On structural equation modeling with data that are not missing completely at random

A general latent variable model is given which includes the specification of a missing data mechanism. This framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data. Here, missing completely at random is not a prerequisite for unbiased estimation in large samples, as when using the traditional listwise or pairwise present data approaches. The theory is connected with old and new results in the area of selection and factorial invariance. It is pointed out that in many applications, maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software, such as LISREL and LISCOMP. Several sets of artifical data are generated within the general model framework. The proposed estimator is compared to the two traditional ones and found superior.The research of the first author was supported by grant No. SES-8312583 from the National Science Foundation and by a Spencer Foundation grant. We wish to thank Chuen-Rong Chan for drawing the path diagram.     

Methods for Mediation Analysis with Missing Data

Despite wide applications of both mediation models and missing data techniques, formal discussion of mediation analysis with missing data is still rare. We introduce and compare four approaches to dealing with missing data in mediation analysis including listwise deletion, pairwise deletion, multiple imputation (MI), and a two-stage maximum likelihood (TS-ML) method. An R package bmem is developed to implement the four methods for mediation analysis with missing data in the structural equation modeling framework, and two real examples are used to illustrate the application of the four methods. The four methods are evaluated and compared under MCAR, MAR, and MNAR missing data mechanisms through simulation studies. Both MI and TS-ML perform well for MCAR and MAR data regardless of the inclusion of auxiliary variables and for AV-MNAR data with auxiliary variables. Although listwise deletion and pairwise deletion have low power and large parameter estimation bias in many studied conditions, they may provide useful information for exploring missing mechanisms.     

MISSING DATA: A CONCEPTUAL REVIEW FOR APPLIED PSYCHOLOGISTS

There has been conspicuously little research concerning missing data problems in the applied psychology literature. Fortunately, other fields have begun to investigate this issue. These include survey research, marketing, statistics, economics, and biometrics. A review of this literature suggests several trends for applied psychologists. For example, listwise deletion of data is often the least accurate technique to deal with missing data. Other methods for estimating missing data scores may be more accurate and preserve more data for investigators to analyze. Further, the literature reveals that the amount of missing data and the reasons for deletion of data impact how investigators should handle the problem. Finally, there is a great need for more investigation of strategies for dealing with missing data, especially when data are missing in nonrandom or systematic patterns.     

Analysis of Interactions and Nonlinear Effects with Missing Data: A Factored Regression Modeling Approach Using Maximum Likelihood Estimation

Abstract

When estimating multiple regression models with incomplete predictor variables, it is necessary to specify a joint distribution for the predictor variables. A convenient assumption is that this distribution is a multivariate normal distribution, which is also the default in many statistical software packages. This distribution will in general be misspecified if predictors with missing data have nonlinear effects (e.g., x²) or are included in interaction terms (e.g., x·z). In the present article, we introduce a factored regression modeling approach for estimating regression models with missing data that is based on maximum likelihood estimation. In this approach, the model likelihood is factorized into a part that is due to the model of interest and a part that is due to the model for the incomplete predictors. In three simulation studies, we showed that the factored regression modeling approach produced valid estimates of interaction and nonlinear effects in regression models with missing values on categorical or continuous predictor variables under a broad range of conditions. We developed the R package mdmb, which facilitates a user-friendly application of the factored regression modeling approach, and present a real-data example that illustrates the flexibility of the software.     

Estimating Incremental Validity Under Missing Data

A common form of missing data is caused by selection on an observed variable (e.g., Z). If the selection variable was measured and is available, the data are regarded as missing at random (MAR). Selection biases correlation, reliability, and effect size estimates when these estimates are computed on listwise deleted (LD) data sets. On the other hand, maximum likelihood (ML) estimates are generally unbiased and outperform LD in most situations, at least when the data are MAR. The exception is when we estimate the partial correlation. In this situation, LD estimates are unbiased when the cause of missingness is partialled out. In other words, there is no advantage of ML estimates over LD estimates in this situation. We demonstrate that under a MAR condition, even ML estimates may become biased, depending on how partial correlations are computed. Finally, we conclude with recommendations about how future researchers might estimate partial correlations even when the cause of missingness is unknown and, perhaps, unknowable.     

Gender Differences in Attitudes Toward Affirmative Action Programs in Australia: Effects of Beliefs,Interests, and Attitudes Toward Women

This research identified mediators explaining the relationship between gender and attitudes toward affirmative action programs for women. Structural equation modeling was conducted on survey data obtained from academics at an Australian university (n = 198 with listwise deletion of missing data). Findings indicated that the relationship between gender and affirmative action attitudes was mediated by (1) perceptions of affirmative action's impact on material self-interest, (2) belief in the existence of gender discrimination, and (3) traditional attitudes toward women. Implications for organizations, policymakers, and researchers are discussed.     

Encountering Dissimilar Views in Deliberation: Political Knowledge,Attitude Strength,and Opinion Change

Conversing with diverse points of view stands as the central tenet of deliberative democracy, yet empirical evidence has suggested mixed outcomes related to perspective change as a result of deliberative encounters. I propose a difference‐driven model that suggests individual predispositions moderate the processing of dissimilar views when changing policy preferences. My analysis is based on a random sample of over 400 voters at a California‐wide deliberative event, where participants discussed proposals for reforming the state politics. I find that encountering more and different arguments transforms policy attitudes. Yet it is more difficult for people to change their minds on issues about which they hold strong beliefs. Some evidence suggests that different psychologies are at play for people who enter deliberation with substantial or weak political knowledge and for those who deliberate while holding strong or moderate prior opinions. Well‐grounded strong opinions are resistant to change, while well‐grounded moderate opinions are persuadable in deliberative groups. Uninformed positions can become entrenched in like‐minded groups, yet they can be adjusted once participants deliberate with dissimilar views, especially opinions that are held strongly without good informational ground. The findings urge deliberative forums that introduce participants to diverse perspectives to foster a considered public opinion.     

Compensatory Effects of Cognitive Characteristics of the Source,the Message,and the Receiver Upon Attitude Change

This study explored the interactive effects of four cognitive variables (perceived expertise of the source, recipients' initial attitudes, number of arguments, and message sidedness) on attitude change. A 2 × 2 × 2 factorial design was used (N = 236 Canadian students): Results showed a positive and significant main effect of initial attitude on attitude change and three significant compensatory effects of independent variables on attitude change: (a) two-sided messages were more persuasive with fewer arguments; conversely for one-sided messages; (b) high expertise compensated for low number of arguments and conversely; and (c) higher expertise was more persuasive in the case of unfavorable recipients and conversely. Results also showed that when the message was one-sided and the number of arguments was large, low expertise was more persuasive than high expertise on initially opposed recipients, which confirms the cognitive response.     

Prototypicality of arguments and conformity to ingroup norms

Social categorization is claimed to elicit a tendency to conform to ingroup norms, which may result in attitude change after exposure to information on the opinions of other ingroup members. It was hypothesized that the degree to which arguments represented ingroup norms, i.e., were prototypical, would affect their potential influence on attitudes, such that prototypical arguments would be perceived as being of higher quality and would elicit more attitude change. Moreover, prototypical arguments were expected to elicit more argument elaboration. Two experiments were designed to test these predictions. In Experiment 1 subjects were exposed to both a set of pro and a set of contra arguments, while one of the sets was allegedly prototypical of ingroup attitudes. In Experiment 2 subjects were exposed to either prototypical or a-prototypical pro or contra arguments allegedly originating from in- or outgroup. In both studies conformity to ingroup norms was observed. In addition, prototypical ingroup arguments elicited higher quality ratings in the first study. Indications of higher elaboration of prototypical ingroup arguments were found.     

Persuasion by a Single Route: A View From the Unimodel

Major current notions of persuasion depict it as attainable via 2 qualitatively distinct routes: (a) a central or a systematic route in which opinions and attitudes are based on carefully processed arguments in the persuasive message and (b) a peripheral or heuristic route in which they are based on briefly considered heuristics or cues, exogenous to the message. This article offers a single-route reconceptualization that treats these dual routes to persuasion as involving functionally equivalent types of evidence from which persuasive conclusions may be drawn. Previous findings in the dual-process literature are reconsidered in light of this "unimodel," and novel data are presented consistent with its assumptions. Beyond its parsimony and integrative potential, the unimodel offers conceptual, empirical, and practical advantages in the persuasion domain.     

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/.     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated efficiently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.     

Modelling acceptability of the intelligent speed adapter

The Intelligent Speed Adapter (ISA) is an example of in-car telematics equipment that is developed to reduce speeding. It can be introduced as a policy instrument to improve speed limit compliance and with that traffic safety. However, previous research indicated that acceptability of this type of instruments is rather limited. Adopting a policy that may increase acceptability before introduction of the instrument is therefore recommended. To formulate such a policy, understanding the factors that influence ISA acceptability is required. To that effect, a causal model that disentangles the direct and indirect effects of predictor variables on ISA acceptability is developed and estimated. The model includes car drivers’ attitudes, opinions and beliefs on the speeding problem, the policy goal and the policy instrument ISA, as well as speed related behaviour and socio-demographic variables. This paper presents the direct and total effects of these variables on ISA acceptability. The results suggest that explaining clearly how ISA can contribute to attaining various personal and societal goals may be a viable policy to increase ISA acceptability.     

Effects of Scope of Justice,Informant Ethnicity,and Information Frame on Attitudes towards Ethnicity-based Selection

The study ascertained whether attitudes towards ethnicity-based employment selection would be affected by three variables: scope of justice, information frame, and informant ethnicity. Using a scenario approach, results showed that (1) European New Zealanders with greater scope of justice regarding Maori people were less opposed to an employment practice that involves granting preferential treatment to Maori candidates. (2) The frame of information about ethnicity-based selection (i.e. preferential selection vs. diversity-based selection) and informant ethnicity interacted with scope of justice in exerting a significant influence on respondents' views on ethnicity-based selection. Furthermore, individuals' reactions to moral philosophers' justifications for merit or ethnicity-based selection determined their overall attitudes towards each selection practice.     

Handling Missing Covariates in Conditional Mixture Models Under Missing at Random Assumptions

Mixture modeling is a popular method that accounts for unobserved population heterogeneity using multiple latent classes that differ in response patterns. Psychologists use conditional mixture models to incorporate covariates into between-class and/or within-class regressions. Although psychologists often have missing covariate data, conditional mixtures are currently fit with a conditional likelihood, treating covariates as fixed and fully observed. Under this exogenous-x approach, missing covariates are handled primarily via listwise deletion. This sacrifices efficiency and does not allow missingness to depend on observed outcomes. Here we describe a modified joint likelihood approach that (a) allows inference about parameters of the exogenous-x conditional mixture even with nonnormal covariates, unlike a conventional multivariate mixture; (b) retains all cases under missing at random assumptions; (c) yields lower bias and higher efficiency than the exogenous-x approach under a variety of conditions with missing covariates; and (d) is straightforward to implement in available commercial software. The proposed approach is illustrated with an empirical analysis predicting membership in latent classes of conduct problems. Recommendations for practice are discussed.     

An experimental test of the influence of selection procedures on fairness perceptions,attitudes about the organization,and job pursuit intentions

College juniors and seniors (N=184) reviewed a 12-page college recruiting brochure containing information about a large, high-technology corporation. Six versions of the brochure created a 2×3 experimental design in which the compensation and selection practices of the company were manipulated. Compensation and benefit packages were described as being either above or comparable to the industry average. The selection procedure was described as either a biodata inventory, an abstract cognitive test, or an in-basket simulation. We hypothesized that selection procedures would have an indirect, rather than direct, effect on attitudes about the organization and job pursuit intentions. LISREL analyses found good fits for the measurement model (RNI=.97) and the structural model (RNI=.95). All hypothesized paths were significant (p<.05) except for the path between compensation and attitudes about the organization. It is suggested that different selection procedures with comparable levels of validity may have different effects on candidates' attitudes about employers.     

Interdependent sampling and social influence

Most explanations of social influence focus on why individuals might want to agree with the opinions or attitudes of others. The authors propose a different explanation that assumes the attitudes of others influence only the activities and objects individuals are exposed to. For example, individuals are likely to be exposed to activities that their friends enjoy. The authors demonstrate that such influence over sampling behavior is sufficient to produce a social influence effect when individuals form attitudes by learning from experience. Even if the experiences of 2 individuals, when they sample an object or event, are independent random variables, their attitudes will become positively correlated if their sampling processes are interdependent. Interdependent sampling of activities thus provides a different explanation of social influence with distinct empirical and theoretical implications.     

Attitudes toward immigrants: The interactive role of the authoritarian predisposition,social norms,and humanitarian values

This investigation tested whether social norms and endorsement of humanitarian values interact to influence authoritarians' attitudes toward immigrants. Oyamot, Borgida, and Fisher (2006) found correlational evidence for a model in which: (1) clear social norms for attitudes toward an outgroup (favorable or unfavorable) influence the authoritarianism–attitude relationship in the direction of the norm, and (2) in the absence of clear social norms, endorsement of humanitarian–egalitarian values attenuate the intolerant tendencies of authoritarians. The current investigation tested the model in a survey experiment conducted in a diverse adult sample (N = 388). We measured participants' levels of authoritarian predisposition and endorsement of humanitarian values. Participants were then randomly told that Americans in general had either negative, positive, or mixed opinions about immigrants and immigration (social norm condition), and then asked about their attitude toward immigrants. Consistent with the model, authoritarianism was negatively related to attitudes toward immigrants in the negative norm condition. However, authoritarians' tendency toward intolerance was attenuated when they thought that Americans in general had positive opinions about immigrants. Also as predicted, when societal norms were depicted as mixed, authoritarians' attitudes depended upon endorsement of humanitarian values: humanitarian authoritarians held positive attitudes and non-humanitarian authoritarians held the most negative attitudes toward immigrants. Implications for understanding the effects of authoritarian predispositions in varying social contexts are discussed.     

Agency and responsibility: The personal and the political

In this paper, I review arguments according to which harsh criminal punishments and poverty are undeserved and therefore unjust. Such arguments come in different forms. First, one may argue that no one deserves to be poor or be punished, because there is no such thing as desert-entailing moral responsibility. Second, one may argue that poor people in particular do not deserve to remain in poverty or to be punished if they commit crimes, because poor people suffer from psychological problems that undermine their agency and moral responsibility. Third, one may argue that poor and otherwise marginalized people frequently face external obstacles that prevent them from taking alternative courses of action. The first kind of argument has its place in the philosophy seminar. Psychological difficulties may be important to attend to both in personal relationships and when holding ourselves responsible. Nevertheless, I argue that neither type of argument belongs in political contexts. Moral responsibility scepticism ultimately rests on contested intuitions. Labelling certain groups of people particularly irrational, weak-willed, or similar is belittling and disrespectful; such claims are also hard to prove, and may have the opposite effect to the intended one on people's attitudes. Arguments from external obstacles have none of these problems. Such arguments may not take us all the way to criminal justice reform, but in this context, we can supplement them with epistemic arguments and crime prevention arguments.     

Analysis of structural equation model with ignorable missing continuous and polytomous data

The main purpose of this article is to develop a Bayesian approach for structural equation models with ignorable missing continuous and polytomous data. Joint Bayesian estimates of thresholds, structural parameters and latent factor scores are obtained simultaneously. The idea of data augmentation is used to solve the computational difficulties involved. In the posterior analysis, in addition to the real missing data, latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data. An algorithm that embeds the Metropolis-Hastings algorithm within the Gibbs sampler is implemented to produce the Bayesian estimates. A goodness-of-fit statistic for testing the posited model is presented. It is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small. Computational efficiency of the proposed procedure is illustrated by simulation studies and a real example.The work described in this paper was fully supported by a grant from the Research Grants Council of the HKSAR (Project No. CUHK 4088/99H). The authors are greatly indebted to the Editor and anonymous reviewers for valuable comments in improving the paper; and also to D. E. Morisky and J.A. Stein for the use of their AIDS data set.