Taxonic structure of infant reactivity: evidence from a taxometric perspective

Previously, we proposed a theoretical framework that classified infants into qualitative categories of reactivity, rather than on a continuous dimension. The present research used an objective statistical procedure (maximum covariance analysis, or MAXCOV) to determine if a qualitative latent structure, consistent with our theoretical conjectures, would be found to underlie quantitative indices of reactivity to stimuli in a sample of 599 four-month-old infants. Results of the MAXCOV analysis showed clear evidence of a latent discontinuity underlying the behavioral measures of infant reactivity. The base rate of the latent class (or taxon) was estimated at 10%. Infants within the putative high-reactivity taxon, compared with infants not in the taxon, were elevated on measures of behavioral inhibition at 4.5 years. These results provide objective empirical support for a central tenet in our theoretical model by supporting the taxonicity of infant reactivity.     

Deeper into the schizotypy taxon: on the robust nature of maximum covariance analysis

Prior research has suggested that the latent structure of the schizotypy construct (P. E. Meehl, 1990) may be qualitative in nature and have a low base rate (L. Korfine & M. F. Lenzenweger, 1995; M. F. Lenzenweger & L. Korfine, 1992). These studies relied on the application of maximum covariance analysis (MAXCOV) to 8 true-false format items from a schizotypy measure. The current study sought to examine the robustness of those prior findings through MAXCOV analysis of fully quantitative measures of schizotypy. Measures of perceptual aberration, magical ideation, and referential thinking were analyzed using MAXCOV in a sample of 429 persons. The results of these analyses strongly support a latent taxonic structure for schizotypy and a low base rate for the schizotypy taxon. Furthermore, the members of the putative taxon reveal an increased level of deviance on a psychometric measure known to be associated with schizophrenia liability. The possibility that the dichotomous item format of those items analyzed previously with MAXCOV lead to spurious pseudotaxonicity is greatly diminished in light of these results.     

Malingering as a categorical or dimensional construct: the latent structure of feigned psychopathology as measured by the SIRS and MMPI-2

The 6 nonoverlapping primary scales of the Structured Interview of Reported Symptoms (SIRS) were subjected to taxometric analysis in a group of 1,211 criminal and civil examinees in order to investigate the latent structure of feigned psychopathology. Both taxometric procedures used in this study, mean above minus below a cut (MAMBAC) and maximum covariance (MAXCOV), produced dimensional results. A subgroup of participants (n = 711) with valid Minnesota Multiphasic Personality Inventory-2 (MMPI-2) protocols were included in a second round of analyses in which the 6 nonoverlapping primary scales of the SIRS and the Infrequency (F), Infrequency-Psychopathology (Fp), and Dissimulation (Ds) scales of the MMPI-2 served as indicators. Again, the results were more consistent with dimensional latent structure than with taxonic latent structure. On the basis of these findings, it is concluded that feigned psychopathology forms a dimension (levels of fabrication or exaggeration) rather than a taxon (malingering-honest dichotomy) and that malingering is a quantitative distinction rather than a qualitative one. The theoretical and clinical practice implications of these findings are discussed.     

Taxometric analyses of sexual orientation and gender identity

Taxa are nonarbitrary classes whose existence is an empirical question and not a matter of mere semantic convenience. Taxometric procedures detect whether numerical relations between purported indicators of conjectured taxa bear the hallmarks of true taxa. On the basis of theoretical considerations, the current study tested whether taxa underlie sexual orientation and related measures of gender identity. Two taxometric procedures, maximum covariance, making hits maximum (MAXCOV) and mean above minus below a cut (MAMBAC), were applied to Kinsey Scales and measures of childhood gender nonconformity and adult gender identity in a sample of nearly 5,000 members of the Australian Twin Registry. Results suggest that latent taxa underlie these measures. About 12-15% of men and 5-10% of women belong to latent taxa associated with homosexual preference. These percentages are greater than those of individuals who report homosexual preference, however, and hence it appears that an appreciable proportion of individuals in these taxa have heterosexual preference. An understanding of the origins of these latent taxa may be important to understanding the development of sexual orientation and gender identity.     

In search of the psychopathic sexuality taxon: indicator size does matter

Recent research has suggested that a qualitatively distinct subtype of psychopathic sex offender can be identified via taxometric analyses (Harris et al., 2007). In this study we attempted to replicate the hypothesized psychopathic sexuality taxon in a group of 503 male sexual offenders using data from the Psychopathy Checklist-Revised (PCL-R:Hare, 2003) and five coercive and precocious sexuality items. Ambiguous to dimensional results were obtained when, in a replication of the Harris et al. (2007) study,dichotomized indicators were analyzed with summed input maximum covariance (MAXCOV). Clearly dimensional results, however, were obtained when higher correlating and more valid quasi-continuous indicators were analyzed with traditional (input variables not summed) MAXCOV, and both dichotomous and quasi-continuous indicators were analyzed with mean above minus below a cut (MAMBAC) and latent-mode factor analysis (L-Mode). These results suggest that Harris et al. (2007) may have mistaken the random fluctuations of weakly correlating and poorly differentiating indicators for a taxon. Consistent with the vast majority of earlier research,our results suggest that psychopathy (with or without coercive and precocious sexuality) is a dimensional construct.     

Detecting latent taxa: Monte Carlo comparison of taxometric, mixture model, and clustering procedures

A Monte Carlo evaluation of four procedures for detecting taxonicity was conducted using artificial data sets that were either taxonic or nontaxonic. The data sets were analyzed using two of Meehl's taxometric procedures, MAXCOV and MAMBAC, Ward's method for cluster analysis in concert with the cubic clustering criterion and a latent variable mixture modeling technique. Performance of the taxometric procedures and latent variable mixture modeling were clearly superior to that of cluster analysis in detecting taxonicity. Applied researchers are urged to select from the better procedures and to perform consistency tests.     

Assessing Measurement Invariance of the Children's Depression Inventory in Chinese and Italian Primary School Student Samples

This study tested the measurement invariance of Children's Depression Inventory (CDI) and compared its factorial variance/covariance and latent means among Chinese and Italian children. Multigroup confirmatory factor analysis of the original five factors identified by Kovacs revealed that full measurement invariance did not hold. Further analysis showed that 4 of 21 factor loadings, 14 of 26 intercepts, and 12 of 26 item errors were noninvariant. Factor variance and covariance invariant tests revealed significant differences between Chinese and Italian samples. The latent factor mean comparison suggested no significant difference across the two groups. Nevertheless, the finding of partial metric and scalar invariance suggested that observed mean differences on the CDI items cannot be fully explained by the mean differences in the latent factor. These results suggest that researchers and practitioners exercise caution when gauging the size of the true national population differences in depressive symptoms among Italian and Chinese children when assessed via CDI. In addition to providing needed evidence on the use of the CDI in Italian and Chinese children specifically, the methods used in this research can serve more generally as an example for other cross-cultural assessment research to test structural equivalence and measurement invariance of scales and to determine why it is important to do so.     

The latent structure of anxiety sensitivity--revisited

Anxiety sensitivity has been implicated as a risk factor for the development and maintenance of panic and other anxiety disorders. Although researchers have generally assumed that anxiety sensitivity is a dimensional, rather than categorical, variable, recent taxometric research has raised questions concerning the accuracy of this assumption. The present study examined the latent structure of anxiety sensitivity by applying four taxometric procedures (MAXEIG, MAXCOV, MAMBAC, and L-Mode) to data collected from two large nonclinical samples (n = 1,025 and n = 744) using two distinct measures of anxiety sensitivity (Anxiety Sensitivity Profile and Anxiety Sensitivity Index-Revised). In contrast to previous taxometric analyses of anxiety sensitivity, results of the present research provided convergent evidence for a latent anxiety sensitivity dimension. Several potential explanations for the discrepancy between these findings and those of previous research are discussed, as well as the implications of these findings for the conceptualization and measurement of anxiety sensitivity.     

Abstract: A Comparison of the Factor Mixture Model and Taxometric Procedures

Taxometric procedures and the Factor Mixture Model (FMM) have a complimentary set of strengths and weaknesses. Both approaches purport to detect evidence of a latent class structure. Taxometric procedures, popular in psychiatric and psychopathology literature, make no assumptions beyond those needed to compute means and covariances. However, Taxometric procedures assume that observed items are uncorrelated within a class or taxon. This assumption is violated when there are individual differences in the trait underlying the items (i.e., severity differences within class). FMMs can model within-class covariance structures ranging from local independence to multidimensional within-class factor models and permits the specification of more than two classes. FMMs typically rely on normality assumptions for within-class factors and error terms. FMMs are highly parameterized and susceptible to misspecifications of the within-class covariance structure.

The current study compared the Taxometric procedures MAXEIG and the Base-Rate Classification Technique to the FMM in their respective abilities to (1) correctly detect the two-class structure in simulated data, and to (2) correctly assign subjects to classes. Two class data were simulated under conditions of balanced and imbalanced relative class size, high and low class separation, and 1-factor and 2-factor within-class covariance structures. For the 2-factor data, simple and cross-loaded factor loading structures, and positive and negative factor correlations were considered. For the FMM, both correct and incorrect within-class factor structures were fit to the data.

FMMs generally outperformed Taxometric procedures in terms of both class detection and in assigning subjects to classes. Imbalanced relative class size (e.g., a small minority class and a large majority class) negatively impacted both FMM and Taxometric performance while low class separation was much more problematic for Taxometric procedures than the FMM. Comparisons of alterative FMMs based on information criteria generally resulted in correct model choice but deteriorated when small class separation was combined with imbalanced relative class size.     

Trajectories of Youthful Antisocial Behavior: Categories or Continua?

The purpose of this study was to determine whether qualitatively distinct trajectories of antisocial behavior could be identified in 1,708 children (843 boys, 865 girls) from the 1979 National Longitudinal Survey of Youth–Child Data (NLSY-C). Repeated ratings were made on the Behavior Problems Index (BPI: Peterson and Zill Journal of Marriage and the Family, 48, 295–307, 1986) antisocial scale by the mothers of these children when the children were 6, 8, 10, 12, and 14 years of age. Scores on three indicators constructed from the six BPI Antisocial items (callousness, aggression, noncompliance) were then analyzed longitudinally (by summing across the rating periods) and cross-sectionally (by testing each individual rating period) in the full sample as well as in subsamples of boys and girls. Results obtained with the mean above minus below a cut (MAMBAC), maximum covariance (MAXCOV), and latent mode factor analysis (L-Mode) taxometric procedures revealed consistent evidence of continuous latent structure despite the fact Growth Mixture Modeling (GMM) and Latent Class Growth Analysis (LCGA) identified between two and eight trajectories, depending on the stopping rule, in the three antisocial indicators. From these results, it is concluded that the structural model underlying these data is better represented as continuous rather than as categorical. The implications of these results for future research on developmental trajectories of antisocial behavior are discussed.     

Confirmatory Measurement Model Comparisons Using Latent Means

Confirmatory factor analysis (CFA) is often used to verify measurement models derived from classical test theory: the parallel, tau-equivalent, and congeneric test models. In this application, CFA is traditionally applied to the observed covariance or correlation matrix, ignoring the observed mean structure. But CFA is easily extended to allow nonzero observed and latent means. The use of CFA with nonzero latent means in testing six measurement models derived from classical test theory is discussed. Three of these models have not been addressed previously in the context of CFA. The implications of the six models for observed mean and covariance structures are fully described. Three examples of the use of CFA in testing these models are presented. Some advantages and limitations in using CFA with nonzero latent means to verify classical measurement models are discussed.     

Dimensional analyses of taxonic data

Principal component analysis (PCA) and common factor analysis are often used to model latent data structures. Typically, such analyses assume a single population whose correlation or covariance matrix is modelled. However, data may sometimes be unwittingly sampled from mixed populations containing a taxon (nonarbitrary subpopulation) and its complement class. One derives relations between values of PCA parameters within subpopulations and their values in the mixed population. These results are then extended to factor analysis in mixed populations. As relationships between subpopulation and mixed-population principal components and factors sensitively depend on within-subpopulation structures and between-subpopulation differences, naive interpretation of PCA or factor analytic findings can potentially mislead. Several analyses, better suited to the dimensional analysis of admixture data structures, are presented and compared.     

Linear structural equations with latent variables

An interdependent multivariate linear relations model based on manifest, measured variables as well as unmeasured and unmeasurable latent variables is developed. The latent variables include primary or residual common factors of any order as well as unique factors. The model has a simpler parametric structure than previous models, but it is designed to accommodate a wider range of applications via its structural equations, mean structure, covariance structure, and constraints on parameters. The parameters of the model may be estimated by gradient and quasi-Newton methods, or a Gauss-Newton algorithm that obtains least-squares, generalized least-squares, or maximum likelihood estimates. Large sample standard errors and goodness of fit tests are provided. The approach is illustrated by a test theory model and a longitudinal study of intelligence.This investigation was supported in part by a Research Scientist Development Award (KO2-DA00017) and a research grant (DA01070) from the U. S. Public Health Service.     

Using SAS PROC CALIS to fit Level-1 error covariance structures of latent growth models

In the present article, we demonstrates the use of SAS PROC CALIS to fit various types of Level-1 error covariance structures of latent growth models (LGM). Advantages of the SEM approach, on which PROC CALIS is based, include the capabilities of modeling the change over time for latent constructs, measured by multiple indicators; embedding LGM into a larger latent variable model; incorporating measurement models for latent predictors; and better assessing model fit and the flexibility in specifying error covariance structures. The strength of PROC CALIS is always accompanied with technical coding work, which needs to be specifically addressed. We provide a tutorial on the SAS syntax for modeling the growth of a manifest variable and the growth of a latent construct, focusing the documentation on the specification of Level-1 error covariance structures. Illustrations are conducted with the data generated from two given latent growth models. The coding provided is helpful when the growth model has been well determined and the Level-1 error covariance structure is to be identified.     

Optimization-Based Model Fitting for Latent Class and Latent Profile Analyses

Statisticians typically estimate the parameters of latent class and latent profile models using the Expectation-Maximization algorithm. This paper proposes an alternative two-stage approach to model fitting. The first stage uses the modified k-means and hierarchical clustering algorithms to identify the latent classes that best satisfy the conditional independence assumption underlying the latent variable model. The second stage then uses mixture modeling treating the class membership as known. The proposed approach is theoretically justifiable, directly checks the conditional independence assumption, and converges much faster than the full likelihood approach when analyzing high-dimensional data. This paper also develops a new classification rule based on latent variable models. The proposed classification procedure reduces the dimensionality of measured data and explicitly recognizes the heterogeneous nature of the complex disease, which makes it perfect for analyzing high-throughput genomic data. Simulation studies and real data analysis demonstrate the advantages of the proposed method.     

On the power of multivariate latent growth curve models to detect correlated change

We evaluated the statistical power of single-indicator latent growth curve models (LGCMs) to detect correlated change between two variables (covariance of slopes) as a function of sample size, number of longitudinal measurement occasions, and reliability (measurement error variance). Power approximations following the method of Satorra and Saris (1985) were used to evaluate the power to detect slope covariances. Even with large samples (N = 500) and several longitudinal occasions (4 or 5), statistical power to detect covariance of slopes was moderate to low unless growth curve reliability at study onset was above .90. Studies using LGCMs may fail to detect slope correlations because of low power rather than a lack of relationship of change between variables. The present findings allow researchers to make more informed design decisions when planning a longitudinal study and aid in interpreting LGCM results regarding correlated interindividual differences in rates of development.     

Testing pattern hypotheses for covariance matrices

Maximum likelihood estimates of the free parameters, and an asymptotic likelihood-ratio test, are given for the hypothesis that one or more elements of a covariance matrix are zero, and/or that two or more of its elements are equal. The theory applies immediately to a transformation of the covariance matrix by a known nonsingular matrix. Estimation is by Newton's method, starting conveniently from a closed-form least-squares solution.Numerical illustrations include a test for equality of diagonal blocks of a covariance matrix, and estimation of quasi-simplex structures.     

Multilevel models for multiple-baseline data: modeling across-participant variation in autocorrelation and residual variance

Multilevel models (MLM) have been used as a method for analyzing multiple-baseline single-case data. However, some concerns can be raised because the models that have been used assume that the Level-1 error covariance matrix is the same for all participants. The purpose of this study was to extend the application of MLM of single-case data in order to accommodate across-participant variation in the Level-1 residual variance and autocorrelation. This more general model was then used in the analysis of single-case data sets to illustrate the method, to estimate the degree to which the autocorrelation and residual variances differed across participants, and to examine whether inferences about treatment effects were sensitive to whether or not the Level-1 error covariance matrix was allowed to vary across participants. The results from the analyses of five published studies showed that when the Level-1 error covariance matrix was allowed to vary across participants, some relatively large differences in autocorrelation estimates and error variance estimates emerged. The changes in modeling the variance structure did not change the conclusions about which fixed effects were statistically significant in most of the studies, but there was one exception. The fit indices did not consistently support selecting either the more complex covariance structure, which allowed the covariance parameters to vary across participants, or the simpler covariance structure. Given the uncertainty in model specification that may arise when modeling single-case data, researchers should consider conducting sensitivity analyses to examine the degree to which their conclusions are sensitive to modeling choices.     

Latent Growth Curve and Repeated Measures ANOVA Contrasts: What the Models are Telling You

The relationship between the latent growth curve and repeated measures ANOVA models is often misunderstood. Although a number of investigators have looked into the similarities and differences among these models, a cursory reading of the literature can give the impression that they are very different models. Here we show that each model represents a set of contrasts on the occasion means. We demonstrate that the fixed effects parameters of the estimated basis vector latent growth curve model are merely a transformation of the repeated measures ANOVA fixed effects parameters. We further show that differences in fit in models that estimate the same means structure can be due to the different error covariance structures implied by the model. We show these relationships both algebraically and through using data from a simulation.     

Latent variables should remain as such: Evidence from a Monte Carlo study

Use of subject scores as manifest variables to assess the relationship between latent variables produces attenuated estimates. This has been demonstrated for raw scores from classical test theory (CTT) and factor scores derived from factor analysis. Conclusions on scores have not been sufficiently extended to item response theory (IRT) theta estimates, which are still recommended for estimation of relationships between latent variables. This is because IRT estimates appear to have preferable properties compared to CTT, while structural equation modeling (SEM) is often advised as an alternative to scores for estimation of the relationship between latent variables. The present research evaluates the consequences of using subject scores as manifest variables in regression models to test the relationship between latent variables. Raw scores and three methods for obtaining theta estimates were used and compared to latent variable SEM modeling. A Monte Carlo study was designed by manipulating sample size, number of items, type of test, and magnitude of the correlation between latent variables. Results show that, despite the advantage of IRT models in other areas, estimates of the relationship between latent variables are always more accurate when SEM models are used. Recommendations are offered for applied researchers.