Heywood Cases Research Articles

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.

Applying the triads method in the validation of dietary intake using biomarkers

The triads method is applied in validation studies of dietary intake to evaluate the correlation between three measurements (food frequency questionnaire, reference method and biomarker) and the true intake using validity coefficients (Á). The main advantage of this technique is the inclusion of the biomarker, which presents independent errors compared with those of the traditional methods. The method assumes the linearity between the three measurements and the true intake and independence between the three measurement errors. Limitations of this technique include the occurrence of ρ > 1, known as "Heywood case", and the existence of negative correlations, which do not allow the calculation of Á. The objective of this review is to present the concept of the method, describe its application and examine the validation studies of dietary intake that use the triads method. We also conceptualize the "bootstrap" method, used to estimate the confidence intervals of the validity coefficients.

Determinants of Standard Errors of MLEs in Confirmatory Factor Analysis

This paper studies changes of standard errors (SE) of the normal-distribution-based maximum likelihood estimates (MLE) for confirmatory factor models as model parameters vary. Using logical analysis, simplified formulas and numerical verification, monotonic relationships between SEs and factor loadings as well as unique variances are found. Conditions under which monotonic relationships do not exist are also identified. Such functional relationships allow researchers to better understand the problem when significant factor loading estimates are expected but not obtained, and vice versa. What will affect the likelihood for Heywood cases (negative unique variance estimates) is also explicit through these relationships. Empirical findings in the literature are discussed using the obtained results.

Stability of the Factor Structure of the Metabolic Syndrome across Pubertal Development: Confirmatory Factor Analyses of Three Alternative Models

To test the fit and stability of 3 alternative models of the metabolic syndrome's factor structure across 3 developmental stages. With data from the Fels Longitudinal Study, confirmatory factor analyses tested 3 alternative models of the factor structure underlying relationships among 8 metabolic syndrome-associated risks. Models tested were a 1-factor model (A), a 4-factor model (B), and a second-order latent factor model (C). Developmental stages assessed were prepuberty (ages 8-10), puberty (ages 11-15), and postpuberty (ages 16-20). Convergence was achieved for all developmental stages for model A, but the fit was poor throughout (root mean square error of approximation > 0.1). Standardized factor loadings for waist circumference and body mass index were much stronger than those for fasting insulin at all 3 time points. Although prepuberty and postpuberty models converged for models B and C, each model had problems with Heywood cases. The puberty model did not converge for either model B or C. The hypothetical structures commonly used to support the metabolic syndrome concept do not provide adequate fit in a pediatric sample and may be variable by maturation stage. A components-based approach to cardiovascular risk reduction, with emphasis on obesity prevention and control, may be a more appropriate clinical strategy for children and youth than a syndromic approach.

Exploratory Factor Analysis With Small Sample Sizes

Exploratory factor analysis (EFA) is generally regarded as a technique for large sample sizes (N), with N = 50 as a reasonable absolute minimum. This study offers a comprehensive overview of the conditions in which EFA can yield good quality results for N below 50. Simulations were carried out to estimate the minimum required N for different levels of loadings (λ), number of factors (f), and number of variables (p) and to examine the extent to which a small N solution can sustain the presence of small distortions such as interfactor correlations, model error, secondary loadings, unequal loadings, and unequal p/f. Factor recovery was assessed in terms of pattern congruence coefficients, factor score correlations, Heywood cases, and the gap size between eigenvalues. A subsampling study was also conducted on a psychological dataset of individuals who filled in a Big Five Inventory via the Internet. Results showed that when data are well conditioned (i.e., high λ, low f, high p), EFA can yield reliable results for N well below 50, even in the presence of small distortions. Such conditions may be uncommon but should certainly not be ruled out in behavioral research data. * These authors contributed equally to this work

Computational aspects of the minimal residual method in resolving the heywood case

Constructing a factor analysis model by the minimal residual method, we propose a solution to the problem of communalities exceeding 1. In order to determine the quantities, an appropriate lemma is proved which allows us to repair the original proof of the Harman theorem on the minimum of the target of the minimal residual method.

Constrained versus unconstrained estimation in structural equation modeling.

Recently, R. D. Stoel, F. G. Garre, C. Dolan, and G. van den Wittenboer (2006) reviewed approaches for obtaining reference mixture distributions for difference tests when a parameter is on the boundary. The authors of the present study argue that this methodology is incomplete without a discussion of when the mixtures are needed and show that they only become relevant when constrained difference tests are conducted. Because constrained difference tests can hide important model misspecification, a reliable way to assess global model fit under constrained estimation would be needed. Examination of the options for assessing model fit under constrained estimation reveals that no perfect solutions exist, although the conditional approach of releasing a degree of freedom for each active constraint appears to be the most methodologically sound one. The authors discuss pros and cons of constrained and unconstrained estimation and their implementation in 5 popular structural equation modeling packages and argue that unconstrained estimation is a simpler method that is also more informative about sources of misfit. In practice, researchers will have trouble conducting constrained difference tests appropriately, as this requires a commitment to ignore Heywood cases. Consequently, mixture distributions for difference tests are rarely appropriate.

ML estimation for factor analysis: EM or non-EM?

To obtain maximum likelihood (ML) estimation in factor analysis (FA), we propose in this paper a novel and fast conditional maximization (CM) algorithm, which has quadratic and monotone convergence, consisting of a sequence of CM log-likelihood (CML) steps. The main contribution of this algorithm is that the closed form expression for the parameter to be updated in each step can be obtained explicitly, without resorting to any numerical optimization methods. In addition, a new ECME algorithm similar to Liu's (Biometrika 81, 633---648, 1994) one is obtained as a by-product, which turns out to be very close to the simple iteration algorithm proposed by Lawley (Proc. R. Soc. Edinb. 60, 64---82, 1940) but our algorithm is guaranteed to increase log-likelihood at every iteration and hence to converge. Both algorithms inherit the simplicity and stability of EM but their convergence behaviors are much different as revealed in our extensive simulations: (1) In most situations, ECME and EM perform similarly; (2) CM outperforms EM and ECME substantially in all situations, no matter assessed by the CPU time or the number of iterations. Especially for the case close to the well known Heywood case, it accelerates EM by factors of around 100 or more. Also, CM is much more insensitive to the choice of starting values than EM and ECME.

Necessary Conditions for Mean Square Convergence of the Best Linear Factor Predictor

Several sufficient conditions are available for mean square convergence of factor predictors. A necessary and sufficient condition is given in the Heywood case with respect to (confirmatory) factor analysis. This condition generalizes that of Krijnen (2006) and performs better than a signal-to-noise type of condition (Schneeweiss & Mathes, 1995).

Convergence in mean square of factor predictors

Sufficient conditions for mean square convergence of factor predictors in common factor analysis are given by Guttman, by Williams, and by Schneeweiss and Mathes. These conditions do not hold for confirmatory factor analysis or when an error variance equals zero (Heywood cases). Two sufficient conditions are given for the three basic factor predictors and a predictor from rotated principal components analysis to converge to the factors of the model for confirmatory factor analysis, including Heywood cases. For certain model specifications the conditions are necessary. The conditions are sufficient for the existence of a unique true factor. A geometric interpretation is given for factor indeterminacy and mean square convergence of best linear factor prediction.

A Monte Carlo Study Investigating the Impact of Item Parceling on Measures of Fit in Confirmatory Factor Analysis

A simulation study was conducted to examine the effect of item parceling on goodness-of-fit indices at different levels of sample size, number of indicators per factor, factor structure/pattern coefficients, interfactor correlations, and item-level data distribution. Results revealed that the use of item parcels yielded more nonconverged solutions and Heywood cases than individual items. The likelihood of nonconverged solutions and Heywood cases increased as the number of indicators per factor (more items per parcel) decreased. Meanwhile, parcel solutions as compared with item solutions resulted in better fit as measured by the chi-square to degrees-of-freedom ratio, Goodness-of-Fit Index (GFI), Expected Cross-Validation Index (ECVI), and root mean square error of approximation (RMSEA), as well as two incremental fit indices, the Non-Normed Fit Index (NNFI) and Comparative Fit Index (CFI). The same pattern of results was found with data that varied in terms of skewness and kurtosis at the item level. However, the likelihood of nonconverged solutions and Heywood cases was more pronounced when data were extremely skewed/kurtotic at the item level.

The Likehood Function of Conditionally Heteroskedastic Factor Models

We derive the likelihood function and score of factor models with dynamic heteroskedasticity, and the KUHN-TUCKER conditions defining the inequality restricted maximum likelihood estimators that gua- rantee a positive definite covariance matrix. We present three methods to compute the likelihood function, its gradient and factor scores, which are numerically efficient and reliable, and statistically sound. We show that the incidence of zero idiosyncratic variance estimates (HEYWOOD cases) depends on the correlation of a variable with the rest.

Conditions for factor (in)determinacy in factor analysis

The subject of factor indeterminacy has a vast history in factor analysis (Guttman, 1955; Lederman, 1938; Wilson, 1928). It has lead to strong differences in opinion (Steiger, 1979). The current paper gives necessary and sufficient conditions for observability of factors in terms of the parameter matrices and a finite number of variables. Five conditions are given which rigorously define indeterminacy. It is shown that (un)observable factors are (in)determinate. Specifically, the indeterminacy proof by Guttman is extended to Heywood cases. The results are illustrated by two examples and implications for indeterminacy are discussed.

Biochemical markers as additional measurements in dietary validity studies: application of the method of triads with examples from the European Prospective Investigation into Cancer and Nutrition

The validity coefficient of dietary questionnaire measurements can be estimated from a triangular comparison between questionnaire, reference, and biochemical marker measurements with the method of triads. The method assumes that the measurements are linearly related to true intake and have independent random errors. We applied the method of triads to examples from the European Prospective Investigation into Cancer and Nutrition. In some examples, Heywood cases occurred, ie, the estimated validity coefficients were > 1 or the validity coefficients were not estimable. Such results are caused by random sampling fluctuations or violation of the model assumptions. One possible violation is a positive correlation between the random errors of questionnaire and reference measurements. We used a bootstrap method to estimate CIs for the validity coefficients. Validity studies with several hundred subjects, more accurate biochemical indicators of dietary intake, or both, are needed to estimate validity coefficients precisely and avoid complications with the bootstrap method.

An efficient algorithm for joint correspondence analysis

Joint correspondence analysis is a technique for constructing reduced-dimensional representations of pairwise relationships among categorical variables. The technique was proposed by Greenacre as an alternative to multiple correspondence analysis. Joint correspondence analysis differs from multiple correspondence analysis in that it focuses solely on between-variable relationships. Greenacre described one alternating least-squares algorithm for conducting joint correspondence analysis. Another alternating least-squares algorithm is described in this article. The algorithm is guaranteed to converge, and does so in fewer iterations than does the algorithm proposed by Greenacre. A modification of the algorithm for handling Heywood cases is described. The algorithm is illustrated on two data sets.

Algorithms for unweighted least-squares factor analysis

Estimation of the factor model by unweighted least squares (ULS) is distribution free, yields consistent estimates, and is computationally fast if the Minimum Residuals (MinRes) algorithm is employed. MinRes algorithms produce a converging sequence of monotonically decreasing ULS function values. Various suggestions for algorithms of the MinRes type are made for confirmatory as well as for exploratory factor analysis. These suggestions include the implementation of inequality constraints and the prevention of Heywood cases. A simulation study, comparing the bootstrap standard deviations for the parameters with the standard errors from maximum likelihood, indicates that these are virtually equal when the score vectors are sampled from the normal distribution. Two empirical examples demonstrate the usefulness of constrained exploratory and confirmatory factor analysis by ULS used in conjunction with the bootstrap method.

Gramian Matrices in Covariance Structure Models

Covariance structure models frequently contain out-of-range estimates that make no sense from either substantive or statistical points of view. Negative variance estimates are the most well-known of these improper solutions, but correlations that are out of range also occur. Methods to minimize improper estimates have been accomplished by reparameterization and estimation under simple inequality constraints; but these solutions, discussed previously in this journal (Marsh, 1989), do not guarantee that the covariance matrices involved represent variances and covariances of real numbers, as required. A general approach to avoiding improper solutions in structural equation models is proposed. Although this approach does not resolve inadequacies in the data or theoretical model that may generate an improper solution, it solves the long-standing problem of obtaining proper estimates. Index terms: confirmatory factor analysis, EQS, Gramian matrices, Heywood cases, improper solutions, LISREL, structural equation models, underidentification.

The effect of sampling duration on the ability to resolve source types using factor analysis

The effects of sampling duration on source identification using factor analysis method were examined in this paper. Principal factor analyses were performed on the data collected as part of the Philadelphia Regional and Local Sulfur Aerosol Concentrations Study conducted from July to October 1982. The data were mass concentrations of 22 trace elements analyzed by proton induced x-ray emission (PIXE) analysis and their associated analytical uncertainties that are a measure of the precision of the measurements. A weighting scheme is applied to the concentrations of the elements to emphasize the precisely measured elements for factor analysis. Furthermore, a new method was used to eliminate the problem of Heywood cases in the factor analyses. Results show the influence of sampling duration on the number of resolvable factors as well as the amount of variance explained by the factors. The analyses show that reducing the sampling duration can increase the number of sources identified by the model. Idiosyncrasies not found in the total data are shown on the separate analyses on daylight and non-daylight data.

Full-Information Item Factor Analysis

A method of item factor analysis based on Thur stone's multiple-factor model and implemented by marginal maximum likelihood estimation and the EM algorithm is described. Statistical significance of suc cessive factors added to the model is tested by the likelihood ratio criterion. Provisions for effects of guessing on multiple-choice items, and for omitted and not-reached items, are included. Bayes constraints on the factor loadings are found to be necessary to suppress Heywood cases. Numerous applications to simulated and real data are presented to substantiate the accuracy and practical utility of the method. Index terms: Armed Services Vocational Aptitude Bat tery, Beta prior, EM algorithm, Item factor analysis, TESTFACT, Tetrachoric correlation.

Offending estimates in covariance structure analysis: Comments on the causes of and solutions to Heywood cases.

In this article we discuss, illustrate, and compare the relative efficacy of three recommended approaches for handling negative error variance estimates (i.e., Heywood cases): (a) setting the offending estimate to zero, (b) adopting a model parameterization that ensures positive error variance estimates, and (c) using models with equality constraints that ensure nonnegative (but possibly zero) error variance estimates. The three approaches are evaluated in two distinct situations: Heywood cases caused by lack of fit and misspecification error, and Heywood cases induced from sampling fluctuations. The results indicate that in the case of sampling fluctuations the simple approach of setting the offending estimate to zero works reasonably well. In the case of lack of fit and misspecification error, the theoretical difficulties that give rise to negative error variance estimates have no ready-made methodological solutions.