Power of the likelihood ratio test in covariance structure analysis

A procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived. The procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified. Using the noncentral Chi-square distribution, the power of the test is approximated by the asymptotic one for a sequence of local alternatives. The procedure is illustrated by an example. A Monte Carlo experiment also shows how good the approximation is for a specific case.This research was made possible by a grant from the Dutch Organization for Advancement of Pure Research (ZWO). The authors also like to acknowledge the helpful comments and suggestions from the editor and anonymous reviewers.     

Response–reinforcer contiguity versus response-rate–reinforcer-rate covariance in rats' lever pressing: Support for a multiscale view

The multiscale molar view sees behavior as a flow, like a river, extended in time. Matching theory expresses the way activities compete for time. Relative time taken by any activity depends on relative induction. The present experiment tested matching theory applied to concurrent contingent and noncontingent food. As adjunctive activities that compete with operant activity, we recorded hopper head entries and presses on a lever near the food hopper that had no programmed consequences. Eight naïve rats were first exposed to a variable-time 60 s schedule, which across conditions was gradually transformed into a variable-interval 60 s schedule by increasing the proportion of food that was delivered contingent on pressing a lever far from the hopper. Another group of 4 rats that had been trained to press a lever near a food hopper were introduced in the second condition, in which one food delivery was contingent on far-lever pressing. We found induction following a power function to describe pressing on the far lever (operant activity). Matching theory combined with power-function induction also accounted for adjunctive activity. Results with single contingent food deliveries provided little support for the molecular view that behavior consists of discrete responses “strengthened” by immediately following reinforcers.     

Avoidance,induction, and the illusion of reinforcement

Environmental events that impact reproductive success may be called phylogenetically important events (PIEs). Some promote reproductive success, like mates and food; others threaten reproductive success, like predators and injury. Beneficial PIEs induce activities that enhance them, and detrimental PIEs induce activities that mitigate or avoid them. Free-operant avoidance relies on electric shock as a proxy for injury, a PIE. One theory takes avoidance behavior to be reinforced by its reducing shock rate. A more complete explanation is that avoidance both reduces shock rate and is induced by the PIEs it usually prevents. Shocks received act in concert with shock-rate reduction, in a feedback system. Four parametric data sets were analyzed to show that avoidance is induced by received shock rate according to power functions. Avoidance is not reinforced at all; avoidance is induced by its failures. Induction explains not only avoidance itself, but also phenomena unique to avoidance, like warmup and effects of unavoidable shock. Induction explains behavior more generally than reinforcement, because induction explains not only food-maintained operant and nonoperant behavior, but also shock-maintained behavior, including avoidance. Reinforcement fails to explain behavior when reinforcement is defined as strengthening by consequences. Induction erases the distinction between consequences and antecedents.     

Structural analysis of covariance and correlation matrices

A general approach to the analysis of covariance structures is considered, in which the variances and covariances or correlations of the observed variables are directly expressed in terms of the parameters of interest. The statistical problems of identification, estimation and testing of such covariance or correlation structures are discussed.Several different types of covariance structures are considered as special cases of the general model. These include models for sets of congeneric tests, models for confirmatory and exploratory factor analysis, models for estimation of variance and covariance components, regression models with measurement errors, path analysis models, simplex and circumplex models. Many of the different types of covariance structures are illustrated by means of real data.1978 Psychometric Society Presidential Address.This research has been supported by the Bank of Sweden Tercentenary Foundation under the project entitledStructural Equation Models in the Social Sciences, Karl G. Jöreskog, project director.     

A Comparison of Inverse-Wishart Prior Specifications for Covariance Matrices in Multilevel Autoregressive Models

Multilevel autoregressive models are especially suited for modeling between-person differences in within-person processes. Fitting these models with Bayesian techniques requires the specification of prior distributions for all parameters. Often it is desirable to specify prior distributions that have negligible effects on the resulting parameter estimates. However, the conjugate prior distribution for covariance matrices—the Inverse-Wishart distribution—tends to be informative when variances are close to zero. This is problematic for multilevel autoregressive models, because autoregressive parameters are usually small for each individual, so that the variance of these parameters will be small. We performed a simulation study to compare the performance of three Inverse-Wishart prior specifications suggested in the literature, when one or more variances for the random effects in the multilevel autoregressive model are small. Our results show that the prior specification that uses plug-in ML estimates of the variances performs best. We advise to always include a sensitivity analysis for the prior specification for covariance matrices of random parameters, especially in autoregressive models, and to include a data-based prior specification in this analysis. We illustrate such an analysis by means of an empirical application on repeated measures data on worrying and positive affect.     

Exploring Factor Model Parameters across Continuous Variables with Local Structural Equation Models

Using an empirical data set, we investigated variation in factor model parameters across a continuous moderator variable and demonstrated three modeling approaches: multiple-group mean and covariance structure (MGMCS) analyses, local structural equation modeling (LSEM), and moderated factor analysis (MFA). We focused on how to study variation in factor model parameters as a function of continuous variables such as age, socioeconomic status, ability levels, acculturation, and so forth. Specifically, we formalized the LSEM approach in detail as compared with previous work and investigated its statistical properties with an analytical derivation and a simulation study. We also provide code for the easy implementation of LSEM. The illustration of methods was based on cross-sectional cognitive ability data from individuals ranging in age from 4 to 23 years. Variations in factor loadings across age were examined with regard to the age differentiation hypothesis. LSEM and MFA converged with respect to the conclusions. When there was a broad age range within groups and varying relations between the indicator variables and the common factor across age, MGMCS produced distorted parameter estimates. We discuss the pros of LSEM compared with MFA and recommend using the two tools as complementary approaches for investigating moderation in factor model parameters.     

Goodness-of-fit and confidence intervals of approximate models

If the model for the data are strictly speaking incorrect, then how can one test whether the model fits? Standard goodness-of-fit (GOF) tests rely on strictly correct or incorrect models. But in practice the correct model is not assumed to be available. It would still be of interest to determine how good or how bad the approximation is. But how can this be achieved? If it is determined that a model is a good approximation and hence a good explanation of the data, how can reliable confidence intervals be constructed? In this paper, an attempt is made to answer the above questions. Several GOF tests and methods of constructing confidence intervals are evaluated both in a simulation and with real data from the internet-based daily news memory test.     

Nonparametric Estimation of Standard Errors in Covariance Analysis Using the Infinitesimal Jackknife

The infinitesimal jackknife provides a simple general method for estimating standard errors in covariance structure analysis. Beyond its simplicity and generality what makes the infinitesimal jackknife method attractive is that essentially no assumptions are required to produce consistent standard error estimates, not even the requirement that the population sampled has the covariance structure assumed. Commonly used covariance structure analysis software uses parametric methods for estimating parameters and standard errors. When the population sampled has the covariance structure assumed, but fails to have the distributional form assumed, the parameter estimates usually remain consistent, but the standard error estimates do not. This has motivated the introduction of a variety of nonparametric standard error estimates that are consistent when the population sampled fails to have the distributional form assumed. The only distributional assumption these require is that the covariance structure be correctly specified. As noted, even this assumption is not required for the infinitesimal jackknife. The relation between the infinitesimal jackknife and other nonparametric standard error estimators is discussed. An advantage of the infinitesimal jackknife over the jackknife and the bootstrap is that it requires only one analysis to produce standard error estimates rather than one for every jackknife or bootstrap sample.     

模拟互变问题解决认知策略的实验研究

本实验通过对模拟互变问题解决认知策略的定性和定量分析,研究认知策略的类型及其信息加工程序。结果表明,存在尝试性策略、分析性策略和直觉性策略三类策略。最佳策略属于分析性策略。