Three-mode orthomax rotation

Factor analysis and principal components analysis (PCA) are often followed by an orthomax rotation to rotate a loading matrix to simple structure. The simple structure is usually defined in terms of the simplicity of the columns of the loading matrix. In Three-mode PCA, rotational freedom of the so called core (a three-way array relating components for the three different modes) can be used similarly to find a simple structure of the core. Simple structure of the core can be defined with respect to all three modes simultaneously, possibly with different emphases on the different modes. The present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally, this approach relies on repeated (two-way) orthomax applied to supermatrices containing the frontal, lateral or horizontal slabs, respectively. The procedure is illustrated by means of a number of exemplary analyses. As a by-product, application of the Three-mode Orthomax procedures to two-way arrays is shown to reveal interesting relations with and interpretations of existing two-way simple structure rotation techniques.This research has been made possible by a fellowship from the Royal Netherlands Academy of Arts and Sciences to the author. The author is obliged to Jos ten Berge and two anonymous reviewers for useful comments on an earlier version of this paper.     

Maximum likelihood additivity analysis

A maximum likelihood estimation procedure was developed to fit unweighted and weighted additive models to conjoint data obtained by the categorical rating, the pair comparison or the directional ranking method. The scoring algorithm used to fit the models was found to be both reliable and efficient, and the program MAXADD is capable of handling up to 300 parameters to be estimated. Practical uses of the procedure are reported to demonstrate various advantages of the procedure as a statistical method.The research reported here was supported by Grant A6394 to the author from the Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Iowa City, Iowa, in May, 1980.Thanks are due to Jim Ramsay, Justine Sergent and anonymous reviewers for their helpful comments.Two MAXADD programs which perform the computations discussed in this paper may be obtained from the author.     

Reanalysis of the equation for simple action

De Villiers and Herrnstein (1976) have shown that the equation for simple action, derived from the matching law, predicts change in behavioral output for some 40 experiments in which the value of a single source of reinforcement has been varied. Using only the positive-reinforcement studies they cite that used five or more different reinforcement values, we found the high percentage of variance they report accommodated by this equation (94%) is predicated on instances of averaging rates of behavioral output before making a least-squares fit of the equation. In our reanalysis, which minimizes rate averaging, only 78% of the data variance is accommodated. This diminished data-variance accommodation can be improved by adding parameters that permit the equation's scaling constant to change as a function of reinforcement. Although these parameters permit acceptable levels of accommodation of data variance, they correspond to no obvious psychological processes. These findings support the view that the equation for simple action is an inadequate model for behavioral output.     

Preference for unsegmented interreinforcement intervals in concurrent chains

Five pigeons were trained under concurrent-chain schedules in which a pair of independent, concurrent variable-interval 60-s schedules were presented in the initial link and either both variable-interval or both fixed-interval schedules were presented in the terminal link. Except for the baseline, one of the terminal-link schedules was always a two-component chained schedule and the other was either a simple or a tandem schedule of equal mean interreinforcement interval. The values of the fixed-interval schedules were either 15 s or 60 s; that of the variable-interval schedules was always 60 s. A 1.5-s changeover delay operated during the initial link in some conditions. The pigeons preferred a simple or a tandem schedule to a chain. For the fixed-interval schedules, this preference was greater when the fixed interval was 60 s than when it was 15 s. For the variable-interval schedules, the preferences were less pronounced and occurred only when the changeover delay was in effect. For a given type of schedule and interreinforcement interval, similar preferences were obtained whether the nonchained schedule was a tandem or simple schedule. The changeover delay generally inflated preference and lowered the changeover rate, especially when the terminal-link schedules were either short (15 s) or aperiodic (variable-interval). The results were consistent with the notion that segmenting the interreinforcement interval of a schedule into a chain lowers the preference for it.     

Susceptibility of spatial and verbal working memory to demands of the central executive

We used a dual-task paradigm to examine the degree to which domain-specific spatial and verbal subsystems depend on the domain-general central executive. Forty participants were asked to retain spatial or verbal information while performing a concurrent secondary task related to simple arithmetic. The secondary tasks consisted of three cognitive processes: single-digit addition, a digit-carrying operation, and digit reading. The single-digit addition and carry operation include central executive functioning, while digit reading relies solely on the phonological loop. The single-digit addition caused a performance decrement on the spatial working memory task, while the digit reading impaired performance on the verbal working memory task. The carry operation interfered with recall accuracy on both working memory tasks. The spatial working memory task was significantly correlated with the verbal working memory task only when the secondary task was more demanding on the central executive. Our results suggest that spatial working memory rather than verbal working memory is susceptible to failure of central executive functioning and that the central executive plays an important role in regulating the cognitive demands of different domains.     

Types of \mathsf{I} -Free Hereditary Right Maximal Terms

The implicational fragment of the relevance logic “ticket entailment” is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of T_→ are in normal form and built in all but one case from and only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define FIT_→ – a Fitch-style calculus (related to FT_→) for the implicational fragment of ticket entailment.     

A case of extreme simplicity of the core matrix in three-mode principal components analysis

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed. Constructive comments from anonymous reviewers are gratefully acknowledged.     

Quasivarieties Generated by Simple MV-algebras

In this paper we show that the quasivariety generated by an infinite simple MV-algebra only depends on the rationals which it contains. We extend this property to arbitrary families of simple MV-algebras.     

Empirical bayes estimates of domain scores under binomial and hypergeometric distributions for test scores

We introduce two simple empirical approximate Bayes estimators (EABEs)— and —for estimating domain scores under binomial and hypergeometric distributions, respectively. Both EABEs (derived from corresponding marginal distributions of observed test scorex without relying on knowledge of prior domain score distributions) have been proven to hold -asymptotic optimality in Robbins' sense of convergence in mean. We found that, where and are the monotonized versions of and under Van Houwelingen's monotonization method, respectively, the convergence rate of the overall expected loss of Bayes risk in either or depends on test length, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, and outperform their maximum likelihood counterparts over the middle range of domain scales. In terms of mean-squared error, we also found that: (a) given a unimodal prior distribution of domain scores, performs better than both and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterogeneity; (b) performs as well as when prior distribution is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimodal prior distribution exists.The authors are indebted to both anonymous reviewers, especially Reviewer 2, and the Editor for their invaluable comments and suggestions. Thanks are also due to Yuan-Chin Chang and Chin-Fu Hsiao for their help with our simulation and programming work.     

Session duration and the VI response function: Within-session prospective and retrospective effects

Two experiments examined the effects of session duration on responding during simple variable-interval schedules. In Experiment 1, rats were exposed to a series of simple variable-interval schedules differing in both session duration (10 min or 30 min) and scheduled reinforcement rate (7.5 s, 15 s, 30 s, and 480 s). The functions relating response rate to reinforcement rate were predominantly monotonic for the short (10-min) sessions but were predominantly bitonic for the long (30-min) sessions, when data from the entire session were considered. Examination of responding within sessions suggested that differences in the whole-session data were produced by a combination of prospective processes (i.e., processes based on events scheduled to occur later in the session) and retrospective processes (i.e., processes based on events that had already occurred in the session). In Experiment 2, rats were exposed to a modified discrimination procedure in which pellet flavor (standard or banana) predicted session duration (10 min or 30 min). All rats came to respond faster during the short (10-min) sessions than during the first 10 min of the long sessions. As in Experiment 1, the results seemed to reflect the simultaneous operation of both prospective and retrospective processes. The results shed light on the recent controversy over the form of the variable-interval response function by identifying one variable (session duration) and two types of processes (prospective and retrospective) that influence responding on these schedules.