Linear relations amongk sets of variables

A family of solutions for linear relations amongk sets of variables is proposed. It is shown how these solutions apply fork=2, and how they can be generalized from there tok3.The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.     

Statistical inference based on ranks

This paper develops a unified approach, based on ranks, to the statistical analysis of data arising from complex experimental designs. In this way we answer a major objection to the use of rank procedures as a major methodology in data analysis. We show that the rank procedures, including testing, estimation and multiple comparisons, are generated in a natural way from a robust measure of scale. The rank methods closely parallel the familiar methods of least squares, so that estimates and tests have natural interpretations.This research was supported in part by grant MCS76-07292 from the National Science Foundation.     

Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: II: A search procedure to locate the greatest lower bound

Finding the greatest lower bound for the reliability of the total score on a test comprisingn non-homogenous items with dispersion matrix Σ _x is equivalent to maximizing the trace of a diagonal matrix Σ _E with elements θ _I , subject to Σ _E and Σ _T =Σ _x − Σ _E being non-negative definite. The casesn=2 andn=3 are solved explicity. A computer search in the space of the θ _i is developed for the general case. When Guttman's λ₄ (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ _i makes the rank of Σ _T less thann − 1. Numerical examples of various bounds are given. Present affiliation of the first author: St. Hild's College of Education, Durham City, England.     

SELECTIVE ASSOCIATIONS PRODUCED SOLELY WITH APPETITIVE CONTINGENCIES: THE STIMULUS-REINFORCER INTERACTION REVISITED

In studies reporting stimulus-reinforcer interactions in traditional conditioning paradigms, when a tone-light compound was associated with food the light gained stimulus control, but when the compound was paired with shock avoidance the tone gained control. However, the physical nature of the reinforcerrelated events (food vs. shock) presented in the presence of the tone-light compound was always confounded with the conditioned hedonic value of the compound's presence relative to its absence. When the compound was paired with shock, its presence was negative relative to its absence (which was shock-free). In contrast, when the compound was paired with food, its presence was positive relative to its absence (which was food-free). The present experiment dealt with this confounding effect by conditioning a tone-light compound to be positive or negative, relative to its absence, solely with food reinforcement. One group of rats received food for responding in the presence of the tone-light compound and no food in its absence. The other group also responded in the presence of the compound, but received food only in its absence. These rats were trained on a chained schedule in which responding in the presence of the tone-light compound produced a terminal link signaled by the absence of the compound; responding ceased in the terminal link because it delayed food delivery. In a test session to assess stimulus control by the elements of the compound, tone and light were presented separately under extinction conditions. Rats that had been exposed to a positive correlation between food and the compound emitted almost double the responses in the presence of the light as in the presence of the tone. In comparison, rats that had been exposed to a negative correlation emitted only two thirds as many responses in the presence of the light as in the presence of the tone. Because this selective association was produced using only food, it appears that the contingencies under which a reinforcer is presented, rather than (or as well as) its physical properties, can generate the selective associations previously attributed to “stimulus-reinforcer interactions.” This could mean that regardless of the class of reinforcer that ultimately maintains responding (appetitive or aversive), the contingency-generated hedonic value of the compound stimulus may influence the dominant modality of stimulus control.     

Paired comparison,triple comparison,and ranking experiments as generalized linear models,and their implementation on GLIM

A wide variety of paired comparison, triple comparison, and ranking experiments may be viewed as generalized linear models. These include paired comparison models based on both the Bradley-Terry and Thurstone-Mosteller approaches, as well as extensions of these models that allow for ties, order of presentation effects, and the presence of covariates. Moreover, the triple comparison model of Pendergrass and Bradley, as well as models for complete rankings of more than three items, can also be represented as generalized linear models. All such models can be easily fit by maximum likelihood, using the widely available GLIM computer package. Additionally, GLIM enables the computation of likelihood ratio statistics for testing many hypotheses of interest. Examples are presented that cover a variety of cases, along with their implementation on GLIM.     

Confirmation of linear system theory prediction: Changes in Herrnstein's k as a function of changes in reinforcer magnitude

Eight human subjects pressed a lever on a range of variable-interval schedules for 0.25¢ to 35.0¢ per reinforcement. Herrnstein's hyperbola described seven of the eight subjects' response-rate data well. For all subjects, the y-asymptote of the hyperbola increased with increasing reinforcer magnitude and its reciprocal was a linear function of the reciprocal of reinforcer magnitude. These results confirm predictions made by linear system theory; they contradict formal properties of Herrnstein's account and of six other mathematical accounts of single-alternative responding.     

Perceived self-efficacy by Under-10 tennis players when scaling the equipment and play area

The modification of child-sports aims to develop an optimal learning environment that facilitates the emergence of desirable psychological outcomes (e.g., self-efficacy). The aim of the study was to assess the effect of reducing net height and court size on self-efficacy and shot-efficacy of U-10 tennis players in a real-game context. Twenty U-10 tennis players (M = 9.46, SD = 0.66 years of age; M = 3.65, SD = 1.53 years of tennis experience) played two round-robin tournaments one week apart in the same order and schedule. The first tournament was played under the International Tennis Federation’s Tennis 10s regulation at green stage (GT). Afterwards, the modified tournament (MT) was played with the same regulation GT, however, net height (0.91 m–0.80 m) and court size (23.77 m × 8.23 m–18.00 m × 8.23 m) were reduced. Results accomplished using Bayesian and Frequentist inferences showed an increase in players’ self-efficacy when serving in MT than GT (BF₁₀ = 4.796; δ = −0.576; and p = .011). This is increase may be due to a reduction in their serving faults in MT (BF₁₀ = 6.169; δ = −0.591; and p = .010). Therefore, reducing net height and court size enhances the serve performance and self-efficacy and thus promotes positive tennis experiences.     

Generalized latent trait models

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.     

Bayesian inference for finite mixtures of generalized linear models with random effects

We present an hierarchical Bayes approach to modeling parameter heterogeneity in generalized linear models. The model assumes that there are relevant subpopulations and that within each subpopulation the individual-level regression coefficients have a multivariate normal distribution. However, class membership is not known a priori, so the heterogeneity in the regression coefficients becomes a finite mixture of normal distributions. This approach combines the flexibility of semiparametric, latent class models that assume common parameters for each sub-population and the parsimony of random effects models that assume normal distributions for the regression parameters. The number of subpopulations is selected to maximize the posterior probability of the model being true. Simulations are presented which document the performance of the methodology for synthetic data with known heterogeneity and number of sub-populations. An application is presented concerning preferences for various aspects of personal computers.