c) bootstrap confidence intervals based on the Studentized statistics are recommended; (d) if structural hypothesis about the population covariance matrix is taken into account then the bootstrap distribution of the normal theory likelihood ratio test statistic is close to the corresponding sampling distribution with slightly heavier right tail.This study was carried out in part under the ISM cooperative research program (91-ISM · CRP-85, 92-ISM · CRP-102). The authors would like to thank the editor and three reviewers for their helpful comments and suggestions which improved the quality of this paper considerably.     

Identification of inconsistent variates in factor analysis

When some of observed variates do not conform to the model under consideration, they will have a serious effect on the results of statistical analysis. In factor analysis the model with inconsistent variates may result in improper solutions. In this article a useful method for identifying a variate as inconsistent is proposed in factor analysis. The procedure is based on the likelihood principle. Several statistical properties such as the effect of misspecified hypotheses, the problem of multiple comparisons, and robustness to violation of distributional assumptions are investigated. The procedure is illustrated by some examples.     

Social reinforcement of operant behavior in rats: a methodological note.

An apparatus was developed to study social reinforcement in the rat. Four Long-Evans female rats were trained to press a lever via shaping, with the reinforcer being access to a castrated male rat. Responding under a fixed-ratio schedule and in extinction was also observed. Social access was found to be an effective reinforcer. When social reinforcement was compared with food reinforcement under similar conditions of deprivation and reinforcer duration, no significant differences were observed.     

A probabilistic multidimensional scaling with unique axes1

Abstract: A probabilistic multidimensional scaling model is proposed. The model assumes that the coordinates of each stimulus are normally distributed with variance Σ_i = diag(σ²₁, … σ²_Ri). The advantage of this model is that axes are determined uniquely. The distribution of the distance between two stimuli is obtained by polar coordinates transformation. The method of maximum likelihood estimation for means and variances using the EM algorithm is discussed. Further, simulated annealing is suggested as a means of obtaining initial values in order to avoid local maxima. A simulation study shows that the estimates are accurate, and a numerical example concerning the location of Japanese cities shows that natural axes can be obtained without introducing individual parameters.     

韦伯比的拟合公式

本文以神经生理学为基础,结合感官具有调节和比较两功能的实用性,建立起刺激在大脑皮层上引起感知效应Ｅ的数学模型：Ｅ＝Ｂｌｏｇ（１＋ｋＩ）,其中ｋ＝Ｉ￣（－ｂ）／ａ。从此出发,可推出中等以上刺激的差别阈判据：△ＥＭ＝恒量。将此判据推广到弱刺激,即得到差别阈的普适规律：△Ｉ／（Ｉ＋ａＩ￣ｂ）＝Ｎ,亦即韦伯定律的修订公式。从弱到强的广大刺激范围内,它都能与实验数据拟合良好。     

A molar theory of reinforcement schedules

Behavior of subjects exposed to concurrent and individual interval and ratio schedules of reinforcement may be described in terms of a set of expressions relating the value of responses to their durations, a feedback equation relating reinforcement to response duration, and the assumption that subjects allocate their time among various responses so as to maximize value.     

Delayed reinforcement of fixed-ratio performance without mediating exteroceptive conditioned reinforcement

The performance of pigeons was studied under conditions in which the completion of a fixed-ratio requirement was not contiguous with the presentation of a reinforcer. Timein and timeout periods alternated throughout the experimental sessions. Responses made by an experimental bird during the timein period were accumulated, and when a fixed-ratio requirement had been met, grain was presented to the experimental bird and a yoked control following their first response in the next timein period. Across most manipulations of the fixed-ratio requirement and of the duration of the timeout period, the response rates of the experimental birds were considerably higher than those of their controls, suggesting that the response-reinforcer dependency controlled the behavior of the experimental bird in the absence of a close temporal association between responding on the ratio schedule and reinforcer presentations.     

A new computational method to fit the weighted euclidean distance model

This paper describes a computational method for weighted euclidean distance scaling which combines aspects of an analytic solution with an approach using loss functions. We justify this new method by giving a simplified treatment of the algebraic properties of a transformed version of the weighted distance model. The new algorithm is much faster than INDSCAL yet less arbitrary than other analytic procedures. The procedure, which we call SUMSCAL (subjectivemetricscaling), gives essentially the same solutions as INDSCAL for two moderate-size data sets tested.Comments by J. Douglas Carroll and J. B. Kruskal have been very helpful in preparing this paper.     

Alternative fixed-ratio fixed-interval schedules of reinforcement

Five rats were trained under alternative fixed-ratio fixed-interval schedules, in which food reinforcement was provided for the completion of either a fixed-ratio or a fixed-interval requirement, whichever was met first. Overall response rate and running rate (the rate of responding after the postreinforcement pause) decreased for all subjects as the fixed-ratio value increased. As the proportion of reinforcements obtained from the fixed-ratio component increased and the alternative schedule approached a simple fixed ratio, overall response rate and running rate both increased; conversely, as the proportion of reinforcements obtained from the fixed-interval component increased and the alternative schedule approached a simple fixed interval, response rates decreased. Postreinforcement pause length increased linearly as the average time between reinforcements increased, regardless of the schedule parameters. A break-run pattern of responding was predominant at low- and medium-valued fixed ratios. All subjects displayed at least occasional positively accelerated responding within interreinforcement intervals at higher fixed-ratio values.     

A stochastic multidimensional scaling procedure for the empirical determination of convex indifference curves for preference/choice analysis

The vast majority of existing multidimensional scaling (MDS) procedures devised for the analysis of paired comparison preference/choice judgments are typically based on either scalar product (i.e., vector) or unfolding (i.e., ideal-point) models. Such methods tend to ignore many of the essential components of microeconomic theory including convex indifference curves, constrained utility maximization, demand functions, et cetera. This paper presents a new stochastic MDS procedure called MICROSCALE that attempts to operationalize many of these traditional microeconomic concepts. First, we briefly review several existing MDS models that operate on paired comparisons data, noting the particular nature of the utility functions implied by each class of models. These utility assumptions are then directly contrasted to those of microeconomic theory. The new maximum likelihood based procedure, MICROSCALE, is presented, as well as the technical details of the estimation procedure. The results of a Monte Carlo analysis investigating the performance of the algorithm as a number of model, data, and error factors are experimentally manipulated are provided. Finally, an illustration in consumer psychology concerning a convenience sample of thirty consumers providing paired comparisons judgments for some fourteen brands of over-the-counter analgesics is discussed.