On Insensitivity of the Chi-Square Model Test to Nonlinear Misspecification in Structural Equation Models

In this paper, we show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable to detect the presence of nonlinear terms in the model. As an example, we consider a regression model with latent variables and interactions terms. Not only the model test has zero power against that type of misspecifications, but even the theoretical (chi-square) distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model. We explain this phenomenon by exploiting results on asymptotic robustness in structural equation models. The importance of this paper is to warn against the conclusion that if a proposed linear model fits the data well according to the chi-quare goodness-of-fit test, then the underlying model is linear indeed; it will be shown that the underlying model may, in fact, be severely nonlinear. In addition, the present paper shows that such insensitivity to nonlinear terms is only a particular instance of a more general problem, namely, the incapacity of the classical chi-square goodness-of-fit test to detect deviations from zero correlation among exogenous regressors (either being them observable, or latent) when the structural part of the model is just saturated.     

Testing Process Factor Analysis Models Using the Parametric Bootstrap

Process factor analysis (PFA) is a latent variable model for intensive longitudinal data. It combines P-technique factor analysis and time series analysis. The goodness-of-fit test in PFA is currently unavailable. In the paper, we propose a parametric bootstrap method for assessing model fit in PFA. We illustrate the test with an empirical data set in which 22 participants rated their effects everyday over a period of 90 days. We also explore Type I error and power of the parametric bootstrap test with simulated data.     

Computing the real solutions of Fleishman's equations for simulating non-normal data

Fleishman's power method is frequently used to simulate non-normal data with a desired skewness and kurtosis. Fleishman's method requires solving a system of nonlinear equations to find the third-order polynomial weights that transform a standard normal variable into a non-normal variable with desired moments. Most users of the power method seem unaware that Fleishman's equations have multiple solutions for typical combinations of skewness and kurtosis. Furthermore, researchers lack a simple method for exploring the multiple solutions of Fleishman's equations, so most applications only consider a single solution. In this paper, we propose novel methods for finding all real-valued solutions of Fleishman's equations. Additionally, we characterize the solutions in terms of differences in higher order moments. Our theoretical analysis of the power method reveals that there typically exists two solutions of Fleishman's equations that have noteworthy differences in higher order moments. Using simulated examples, we demonstrate that these differences can have remarkable effects on the shape of the non-normal distribution, as well as the sampling distributions of statistics calculated from the data. Some considerations for choosing a solution are discussed, and some recommendations for improved reporting standards are provided.     

Pairwise Likelihood Ratio Tests and Model Selection Criteria for Structural Equation Models with Ordinal Variables

Correlated multivariate ordinal data can be analysed with structural equation models. Parameter estimation has been tackled in the literature using limited-information methods including three-stage least squares and pseudo-likelihood estimation methods such as pairwise maximum likelihood estimation. In this paper, two likelihood ratio test statistics and their asymptotic distributions are derived for testing overall goodness-of-fit and nested models, respectively, under the estimation framework of pairwise maximum likelihood estimation. Simulation results show a satisfactory performance of type I error and power for the proposed test statistics and also suggest that the performance of the proposed test statistics is similar to that of the test statistics derived under the three-stage diagonally weighted and unweighted least squares. Furthermore, the corresponding, under the pairwise framework, model selection criteria, AIC and BIC, show satisfactory results in selecting the right model in our simulation examples. The derivation of the likelihood ratio test statistics and model selection criteria under the pairwise framework together with pairwise estimation provide a flexible framework for fitting and testing structural equation models for ordinal as well as for other types of data. The test statistics derived and the model selection criteria are used on data on ‘trust in the police’ selected from the 2010 European Social Survey. The proposed test statistics and the model selection criteria have been implemented in the R package lavaan.     

Non-linearity,Complexity and Limited Measurement in the Relationship Between Satisfaction with Specific Life Domains and Satisfaction with Life as a Whole

In this article we defend that the adoption of a non-linear approach, theoretically framed on complexity theories can make some contribution to the bottom-up approach, which explains the levels of satisfaction with life as a whole through the combination of the levels of satisfaction in different life domains. Two approaches have been tested: (Rojas in J Happiness Stud 7:467–497, 2006) constant elasticity of substitution model and the model with quadratic terms and interaction effects (González et al. in Soc Indic Res 80:267–295, 2006; González et al. in Qual Quant 42:1–21, 2008). In order to prevent obtaining false non-linear relationships they have been analysed twice taking into account or not limited measurement of satisfaction with life as a whole. Results show that: (a) any of the two non-linear models fits better than the linear one; (b) any of the models failing to take into account limited measurement fits worse; (c) the non-linear model with quadratic terms and interaction effects fits better than Rojas’. The implications for the study of psychological well-being are discussed.     

Application of a Multidimensional Nested Logit Model to Multiple-Choice Test Items

Nested logit models have been presented as an alternative to multinomial logistic models for multiple-choice test items (Suh and Bolt in Psychometrika 75:454–473, 2010) and possess a mathematical structure that naturally lends itself to evaluating the incremental information provided by attending to distractor selection in scoring. One potential concern in attending to distractors is the possibility that distractor selection reflects a different trait/ability than that underlying the correct response. This paper illustrates a multidimensional extension of a nested logit item response model that can be used to evaluate such distinctions and also defines a new framework for incorporating collateral information from distractor selection when differences exist. The approach is demonstrated in application to questions faced by a university testing center over whether to incorporate distractor selection into the scoring of its multiple-choice tests. Several empirical examples are presented.     

Human memory reconsolidation can be explained using the temporal context model

Recent work by Hupbach, Gomez, Hardt, and Nadel (Learning & Memory, 14, 47–53, 2007) and Hupbach, Gomez, and Nadel (Memory, 17, 502–510, 2009) suggests that episodic memory for a previously studied list can be updated to include new items, if participants are reminded of the earlier list just prior to learning a new list. The key finding from the Hupbach studies was an asymmetric pattern of intrusions, whereby participants intruded numerous items from the second list when trying to recall the first list, but not viceversa. Hupbach et al. (2007; 2009) explained this pattern in terms of a cellular reconsolidation process, whereby first-list memory is rendered labile by the reminder and the labile memory is then updated to include items from the second list. Here, we show that the temporal context model of memory, which lacks a cellular reconsolidation process, can account for the asymmetric intrusion effect, using well-established principles of contextual reinstatement and item–context binding.     

Statistical Tests for Comparing Possibly Misspecified and Nonnested Models

Model selection criteria (MSC) involves selecting the model with the best estimated goodness-of-fit to the data generating process. Following the method of Vuong (1989), a large sample Model Selection Test (MST), is introduced that can be used in conjunction with most existing MSC procedures to decide if the estimated goodness-of-fit for one model is significantly different from the estimated goodness-of-fit for another model. The MST extends the classical generalized likelihood ratio test, is valid in the presence of model misspecification, and is applicable to situations involving nonnested probability models. Simulation studies designed to illustrate the concept of the MST and its conservative decision rule (relative to the MSC method) are also presented. Copyright 2000 Academic Press.     

A formal ideal-based account of typicality

Inspired by Barsalou’s (Journal of Experimental Psychology: Learning, Memory, and Cognition, 11, 629–654, 1985) proposal that categories can be represented by ideals, we develop and test a computational model, the ideal dimension model (IDM). The IDM is tested in its account of the typicality gradient for 11 superordinate natural language concepts and, using Bayesian model evaluation, contrasted with a standard exemplar model and a central prototype model. The IDM is found to capture typicality better than do the exemplar model and the central tendency prototype model, in terms of both goodness of fit and generalizability. The present findings challenge the dominant view that exemplar representations are most successful and present compelling evidence that superordinate natural language categories can be represented using an abstract summary, in the form of ideal representations. Supplemental appendices for this article can be downloaded from .     

Anticipation of cognitive demand during decision-making

An abundance of evidence indicates that action selection is guided, at least in certain contexts, by anticipation of action outcomes. In one particularly clear demonstration of this principle, Bechara and colleagues, studying a gambling task, observed phasic skin conductance responses just prior to actions associated with a relatively high risk of monetary loss (Bechara et al. in J Neurosci 19:5473–5481, 1999; Bechara et al. in Science 275:1293–1295, 1997; Bechara et al. in Cereb Cortex 6:215–225, 1996). In the present work, we tested for the same effect in a paradigm where choices resulted not in differential monetary outcomes, but in differential requirements for subsequent mental effort. In two experiments, we observed an anticipatory skin conductance response prior to actions resulting in a high level of cognitive demand. This finding indicates that requirements for effortful cognitive control are anticipated during action selection. We argue, based on convergent evidence, that such anticipation may not only trigger preparation; it may also play a direct role in effort-based decision-making.     

Extracting meaning from past affective experiences: The importance of peaks,ends, and specific emotions

This article reviews existing empirical research on the peak-and-end rule. This rule states that people's global evaluations of past affective episodes can be well predicted by the affect experienced during just two moments: the moment of peak affect intensity and the ending. One consequence of the peak-and-end rule is that the duration of affective episodes is largely neglected. Evidence supporting the peak-and-end rule is robust, but qualified. New directions for future work in this emerging area of study are outlined. In particular, the personal meanings associated with specific moments and with specific emotions should be assessed. It is hypothesised that moments rich with self-relevant information will dominate people's global evaluations of past affective episodes. The article concludes with a discussion of ways to measure and optimise objective happiness.     

Stepwise variable selection in factor analysis

It is very important to choose appropriate variables to be analyzed in multivariate analysis when there are many observed variables such as those in a questionnaire. What is actually done in scale construction with factor analysis is nothing but variable selection.In this paper, we take several goodness-of-fit statistics as measures of variable selection and develop backward elimination and forward selection procedures in exploratory factor analysis. Once factor analysis is done for a certain numberp of observed variables (thep-variable model is labeled the current model), simple formulas for predicted fit measures such as chi-square, GFI, CFI, IFI and RMSEA, developed in the field of the structural equation modeling, are provided for all models obtained by adding an external variable (so that the number of variables isp + 1) and for those by deleting an internal variable (so that the number isp – 1), provided that the number of factors is held constant.A programSEFA (Stepwise variable selection in Exploratory Factor Analysis) is developed to actually obtain a list of the fit measures for all such models. The list is very useful in determining which variable should be dropped from the current model to improve the fit of the current model. It is also useful in finding a suitable variable that may be added to the current model. A model with more appropriate variables makes more stable inference in general.The criteria traditionally often used for variable selection is magnitude of communalities. This criteria gives a different choice of variables and does not improve fit of the model in most cases.The URL of the programSEFA is http://koko15.hus.osaka-u.ac.jp/~harada/factor/stepwise/.     

Graded response model based on the logistic positive exponent family of models for dichotomous responses

Samejima (Psychometrika 65:319–335, 2000) proposed the logistic positive exponent family of models (LPEF) for dichotomous responses in the unidimensional latent space. The objective of the present paper is to propose and discuss a graded response model that is expanded from the LPEF, in the context of item response theory (IRT). This specific graded response model belongs to the general framework of graded response model (Samejima, Psychometrika Monograph, No. 17, 1969 and No. 18, 1972; Handbook of modern item response theory, Springer, New York, 1997; Encyclopedia of Social Measurement, Academic Press, San Diego, 2004), and, in particular to the heterogeneous case (Samejima, Psychometrika Monograph, No. 18, 1972). Thus, the model can deal with any number of ordered polytomous responses, such as letter grades (e.g., A, B, C, D, F), etc. For brevity, hereafter, the model will be called the LPEF graded response model, or LPEFG. This model reflects the opposing two principles contained in the LPEF for dichotomous responses, with the logistic model (Birnbaum, Statistical theories of mental test scores, Addison Wesley, Reading, 1968) as their transition, which provide a reasonable rationale for partial credits in LPEFG, among others.     

Tests of Homoscedasticity,Normality, and Missing Completely at Random for Incomplete Multivariate Data

Test of homogeneity of covariances (or homoscedasticity) among several groups has many applications in statistical analysis. In the context of incomplete data analysis, tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random (MCAR). These tests of MCAR require large sample sizes n and/or large group sample sizes n _i , and they usually fail when applied to nonnormal data. Hawkins (Technometrics 23:105–110, 1981) proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when n _i are small. This paper proposes a modification of this test for complete data to improve its performance, and extends its application to test of homoscedasticity and MCAR when data are multivariate normal and incomplete. Moreover, it is shown that the statistic used in the Hawkins test in conjunction with a nonparametric k-sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and nonnormal data. It is explained how a combination of the proposed normal-theory Hawkins test and the nonparametric test can be employed to test for homoscedasticity, MCAR, and multivariate normality. Simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of Type I error rejection rates. Also, a power study of the proposed tests indicates good power. The proposed methods use appropriate missing data imputations to impute missing data. Methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods. Examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups.     

When a good fit can be bad

How should we select among computational models of cognition? Although it is commonplace to measure how well each model fits the data, this is insufficient. Good fits can be misleading because they can result from properties of the model that have nothing to do with it being a close approximation to the cognitive process of interest (e.g. overfitting). Selection methods are introduced that factor in these properties when measuring fit. Their success in outperforming standard goodness-of-fit measures stems from a focus on measuring the generalizability of a model's data-fitting abilities, which should be the goal of model selection.     

Analyzing experimental data using the Rasch model

We present a method for studying experimental data based on a psychometric model, the “Rasch model” (Rasch, 1966; Thissen & Steinberg, 1986). We illustrate the method with the use of a data set in the field of concept research. More specifically, we investigate whether a conjunctive concept can be seen as an additive combination of its constituents. High correlations between model and data are obtained, but a formal goodness-of-fit test indicates that the model does not completely account for the data. We then alter the Rasch model in such a way as to capture our idea of why the model deviates from the data. This results in higher correlations and a strong increase in goodness-of-fit. It is concluded that our ideas, as incorporated in the model, adequately summarize the data. More generally, this research illustrates that applying the Rasch model and altering it according to one’s hypotheses is an excellent way to analyze experimental data.     

The Invisible Foole

I review the classic skeptical challenges of Foole in Leviathan and the Lydian Shepherd in Republic against the prudential rationality of justice. Attempts to meet these challenges contribute to the reconciliation project (Kavka in Hobbesian moral and political theory, 1986) that tries to establish that morality is compatible with rational prudence. I present a new Invisible Foole challenge against the prudential rationality of justice. Like the Lydian Shepherd, the Invisible Foole can violate justice offensively (Kavka, Hobbesian moral and political theory, 1986; Law and Philosophy, 14:5–34, 1995) without harming his reputation for justice. And like the Foole, the Invisible Foole dismisses the possibility that being just preserves goods intrinsic to justice, and will be just only if he fears that others will punish his injustice by withholding the external goods like labor and material goods that he would otherwise receive for their performance in covenants. I argue that given a plausible folk-theorem interpretation, Hobbes’ response to the Foole’s challenge is inconclusive, and depends crucially upon common knowledge assumptions that may or may not obtain in actual societies. I present two analogous folk-theorem arguments in response to the Invisible Foole’s challenge, one using the idea that the Invisible Foole’s power of concealment might be transitory, and the other using the idea that members of society might stop performing in covenants with anyone, thus punishing the Invisible Foole indirectly, if the Invisible Foole commits sufficiently many injustices.     

Comparing the Argumentum Model of Topics to Other Contemporary Approaches to Argument Schemes: The Procedural and Material Components

This paper focuses on the inferential configuration of arguments, generally referred to as argument scheme. After outlining our approach, denominated Argumentum Model of Topics (AMT, see Rigotti and Greco Morasso 2006, 2009; Rigotti 2006, 2008, 2009), we compare it to other modern and contemporary approaches, to eventually illustrate some advantages offered by it. In spite of the evident connection with the tradition of topics, emerging also from AMT’s denomination, its involvement in the contemporary dialogue on argument schemes should not be overlooked. The model builds in particular on the theoretical and methodological perspective of pragma-dialectics in its extended version, reconciling dialectic and rhetoric; nevertheless, it also takes into account numerous other contributions to the study of argument schemes. Aiming at a representation of argument schemes able to monitor the inferential cohesion and completeness of arguments, AMT focuses on two components of argument scheme that could be distinguished, readapting pragma-dialectical terms, as procedural and material respectively. The procedural component is based on the semantic-ontological structure, which generates the inferential connection from which the logical form of the argument is derived. The material component integrates into the argument scheme the implicit and explicit premises bound to the contextual common ground (Rigotti 2006). In this paper, the comparison of the AMT to other approaches focuses on the inferential configuration of arguments and not on the typologies of argument schemes and on the principles they are based on, which the authors intend to tackle in a further paper.