Comparing variances of correlated variables

A test is proposed for the equality of the variances ofk 2 correlated variables. Pitman's test fork = 2 reduces the null hypothesis to zero correlation between their sum and their difference. Its extension, eliminating nuisance parameters by a bootstrap procedure, is valid for any correlation structure between thek normally distributed variables. A Monte Carlo study for several combinations of sample sizes and number of variables is presented, comparing the level and power of the new method with previously published tests. Some nonnormal data are included, for which the empirical level tends to be slightly higher than the nominal one. The results show that our method is close in power to the asymptotic tests which are extremely sensitive to nonnormality, yet it is robust and much more powerful than other robust tests.This research was supported by the fund for the promotion of research at the Technion.     

The nonequivalence of behavioral and mathematical equivalence.

Sidman and his colleagues derived behavioral tests for stimulus equivalence from the axiom in logic and mathematics that defines a relation of equivalence. The analogy has generated abundant research in which match-to-sample methods have been used almost exclusively to study interesting and complex stimulus control phenomena. It has also stimulated considerable discussion regarding interpretation of the analogy and speculation as to its validity and generality. This article reexamines the Sidman stimulus equivalence analogy in the context of a broader consideration of the mathematical axiom than was included in the original presentation of the analogy and some of the data that have accumulated in the interim. We propose that (a) mathematical and behavioral examples of equivalence relations differ substantially, (b) terminology is being used in ways that can lead to erroneous conclusions about the nature of the stimulus control that develops in stimulus equivalence experiments, and (c) complete analyses of equivalence and other types of stimulus-stimulus relations require more than a simple invocation of the analogy. Implications of our analysis for resolving current issues and prompting new research are discussed.     

An alternative two stage least squares (2SLS) estimator for latent variable equations

The Maximum-likelihood estimator dominates the estimation of general structural equation models. Noniterative, equation-by-equation estimators for factor analysis have received some attention, but little has been done on such estimators for latent variable equations. I propose an alternative 2SLS estimator of the parameters in LISREL type models and contrast it with the existing ones. The new 2SLS estimator allows observed and latent variables to originate from nonnormal distributions, is consistent, has a known asymptotic covariance matrix, and is estimable with standard statistical software. Diagnostics for evaluating instrumental variables are described. An empirical example illustrates the estimator. I gratefully acknowledge support for this research from the Sociology Program of the National Science Foundation (SES-9121564) and the Center for Advanced Study in the Behavioral Sciences, Stanford, California. This paper was presented at the Interdisciplinary Consortium for Statistical Applications at Indiana University at Bloomington (March 2, 1994) and at the RMD Conference on Causal Modeling at Purdue University, West Lafayette, Indiana (March 3-5, 1994).     

Correspondence analysis and association models constrained by a conditional independence graph

The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered.Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.Thanks are due to anonymous referees who substantially improved the original draft of this paper.     

Relations between baseline contingencies and equivalence probe performances

Following the emergence of two three-member equivalence classes (A1B1C1 and A2B2C2), 5 college students were exposed to one or more changes in the reinforcement contingencies controlling baseline conditional discriminations. AC relations were either reversed (i.e., C2 was reinforced and C1 punished when A1 was the sample; C1 was reinforced and C2 punished when A2 was the sample) or arranged randomly (i.e., C2 and C1 were reinforced and punished equally often in the presence of A1 and A2). In a third condition, the original AB and AC relations were reversed. Results showed that although baseline conditional discrimination performances were under the control of reinforcement contingencies, and performances on symmetry trials varied with baseline responding for 3 of 4 subjects when contingencies were reversed, performances on transitivity probes remained consistent with the initial equivalence class. These inconsistencies between probe and baseline performances were striking because conditional discriminations are thought to be the determinants of equivalence class performance. Similarly, the contrast between performances on symmetry and transitivity probes was of theoretical interest because equivalence classes are defined by congruent patterns of responding on probe trials.     

Conditional discrimination in mentally retarded adults: the development of generalized skills.

The development of generalized conditional discrimination skills was examined in adults with retardation. Two subjects with histories of failure to acquire arbitrary matching under trial-and-error procedures were successful under procedures that trained one or more prerequisite skills. The successive discrimination between the sample stimuli was established by training the subjects to name the stimuli. The simultaneous discrimination between the comparison stimuli was established using either (a) standard simple discrimination training with reversals or (b) a procedure in which each of the two sample-comparison relations in the conditional discrimination was presented in blocks of trials, with the size of the blocks decreasing gradually until sample presentation was randomized. The amount of prerequisite training required varied across subjects and across successive conditional discriminations. After acquiring either two or three conditional discriminations with component training, both subjects learned new conditional discriminations under trial-and-error procedures. In general, each successive conditional discrimination was acquired more rapidly. Tests showed that conditional responding had become a generalized skill. Symmetry was shown for almost all trained relations. Symmetry trial samples were ultimately named the same as the stimuli to which they were related in training.     

A note on distributional properties of the Jöreskog-Sörbom fit indices

In introducing the LISREL model for systems of linear structural equations, Jöreskog and Sörbom proposed two goodness-of-fit indices, GFI and AGFI. Their asymptotic distributions and some statistical properties are discussed.The authors would like to acknowledge the helpful comments and suggestions from the Associate Editor and two anonymous reviewers.     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

Conditioned reinforcement by conditional discriminative stimuli.

A concurrent-chains schedule was used to examine how a delay to conditional discriminative stimuli affects conditioned reinforcement strength. Pigeons' key-peck responses in the initial link produced either of two terminal links according to independent variable-interval 30-s schedules. Each terminal link involved an identical successive conditional discrimination and was segmented into three links: a delay interval (green), a color conditional discriminative stimulus (blue or red), and a line conditional discriminative stimulus (vertical or horizontal lines). Food delivery occurred 45 s after entering the terminal link with a probability of .5, but its conditional probability (1.0 or 0) depended on the combination of the color and the line stimuli. One of the color stimuli occurred independently of further responding, 5 s after entry into the right terminal link, but it occurred 35 s after entry into the left terminal link. One of the line stimuli occurred independently of responding 40 s after entry into either terminal link, synchronized with the offset of the color stimulus. The initial-link relative response rate for the right was consistently higher in comparison with a control condition in which the color stimuli occurred 20 s after entry into either terminal link. The preference for the short delay to the color conditional discriminative stimuli suggests the possibility of conditioned reinforcement by information about the relation between the line conditional discriminative stimuli and the outcomes.     

Trend in correlated proportions

A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.Supported by a grant from the MacArthur Foundation and National Science Foundation Grant BNS85-11774.The authors are indebted to James Heckman for calling our attention to the Clark algorithm.