A survey of residual analysis and a new test of residual trend

A survey of residual analysis in behavior‐analytic research reveals that existing methods are problematic in one way or another. A new test for residual trends is proposed that avoids the problematic features of the existing methods. It entails fitting cubic polynomials to sets of residuals and comparing their effect sizes to those that would be expected if the sets of residuals were random. To this end, sampling distributions of effect sizes for fits of a cubic polynomial to random data were obtained by generating sets of random standardized residuals of various sizes, n. A cubic polynomial was then fitted to each set of residuals and its effect size was calculated. This yielded a sampling distribution of effect sizes for each n. To test for a residual trend in experimental data, the median effect size of cubic‐polynomial fits to sets of experimental residuals can be compared to the median of the corresponding sampling distribution of effect sizes for random residuals using a sign test. An example from the literature, which entailed comparing mathematical and computational models of continuous choice, is used to illustrate the utility of the test.     

Standard errors of fit indices using residuals in structural equation modeling

The asymptotic standard errors of the correlation residuals and Bentler's standardized residuals in covariance structures are derived based on the asymptotic covariance matrix of raw covariance residuals. Using these results, approximations of the asymptotic standard errors of the root mean square residuals for unstandardized or standardized residuals are derived by the delta method. Further, in mean structures, approximations of the asymptotic standard errors of residuals, standardized residuals and their summary statistics are derived in a similar manner. Simulations are carried out, which show that the asymptotic standard errors of the various types of residuals and the root mean square residuals in covariance, correlation and mean structures are close to actual ones.The author is indebted to the reviewers for their comments and suggestions which have led to an improvement of this work.     

Knock,knock, anybody home? Psychological availability as link between work and relationship

This research investigated how negative and positive residuals of the workday spill over into the partner relationship. To unravel the mechanism of this spillover, this study introduces the concept of “psychological availability,” which refers to the individual's ability and motivation to direct psychological resources at the partner. A survey was conducted among 313 Dutch dual‐earner couples with children, using dyadic data analysis to analyze both partners simultaneously in 1 model. Psychological availability mediated the link between negative and positive work‐related residuals (i.e., work‐related negative mood, exhaustion, rumination and work‐related positive mood and vigor, respectively) and marital behavior. The findings suggest that partners' workday residuals spill over into the marital relationship through being more or less psychologically available for each other.     

When fit indices and residuals are incompatible

Standard chi-square-based fit indices for factor analysis and related models have a little known property: They are more sensitive to misfit when unique variances are small than when they are large. Consequently, very small correlation residuals indicating excellent fit can be accompanied by indications of bad fit by the fit indices when unique variances are small. An empirical example of this incompatibility between residuals and fit indices is provided. For illustrative purposes, an artificial example is provided that yields exactly the same correlation residuals as the empirical example but has larger unique variances. For this example, the fit indices indicate excellent fit. A theoretical explanation for this phenomenon is provided using relationships between unique variances and eigenvalues of the fitted correlation matrix.     

Modeling Common Traits and Method Effects in Multitrait-Multimethod Analysis

Method effects often occur when constructs are measured by different methods. In traditional multitrait-multimethod (MTMM) models method effects are regarded as residuals, which implies a mean method effect of zero and no correlation between trait and method effects. Furthermore, in some recent MTMM models, traits are modeled to be specific to a certain method. However, often we are not interested in a method-specific trait but in a trait that is common to all methods. Here we present the Method Effect model with common trait factors, which allows modeling “common” trait factors and method factors that represent method “effects” rather than residuals. The common trait factors are defined as the mean of the true-score variables of all variables measuring the same trait and the method factors are defined as differences between true-score variables and means of true-score variables. Because the model allows estimating mean method effects, correlations between method factors, and correlations between trait and method factors, new research questions may be investigated. The application of the model is demonstrated by 2 examples studying the effect of negative, as compared with positive, item wording for the measurement of mood states.     

Working in East German Socialism in 1980 and in Capitalism 15 Years Later: A Trend Analysis of a Transitional Economy's Working Conditions

Many studies document the changes that have taken place in the new German states, the former German Democratic Republic (GDR), since the end of socialism. Most research looks at the changes that took place after the unification of East and West Germany, but little is known about the differences between the present, somewhat settled situation in the new German states and the stable situation in the GDR before the system change.
The goal of this study was to enlarge our knowledge on these differences. With a trend analysis, aspects of work in the GDR in 1980 ( n =337) were compared with the new German states in 1995 ( n =168). Results showed that there was more job control and complexity, more activity in work improvement and better work organisation in the new German states than in the GDR. There was no difference in stress variables and social support by colleagues between both groups. Relationships with supervisors and appreciation for suggestions for work improvement were better in the GDR than after the introduction of capitalism.     

Gender effects in gaming research: a case for regression residuals?

Numerous recent studies have examined the impact of video gaming on various dependent variables, including the players' affective reactions, positive as well as detrimental cognitive effects, and real-world aggression. These target variables are typically analyzed as a function of game characteristics and player attributes-especially gender. However, findings on the uneven distribution of gaming experience between males and females, on the one hand, and the effect of gaming experience on several target variables, on the other hand, point at a possible confound when gaming experiments are analyzed with a standard analysis of variance. This study uses simulated data to exemplify analysis of regression residuals as a potentially beneficial data analysis strategy for such datasets. As the actual impact of gaming experience on each of the various dependent variables differs, the ultimate benefits of analysis of regression residuals entirely depend on the research question, but it offers a powerful statistical approach to video game research whenever gaming experience is a confounding factor.     

An Ecological Analysis of the Impact of the U.S. Supreme Court's 1973 Abortion Decision

The ecological characteristics of states were correlated with measures of the distribution of abortions among states before and after the Supreme Court's abortion decision. Techniques of path analysis were used to separate these conelations into direct effects and effects mediated through state abortion laws. The Court decision produced a gradual decrease in the effect of abortion laws on the distribution of abortions, and in the indirect effects of ecological variables. The major trend in the direct effects of ecological variables was a gradual increase in the relationship between population size and number of abortions. Differences reflected in state laws continued to influence abortion rates per 1,000 women of reproductive age and abortion ratios per 1,000 live births.     

"SOME THINGS YOU LEARN AREN'T SO":

When interpreting an interaction in the analysis of variance (ANOVA), many active researchers (and, in turn, students) often Ignore the residuals defining the interaction Although this problem has been noted previously, a appears that many users of ANOVA remain uncertain about the proper understanding of interaction effects To clear up this problem, we review the way in which the ANOVA model enables us to take apart a table of group means or the individual measurements contributing to the means to reveal the underlying components We also show how (using only published data) to compute a contrast on the question that may be of primary interest and illustrate for interpreting tables of residuals We conclude with an exercise to check on students understanding of ANOVA and to encourage increased precision in the specification of research results     

Analysis of residuals for the multionmial item response model

Using the item response model as developed on the multinomial distribution, asymptotic variances are obtained for residuals associated with response patterns and first-, and second-order marginal frequencies of manifest variables. When the model does not fit well, an examination of these residuals may reveal the source of the poor fit. Finally, a limited-information test of fit for the model is developed by using residuals defined for the first-, and second-order marginals. Model evaluation based on residuals for these marginals is particularly useful when the response pattern frequencies are sparse.The author would like to thank Yasuo Amemiya and Joseph Lucke for helpful suggestions. This research was supported by a Research Incentive Grant from Arizona State University.     

A note on the reflection of signs in the extraction of centroid factors

This note suggests that the reflection of residuals in the centroid method of factor analysis should be continued, whenever possible, after all the sums of the columns in the correlation matrix, excluding diagonal values, are positive. A criterion is given for determining whether further reflection is possible in such cases.     

Are assessment center behaviors' meanings consistent across exercises? A measurement invariance approach

To examine the appropriateness of a Multi‐Trait–Multi‐Method framework for testing construct validity of Assessment Centers (ACs) and get practical implications for the improved AC design, degree to which the AC dimension‐related performance behaviors consistently manifest across multiple AC rating situations was investigated. The present study used a large sample (N = 5,006) to apply a measurement invariance analysis. AC rating situations generally produced consistent factor loadings for items on AC dimensions, item residuals, dimension factor variances, and covariance between dimensions. The AC rating situation of interview tended to produce higher ratings and less item residuals. These findings support the consistency in constructs assessed across different AC rating situations, while some exercises may be better for teasing apart particular dimensions than others.     

Non-g residuals of the SAT and ACT predict specific abilities

This research examined whether non-g residuals of the SAT and ACT subtests, obtained after removing g, predicted specific abilities. Non-g residuals of the verbal and math subtests of the SAT and ACT were correlated with academic (verbal and math) and non-academic abilities (speed and shop), both based on the Armed Services Vocational Aptitude Battery. Non-g residuals of the SAT and ACT math subtests were positively related to math ability and negatively to verbal ability, whereas the opposite pattern was found for the verbal subtests. Non-g residuals of both sets of subtests were weakly related to non-academic abilities. The results support an investment theory of skills and abilities: Investing in skills in one area (e.g., math) improves abilities in that area but lowers abilities in competing areas (e.g., verbal).     

Additive Multilevel Item Structure Models with Random Residuals: Item Modeling for Explanation and Item Generation

An additive multilevel item structure (AMIS) model with random residuals is proposed. The model includes multilevel latent regressions of item discrimination and item difficulty parameters on covariates at both item and item category levels with random residuals at both levels. The AMIS model is useful for explanation purposes and also for prediction purposes as in an item generation context. The parameters can be estimated with an alternating imputation posterior algorithm that makes use of adaptive quadrature, and the performance of this algorithm is evaluated in a simulation study.     

Contextualized Personality Questionnaires: A Case for Copulas in Structural Equation Models for Categorical Data

For structural equation models (SEMs) with categorical data, correlated measurement residuals are not easily implemented. The problem lies mainly in the absence of a categorical analogue to the multivariate normal distribution and the absence of closed form formulas in SEMs for categorical data. We present a novel technique to handle measurement residuals that keeps the attractive SEM mainframe intact yet adds flexibility in dependence modeling without excessive computational burden. The technique is based upon the concept of copula functions and is introduced with a data set of ordinal responses originating from a contextualized personality study on aggression. Focus is on models arising in a multitrait-multimethod context, where the flexibility in dependence structures allows for method effects that can vary across the latent trait dimension. The empirical application illustrates that ignoring design-implied correlated measurement residuals can potentially influence study results and conclusions in both a quantitative as well as a qualitative way. Model parameter estimates can be biased, but more important, model inferences can be heavily distorted.     

The effect of interstimulus procedures on salt taste intensity functions

Taste intensity functions were constructed for the salt taste, using variations of the method of direct magnitude estimation, to study the effects of adaptation to residuals from prior tastings. The prediction that mouthrinsing would reduce residuals, lower the level of adaptation, and thus yield lower exponents was confirmed. The effect of the residuals tended to be reduced, tiowever, when the power functions were adjusted for such threshold changes. Power function deviations from linearity were also discussed. The literature on intensity scaling for the salt taste was examined and seen to support these findings. It was concluded that the interstimulus procedure, by altering residual stimulus levels, played a major role in determining salt taste intensity functions.     

Bayesian inference for graphical factor analysis models

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years. We acknowledge support from M.U.R.S.T. of Italy and from the European Science Foundation H.S.S.S. Network. We are grateful to the referees and the Editor for many useful suggestions and comments which led to a substantial improvement of the paper. We also thank Nanny Wermuth for stimulating discussions and Kenneth A. Bollen for kindly providing us with the data-set.     

Assessing Item Fit for Unidimensional Item Response Theory Models Using Residuals from Estimated Item Response Functions

Residual analysis (e.g. Hambleton & Swaminathan, Item response theory: principles and applications, Kluwer Academic, Boston, 1985; Hambleton, Swaminathan, & Rogers, Fundamentals of item response theory, Sage, Newbury Park, 1991) is a popular method to assess fit of item response theory (IRT) models. We suggest a form of residual analysis that may be applied to assess item fit for unidimensional IRT models. The residual analysis consists of a comparison of the maximum-likelihood estimate of the item characteristic curve with an alternative ratio estimate of the item characteristic curve. The large sample distribution of the residual is proved to be standardized normal when the IRT model fits the data. We compare the performance of our suggested residual to the standardized residual of Hambleton et al. (Fundamentals of item response theory, Sage, Newbury Park, 1991) in a detailed simulation study. We then calculate our suggested residuals using data from an operational test. The residuals appear to be useful in assessing the item fit for unidimensional IRT models.     

A note on the minres method

This paper discusses the advantages and problems related to factor analysis by minimizing residuals (minres). It is shown that this method fails if the starting point of iterations is not well chosen. A suitable starting point is suggested.     

Correspondence analysis used complementary to loglinear analysis

Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.