When Good Loadings Go Bad: Robustness in Factor Analysis

Structural misspecifications in factor analysis include using the wrong number of factors and omitting cross loadings or correlated errors. The impact of these errors on factor loading estimates is understudied. Factor loadings underlie our assessments of the validity and reliability of indicators. Thus knowing how structural misspecifications affect a factor loading is a key issue. This paper develops analytic conditions of when misspecifications affect Bollen's (1996) model implied instrumental variable, two stage least squares (MIIV-2SLS) estimator of a factor loading. It shows that if an indicator equation is correctly specified, then correlated errors among other measures, mixing up causal indicators with reflective, omitting cross loadings, and omitting direct effects between indicators leave the MIIV-2SLS estimator of the factor loading unchanged. Alternatively, if the indicator or the scaling indicator equation is misspecified, then the loading is unlikely to be robust. The results are illustrated with hypothetical and empirical examples

Model Identification and Computer Algebra

Multiequation models that contain observed or latent variables are common in the social sciences. To determine whether unique parameter values exist for such models, one needs to assess model identification. In practice analysts rely on empirical checks that evaluate the singularity of the information matrix evaluated at sample estimates of parameters. The discrepancy between estimates and population values, the limitations of numerical assessments of ranks, and the difference between local and global identification make this practice less than perfect. In this paper we outline how to use computer algebra systems (CAS) to determine the local and global identification of multiequation models with or without latent variables. We demonstrate a symbolic CAS approach to local identification and develop a CAS approach to obtain explicit algebraic solutions for each of the model parameters. We illustrate the procedures with several examples, including a new proof of the identification of a model for handling missing data using auxiliary variables. We present an identification procedure for Structural Equation Models that makes use of CAS and that is a useful complement to current methods

A Note on Algebraic Solutions to Identification

Algebraic methods to establish the identification of structural equation models remains a viable option. However, sometimes it is unclear whether the algebraic solution establishes identification. One example is when there is more than one way to solve for the parameter, but one way leads to a single value and a second way leads to a function with more than one value. This note proves that one explicit and unique solution is sufficient for model identification even when other explicit solutions permit more than one solution. The results are illustrated with an example. The results are useful to attempts to use algebraic means to address model identification

Three Cs in measurement models: Causal indicators, composite indicators, and covariates.

In the last two decades attention to causal (and formative) indicators has grown. Accompanying this growth has been the belief that we can classify indicators into two categories, effect (reflective) indicators and causal (formative) indicators. This paper argues that the dichotomous view is too simple. Instead, there are effect indicators and three types of variables on which a latent variable depends: causal indicators, composite (formative) indicators, and covariates (the “three Cs”). Causal indicators have conceptual unity and their effects on latent variables are structural. Covariates are not concept measures, but are variables to control to avoid bias in estimating the relations between measures and latent variable(s). Composite (formative) indicators form exact linear combinations of variables that need not share a concept. Their coefficients are weights rather than structural effects and composites are a matter of convenience. The failure to distinguish the “three Cs” has led to confusion and questions such as: are causal and formative indicators different names for the same indicator type? Should an equation with causal or formative indicators have an error term? Are the coefficients of causal indicators less stable than effect indicators? Distinguishing between causal and composite indicators and covariates goes a long way toward eliminating this confusion. We emphasize the key role that subject matter expertise plays in making these distinctions. We provide new guidelines for working with these variable types, including identification of models, scaling latent variables, parameter estimation, and validity assessment. A running empirical example on self-perceived health illustrates our major points

10. Using Instrumental Variable Tests to Evaluate Model Specification in Latent Variable Structural Equation Models

Structural Equation Modeling with latent variables (SEM) is a powerful tool for social and behavioral scientists, combining many of the strengths of psychometrics and econometrics into a single framework. The most common estimator for SEM is the full-information maximum likelihood estimator (ML), but there is continuing interest in limited information estimators because of their distributional robustness and their greater resistance to structural specification errors. However, the literature discussing model fit for limited information estimators for latent variable models is sparse compared to that for full information estimators. We address this shortcoming by providing several specification tests based on the 2SLS estimator for latent variable structural equation models developed by Bollen (1996). We explain how these tests can be used to not only identify a misspecified model, but to help diagnose the source of misspecification within a model. We present and discuss results from a Monte Carlo experiment designed to evaluate the finite sample properties of these tests. Our findings suggest that the 2SLS tests successfully identify most misspecified models, even those with modest misspecification, and that they provide researchers with information that can help diagnose the source of misspecification

Relationships with God among Young Adults: Validating a Measurement Model with Four Dimensions

Experiencing a relationship with God is widely acknowledged as an important aspect of personal religiosity for both affiliated and unaffiliated young adults, but surprisingly few attempts have been made to develop measures appropriate to its latent, multidimensional quality. This paper presents a new model for measuring relationships with God based on religious role theory, attachment to God theory, and insights from interview-based studies, which allows for a wider array of dimensions than have been considered in prior work: anger, anxiety, intimacy, and consistency. To test our model's internal validity, we use confirmatory factor analysis with nationally representative data. To test its external validity, we (1) use difference-in-means tests across gender, race/ethnicity, geographical region, and religious affiliation; and (2) analyze correlations between our four new dimensions and four other commonly used measures of religiosity, thereby demonstrating both our model's validity and value for future studies of personal religiosity

Evaluating measurement error in readings of blood pressure for adolescents and young adults

Readings of blood pressure are known to be subject to measurement error, but the optimal method for combining multiple readings is unknown. This study assesses different sources of measurement error in blood pressure readings and assesses methods for combining multiple readings using data from a sample of adolescents/young adults who were part of a longitudinal epidemiological study based in Cebu, Philippines. Three sets of blood pressure readings were collected at 2-year intervals for 2127 adolescents and young adults as part of the Cebu National Longitudinal Health and Nutrition Study. Multi-trait, multi-method (MTMM) structural equation models in different groups were used to decompose measurement error in the blood pressure readings into systematic and random components and to examine patterns in the measurement across males and females and over time. The results reveal differences in the measurement properties of blood pressure readings by sex and over time that suggest the combination of multiple readings should be handled separately for these groups at different time points. The results indicate that an average (mean) of the blood pressure readings has high validity relative to a more complicated factor-score-based linear combination of the readings

Are Gestational Age, Birth Weight, and Birth Length Indicators of Favorable Fetal Growth Conditions? A Structural Equation Analysis of Filipino Infants

The fetal origin hypothesis emphasizes the life-long health impacts of prenatal conditions. Birth weight, birth length, and gestational age are indicators of the fetal environment. However, these variables often have missing data and are subject to random and systematic errors caused by delays in measurement, differences in measurement instruments, and human error. With data from the Cebu (Philippines) Longitudinal Health and Nutrition Survey, we use structural equation models (SEMs), to explore random and systematic errors in these birth outcome measures, to analyze how maternal characteristics relate to birth outcomes, and to take account of missing data. We assess whether birth weight, birth length, and gestational age are influenced by a single latent variable that we call Favorable Fetal Growth Conditions (FFGC) and if so, which variable is most closely related to FFGC. We find that a model with FFGC as a latent variable fits as well as a less parsimonious model that has birth weight, birth length, and gestational age as distinct individual variables. We also demonstrate that birth weight is more reliably measured than is gestational age. FFGC were significantly influenced by taller maternal stature, better nutritional stores indexed by maternal arm fat and muscle area during pregnancy, higher birth order, avoidance of smoking and maternal age 20-35 years. Effects of maternal characteristics on newborn weight, length and gestational age were largely indirect, operating through FFCG