Fungible parameter estimates in structural equation modeling.

Extending work by Waller (2008) on fungible regression coefficients, we propose a method for computation of fungible parameter estimates in structural equation modeling. Such estimates are defined as distinct alternative solutions for parameter estimates, where all fungible solutions yield identical model fit that is only slightly worse than the fit provided by optimal estimates. When such alternative estimates are found to be highly discrepant from optimal estimates, then substantive interpretation based on optimal estimates is called into question. We present a computational method and 3 illustrations showing the potential impact of this approach in applied research, and we discuss implications and issues for further research. (PsycINFO Database Record

Advances in Modeling Model Discrepancy: Comment on Wu and Browne (2015).

Wu and Browne (Psychometrika, 79, 2015) have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent. Their contribution is related to ongoing developments in the field of Uncertainty Quantification (UQ) on modeling and quantifying effects of model discrepancy. We provide an overview of basic principles of UQ and some relevant developments and we examine the Wu-Browne work in that context. We view the Wu-Browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research.

Hopes and cautions in implementing Bayesian structural equation modeling.

Muthén and Asparouhov (2012) have proposed and demonstrated an approach to model specification and estimation in structural equation modeling (SEM) using Bayesian methods. Their contribution builds on previous work in this area by (a) focusing on the translation of conventional SEM models into a Bayesian framework wherein parameters fixed at zero in a conventional model can be respecified using small-variance priors and (b) implementing their approach in software that is widely accessible. We recognize potential benefits for applied researchers as discussed by Muthén and Asparouhov, and we also see a tradeoff in that effective use of the proposed approach introduces increased demands in terms of expertise of users to navigate new complexities in model specification, parameter estimation, and evaluation of results. We also raise cautions regarding the issues of model modification and model fit. Although we see significant potential value in the use of Bayesian SEM, we also believe that effective use will require an awareness of these complexities.

Sensitivity Analysis in Structural Equation Models: Cases and Their Influence.

The detection of outliers and influential observations is routine practice in linear regression. Despite ongoing extensions and development of case diagnostics in structural equation models (SEM), their application has received limited attention and understanding in practice. The use of case diagnostics informs analysts of the uncertainty of model estimates under different subsets of the data and highlights unusual and important characteristics of certain cases. We present several measures of case influence applicable in SEM and illustrate their implementation, presentation, and interpretation with two empirical examples: (a) a common factor model on verbal and visual ability ( Holzinger & Swineford, 1939 ) and (b) a general structural equation model assessing the effect of industrialization on democracy in a mediating model using country-level data ( Bollen, 1989 ; Bollen & Arminger, 1991 ). Throughout these examples, three issues are emphasized. First, cases may impact different aspects of results as identified by different measures of influence. Second, the important distinction between outliers and influential cases is highlighted. Third, the concept of good and bad cases is introduced-these are influential cases that improve/worsen overall model fit based on their presence in the sample. We conclude with a discussion on the utility of detecting influential cases in SEM and present recommendations for the use of measures of case influence.

Variability in Parameter Estimates and Model Fit Across Repeated Allocations of Items to Parcels.

Different random or purposive allocations of items to parcels within a single sample are thought not to alter structural parameter estimates as long as items are unidimensional and congeneric. If, additionally, numbers of items per parcel and parcels per factor are held fixed across allocations, different allocations of items to parcels within a single sample are thought not to meaningfully alter model fit-at least when items are normally distributed. We show analytically that, although these statements hold in the population, they do not necessarily hold in the sample. We show via a simulation that, even under these conservative conditions, the magnitude of within-sample item-to-parcel-allocation variability in structural parameter estimates and model fit can alter substantive conclusions when sampling error is high (e.g., low N, low item communalities, few items per few parcels). We supply a software tool that facilitates reporting and ameliorating the consequences of item-to-parcel-allocation variability. The tool's utility is demonstrated on an empirical example involving the Neuroticism-Extroversion-Openness (NEO) Personality Inventory and the Computer Assisted Panel Study data set.

Birth order has no effect on intelligence: a reply and extension of previous findings.

We address points raised by Zajonc and Sulloway, who reject findings showing that birth order has no effect on intelligence. Many objections to findings of null birth-order results seem to stem from a misunderstanding of the difference between study designs where birth order is confounded with true causal influences on intelligence across families and designs that control for some of these influences. We discuss some of the consequences of not appreciating the nature of this difference. When between-family confounds are controlled using appropriate study designs and techniques such as multilevel modeling, birth order is shown not to influence intelligence. We conclude with an empirical investigation of the replicability and generalizability of this approach.

Testing differences between nested covariance structure models: Power analysis and null hypotheses.

For comparing nested covariance structure models, the standard procedure is the likelihood ratio test of the difference in fit, where the null hypothesis is that the models fit identically in the population. A procedure for determining statistical power of this test is presented where effect size is based on a specified difference in overall fit of the models. A modification of the standard null hypothesis of zero difference in fit is proposed allowing for testing an interval hypothesis that the difference in fit between models is small, rather than zero. These developments are combined yielding a procedure for estimating power of a test of a null hypothesis of small difference in fit versus an alternative hypothesis of larger difference.

A multilevel approach to the relationship between birth order and intelligence.

Many studies show relationships between birth order and intelligence but use cross-sectional designs or manifest other threats to internal validity. Multilevel analyses with a control variable show that when these threats are removed, two major results emerge: (a) birth order has no significant influence on children's intelligence and (b) earlier reported birth order effects on intelligence are attributable to factors that vary between, not within, families. Analyses on 7- to 8 - and 13- to 14-year-old children from the National Longitudinal Survey of Youth support these conclusions. When hierarchical data structures, age variance of children, and within-family versus between-family variance sources are taken into account, previous research is seen in a new light.

Use of the extreme groups approach: a critical reexamination and new recommendations.

Analysis of continuous variables sometimes proceeds by selecting individuals on the basis of extreme scores of a sample distribution and submitting only those extreme scores to further analysis. This sampling method is known as the extreme groups approach (EGA). EGA is often used to achieve greater statistical power in subsequent hypothesis tests. However, there are several largely unrecognized costs associated with EGA that must be considered. The authors illustrate the effects EGA can have on power, standardized effect size, reliability, model specification, and the interpretability of results. Finally, the authors discuss alternative procedures, as well as possible legitimate uses of EGA. The authors urge researchers, editors, reviewers, and consumers to carefully assess the extent to which EGA is an appropriate tool in their own research and in that of others.

Using Parcels to Convert Path Analysis Models Into Latent Variable Models.

The biasing effects of measurement error in path analysis models can be overcome by the use of latent variable models. In cases where path analysis is used in practice, it is often possible to use parcels as indicators of a latent variable. The purpose of the current study was to compare latent variable models in which parcels were used as indicators of the latent variables, path analysis models of the aggregated variables, and models in which reliability estimates were used to correct for measurement error in path analysis models. Results showed that point estimates of path coefficients were smallest for the path analysis models and largest for the latent variable models. It is concluded that, whenever possible, it is better to use a latent variable model in which parcels are used as indicators than a path analysis model using total scale scores.

Chronic stress and age-related increases in the proinflammatory cytokine IL-6.

Overproduction of IL-6, a proinflammatory cytokine, is associated with a spectrum of age-related conditions including cardiovascular disease, osteoporosis, arthritis, type 2 diabetes, certain cancers, periodontal disease, frailty, and functional decline. To describe the pattern of change in IL-6 over 6 years among older adults undergoing a chronic stressor, this longitudinal community study assessed the relationship between chronic stress and IL-6 production in 119 men and women who were caregiving for a spouse with dementia and 106 noncaregivers, with a mean age at study entry of 70.58 (SD = 8.03) for the full sample. On entry into this portion of the longitudinal study, 28 of the caregivers' spouses had already died, and an additional 50 of the 119 spouses died during the 6 years of this study. Levels of IL-6 and health behaviors associated with IL-6 were measured across 6 years. Caregivers' average rate of increase in IL-6 was about four times as large as that of noncaregivers. Moreover, the mean annual changes in IL-6 among former caregivers did not differ from that of current caregivers even several years after the death of the impaired spouse. There were no systematic group differences in chronic health problems, medications, or health-relevant behaviors that might have accounted for caregivers' steeper IL-6 slope. These data provide evidence of a key mechanism through which chronic stressors may accelerate risk of a host of age-related diseases by prematurely aging the immune response.

Recovery of Weak Common Factors by Maximum Likelihood and Ordinary Least Squares Estimation.

This article examines the relative performance of two commonly used methods of parameter estimation in factor analysis, maximum likelihood (ML) and ordinary least squares (OLS). It is shown that ML will sometimes fail to recover a known population factor structure when OLS succeeds. A simulation study was conducted in which two types of error (model and sampling error) were introduced separately and in combination into correlation matrices generated from known population structures with at least one relatively weak major domain factor. Simulated correlation matrices were factor analyzed using both ML and OLS, and recovery of the relatively weak factor(s) was assessed. In situations with a moderate amount of error, ML often failed to recover the weak factor while OLS succeeded. It is suggested that the correspondence between the assumptions inherent in each method regarding error and the actual nature of error in the data may affect the success of recovery of weak common factors. An example using empirical data is also presented.

2001 Presidential Address: Working with Imperfect Models.

Since the early years of psychological research, investigators in psychology have made use of mathematical models of psychological phenomena. Models are now routinely used to represent and study cognitive processes, the structure of psychological measurements, the structure of correlational relationships among variables, the nature of change over time, and many other topics and phenomena of interest. All of these models, in their attempt to provide a parsimonious representation of psychological phenomena, are wrong to some degree and are thus implausible if taken literally. Such models simply cannot fully represent the complexities of the phenomena of interest and at best provide an approximation of the real world. This imperfection has implications for how we specify, estimate, and evaluate models, and how we interpret results of fitting models to data. Using factor analysis and structural equation models as a context, I examine some implications of model imperfection for our use of models, focusing on formal specification of models; the nature of parameters and parameter estimates; the relevance of discrepancy functions; the issue of sample size; the evaluation, development, and selection of models; and the conduct of simulation studies. The overall perspective is that our use and study of models should be guided by an understanding that our models are imperfect and cannot be made to be exactly correct.

Stress-related modulation of matrix metalloproteinase expression.

Matrix metalloproteinases (MMPs) and the tissue inhibitors of metalloproteinases (TIMPs), whose expression can be controlled by cytokines, play a role in extracellular matrix remodeling in physiological and pathological processes. Using a blister chamber wound model on UV-B-exposed human forearm skin, we examined whether stress or mood-associated neuroendocrine alteration is sufficient to modulate MMP and TIMP expression. We did not find evidence that depressive symptoms were reliably associated with modulation of either MMP or TIMP expression. However, we did find that activation of the hypothalamic-pituitary-adrenal (HPA) and sympathetic-adrenal medullary (SAM) axes can modulate levels of MMPs. A positive association between plasma norepinephrine levels and MMP-2 protein levels, and a negative correlation between plasma cortisol levels and MMP-2 levels were found. The data suggest that activation of the HPA and SAM axes, even in individuals within the normal range of depressive symptoms, could mediate MMP levels and wound healing in blister wounds.

Comments on the Meehl-Waller (2002) procedure for appraisal of path analysis models.

P. E. Meehl and N. G. Waller (2002) proposed an innovative method for assessing path analysis models wherein they subjected a given model, along with a set of alternatives, to risky tests using selected elements of a sample correlation matrix. Although the authors find much common ground with the perspective underlying the Meehl-Waller approach, they suggest that there are aspects of the proposed procedure that require close examination and further development. These include the selection of only one subset of correlations to estimate parameters when multiple solutions are generally available, the fact that the risky tests may test only a subset of parameters rather than the full model of interest, and the potential for different results to be obtained from analysis of equivalent models.

Exploratory factor analysis in behavior genetics research: factor recovery with small sample sizes.

Results of a Monte Carlo study of exploratory factor analysis demonstrate that in studies characterized by low sample sizes the population factor structure can be adequately recovered if communalities are high, model error is low, and few factors are retained. These are conditions likely to be encountered in behavior genetics research involving mean scores obtained from sets of inbred strains. Such studies are often characterized by a large number of measured variables relative to the number of strains used, highly reliable data, and high levels of communality. This combination of characteristics has special consequences for conducting factor analysis and interpreting results. Given that limitations on sample size are often unavoidable, it is recommended that researchers limit the number of expected factors as much as possible.

On the practice of dichotomization of quantitative variables.

The authors examine the practice of dichotomization of quantitative measures, wherein relationships among variables are examined after 1 or more variables have been converted to dichotomous variables by splitting the sample at some point on the scale(s) of measurement. A common form of dichotomization is the median split, where the independent variable is split at the median to form high and low groups, which are then compared with respect to their means on the dependent variable. The consequences of dichotomization for measurement and statistical analyses are illustrated and discussed. The use of dichotomization in practice is described, and justifications that are offered for such usage are examined. The authors present the case that dichotomization is rarely defensible and often will yield misleading results.

Pathological CNS autoimmune disease triggered by traumatic spinal cord injury: implications for autoimmune vaccine therapy.

Lymphocytes respond to myelin proteins after spinal cord injury (SCI) and may contribute to post-traumatic secondary degeneration. However, there is increasing evidence that autoreactive T-lymphocytes may also convey neuroprotection and promote functional recovery after CNS injury. To clarify the role of myelin autoreactive lymphocytes after SCI, we performed contusion injuries in the thoracic spinal cord of transgenic (Tg) mice in which >95% of all CD4+ T-lymphocytes are reactive with myelin basic protein (MBP). We observed significantly impaired recovery of locomotor and reflex function in Tg mice compared with non-Tg (nTg) littermates. Measures of functional impairment in Tg mice correlated with significantly less white matter at the injury site, and morphometric comparisons of injured Tg and nTg spinal cords revealed increased rostrocaudal lesion expansion (i.e., secondary degeneration) in Tg mice. Rostrocaudal to the impact site in SCI-nTg mice, demyelination was restricted to the dorsal funiculus, i.e., axons undergoing Wallerian degeneration. The remaining white matter appeared normal. In contrast, lymphocytes were colocalized with regions of demyelination and axon loss throughout the white matter of SCI-Tg mice. Impaired neurological function and exacerbated neuropathology in SCI-Tg mice were associated with increased intraspinal production of proinflammatory cytokine mRNA; neurotrophin mRNA was not elevated. These data suggest that endogenous MBP-reactive lymphocytes, activated by traumatic SCI, can contribute to tissue injury and impair functional recovery. Any neuroprotection afforded by myelin-reactive T-cells is likely to be an indirect effect mediated by other non-CNS-reactive lymphocytes. Similar to the Tg mice in this study, a subset of humans that are genetically predisposed to autoimmune diseases of the CNS may be adversely affected by vaccine therapies designed to boost autoreactive lymphocyte responses after CNS trauma. Consequently, the safe implementation of such therapies requires that future studies define the mechanisms that control T-cell function within the injured CNS.

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.