# Albert SatorraUniversity Pompeu Fabra | UPF · Department of Economy and Business

Albert Satorra

PhD

## About

116

Publications

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16,911

Citations

Citations since 2016

## Publications

Publications (116)

In 2018, two different approaches have been suggested to solve the estimation problems that have been detected during the analysis of the data of the two-group split ballot multi-trait multi-method (SB-MTMM experiments performed in many countries between 2002 and 2010 in the European Social Survey). One group suggested using the Bayesian estimation...

Saris, Satorra and Coenders (2004) proposed a new approach for estimating the quality of survey questions, which combines the advantages of the following two existing approaches: the multi-trait-multi-method (MTMM) approach and the split-ballot (SB) approach. In practice, this new approach led to frequent occurrences of non-convergence and improper...

In political discourse, the term transversality denotes similarity of opinion on a policy across different segments of the society. We quantify this concept by the index of transversality, defined as the Pearson contingency coefficient of a policy-related categorical variable Y and a segmentation (division) of the population, X (e.g., gender, race...

Two approaches are commonly in use for analyzing panel data: the univariate, which arranges data in long format and estimates just one regression equation; and the multivariate, which arranges data in wide format, and simultaneously estimates a set of regression equations. Although technical articles relating the two approaches exist, they do not s...

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Consider the case in which we have data from repeated surveys covering several geographic areas, and our goal is to characterize these areas on a latent trait that underlies multiple indicators. This characterization occurs, for example, in surveys of information and communication technologies (ICT) conducted by statistical agencies, the objective...

I congratulate Hao Wu and Michael W. Browne (henceforth, WB) on their thought-provoking approach to specification error in moment structure analysis. To my reading, the novel and challenging issue of their paper is to interpret specification error as a (stochastic) second-level variation of the sample covariance matrix, variation that is said to be...

This paper develops a theorem that facilitates computing the degrees of freedom of Wald-type chi-square tests for moment restrictions when there is rank deficiency of key matrices involved in the definition of the test. An if and only if (iff) condition is developed for a simple rule of difference of ranks to be used when computing the desired degr...

Mean corrected higher order sample moments are asymptotically normally distributed. It is shown that both in the literature and popular software the estimates of their asymptotic covariance matrices are incorrect. An introduction to the infinitesimal jackknife is given and it is shown how to use it to correctly estimate the asymptotic covariance ma...

Covariance structure analysis of nonnormal data is important because in practice all data are nonnormal. When applying covariance structure analysis to nonnormal data, it is generally assumed that the asymptotic covariance matrix Γ for the nonredundant terms in the sample covariance matrix S is nonsingular. It is shown this need not be the case, wh...

It is shown that for any full column rank matrix X
0 with more rows than columns there is a neighborhood
$\mathcal{N}$
of X
0 and a continuous function f on
$\mathcal{N}$
such that f(X) is an orthogonal complement of X for all X in
$\mathcal{N}$
. This is used to derive a distribution free goodness of fit test for covariance structure analysi...

It is well known that measurement error in observable variables induces bias in estimates in standard regression analysis and that structural equation models are a typical solution to this problem. Often, multiple indicator equations are subsumed as part of the structural equation model, allowing for consistent estimation of the relevant regression...

We highlight critical conceptual and statistical issues and how to resolve them in conducting Satorra–Bentler (SB) scaled difference chi-square tests. Concerning the original (Satorra & Bentler, 200124.
Satorra , A. and
Bentler , P. M. 2001. A scaled difference chi-square test statistic for moment structure analysis. Psychometrika, 66: 507–514. [...

This article develops a procedure based on copulas to simulate multivariate nonnormal data that satisfy a prespecified variance-covariance matrix. The covariance matrix used can comply with a specific moment structure form (e.g., a factor analysis or a general structural equation model). Thus, the method is particularly useful for Monte Carlo evalu...

Starting with Kenny and Judd (Psychol. Bull. 96:201–210, 1984) several methods have been introduced for analyzing models with interaction terms. In all these methods more information
from the data than just means and covariances is required. In this paper we also use more than just first- and second-order
moments; however, we are aiming to adding j...

Consider the case in which we have data from repeated surveys covering several geographic areas, and our goal is to characterize these areas on a latent trait that underlies multiple indicators. This characterization occurs, for example, in surveys of information and communication technologies (ICT) conducted by statistical agencies, the objective...

This paper introduces a mixture model based on the beta distribution, without pre-established means and variances, to analyze
a large set of Beauty-Contest data obtained from diverse groups of experiments (Bosch-Domènech et al. 2002). This model gives a better fit of the experimental data, and more precision to the hypothesis that a large proportio...

We extend standard methodology for multigroup mean and covariance structure (MACS) analysis to the case where assessment of across-group variation of model parameters is the focus of the study and the data deviate from standard assumptions. The proposed methods are applied to analyze an accounting profitability database covering more than 100,000 f...

When using existing technology, it can be hard or impossible to determine whether two structural equation models that are being considered may be nested. There is also no routine technology for evaluating whether two very different structural models may be equivalent. A simple nesting and equivalence testing (NET) procedure is proposed that uses ra...

A scaled difference test statistic
\(\tilde{T}{}_{d}\)
that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic
\(\tilde{T}_{d}\)
is asymptotically equivalent to the scaled difference test statistic
\(\bar{T}_{d}\)
i...

The book combines longitudinal research and latent variable research, i.e. it explains how longitudinal studies with objectives formulated in terms of latent variables should be performed. The emphasis is on exposing how the methods are applied. Because currently longitudinal research with latent variables follows different approaches with differen...

Assessing the correctness of a structural equation model is essential to avoid drawing incorrect conclusions from empirical research. In the past, the chi-square test was recommended for assessing the correctness of the model but this test has been criticized because of its sensitivity to sample size. As a reaction, an abundance of fit indexes have...

In this paper, we show that for some structural equation models (SEM), the classical chi-square goodness-of-fit test is unable
to detect the presence of nonlinear terms in the model. As an example, we consider a regression model with latent variables
and interactions terms. Not only the model test has zero power against that type of misspecificatio...

A typical structural equation model is intended to reproduce the means, variances, and correlations or covariances among a set of variables based on parameter estimates of a highly restricted model. It is not widely appreciated that the sample statistics being modeled can be quite sensitive to outliers and influential observations leading to bias i...

Most methods for small-area estimation are based on composite estimators derived from design- or model-based methods. A composite estimator is a linear combination of a direct and an indirect estimator with weights that usually depend on unknown parameters which need to be estimated. Although model-based small-area estimators are usually based on r...

Most methods for small-area estimation are based on composite estimators derived from design- or model-based methods. A composite estimator is a linear combination of a direct and an indirect estimator with weights that usually depend on unknown parameters which need to be estimated. Although model-based small-area estimators are usually based on r...

We apply structural equation models to longitudinal data on profits of firms within industries to study the persistence of abnormal returns. We obtain a two-way variance decomposition for abnormal returns: at firm vs. industry levels, and at permanent vs. transitory components. This decomposition enables us to assess the relative importance of the...

A national survey designed for estimating a specific population quantity is sometimes used for estimation of this quantity also for a small area, such as a province. Budget constraints do not allow a greater sample size for the small area, and so other means of improving estimation have to be devised. We investigate such methods and assess them by...

A national survey designed for estimating a specific population quantity is sometimes used for estimation of this quantity also for a small area, such as a province. Budget constraints do not allow a greater sample size for the small area, and so other means of improving estimation have to be devised. We investigate such methods and assess them by...

In this article we propose small area estimators for both the small and large area parameters. When the objective is to estimate parameters at both levels, optimality is achieved by a sample design that combines fixed and proportional allocation. In such a design, one fraction of the sample is distributed proportionally among the small areas and th...

This paper compares five small area estimators. We use Monte Carlo simulation in the context of both artificial and real populations. In addition to the direct and indirect estimators, we consider the optimal composite estimator with population weights, and two composite estimators with estimated weights: one that assumes homogeneity of within area...

In this article we propose using small area estimators to improve the estimates of both the small and large area parameters. When the objective is to estimate parameters at both levels accurately, optimality is achieved by a mixed sample design of fixed and proportional allocations. In the mixed sample design, once a sample size has been determined...

I modify the uniform-price auction rules in allowing the seller to ration bidders. This allows me to provide a strategic foundation for underpricing when the seller has an interest in ownership dispersion. Moreover, many of the so-called "collusive-seeming" equilibria disappear.

In this article we propose small area estimators for both the small and large area parameters. When the objective is to estimate parameters at both levels, optimality is achieved by a sample design that combines fixed and proportional allocation. In such a design, one fraction of the sample is distributed proportionally among the small areas and th...

This paper develops a finite mixture distribution analysis of Beauty-Contest data obtained from diverse groups of experiments. ML estimation using the EM approach provides estimates for the means and variances of the component distributions, which are common to all the groups, and estimates of the mixing proportions, which are specific to each grou...

The use of augmented moment matrices (replacing covariances) allows to carry out mean- and covariance-structure analysis using conventional software for covariance structure analysis. The present paper establishes the algebraic equality of two alternative goodness-of-fit test statistics in normal-theory GLS analysis of augmented moment matrices. In...

The theory of best affine prediction (BAP) is extended to the vector case with possibly singular variance matrix of the predictor
variable. The theory is then applied to derive Thomson’s classical predictor for factor scores, allowing for a singular variance
matrix of the factors. The results are formulated in a free distribution setting. Further,...

This paper compares five small area estimators. We use Monte Carlo simulation in the context of both artificial and real populations. In addition to the direct and indirect estimators, we consider the optimal composite estimator with population weights, and two composite estimators with estimated weights: one that assumes homogeneity of within area...

In the context of structural equation models, we investigate the asymptotic and finite sample size distribution of competing X
2 goodness-offit test statistics. We allow for a) the data to be non-normal, b) the estimation method to be non-optimal, and c) the model to be misspecified. Power of the test is computed distinguishing whether asymptotic r...

This work is part of a project that studies the application of small area composite estimators (a combination of direct and indirect estimators) in regional statistics. We compare three estimators: a direct one based on sample data for each of the Spanish Autonomous Communities, a synthetic (indirect) one that combines Estate wide information with...

Este trabajo es parte de un proyecto que estudia la aplicación de estimadores compuestos (combinación de estimadores directos e indirectos) para áreas pequeñas en estadística regional. Comparamos tres estimadores: uno directo basado en datos muestrales de cada Comunidad Autónoma (CA), otro sintético (indirecto) que combina los datos estatales con i...

Standard methods for analyzing linear-latent variable models rely on the assumption that the observed variables are normally distributed. Normality allows statistical inferences to be carried out based solely on the first-and second-order moments. In general, inferences for nonnormally distributed data require the estimates of matrices of third-and...

This work is part of a project studying the performance of model based estimators in a small area context. We have chosen a simple statistical application in which we estimate the growth rate of accupation for several regions of Spain. We compare three estimators: the direct one based on straightforward results from the survey (which is unbiassed),...

Este trabajo es parte de un proyecto que estudia la aplicación de estimadores compuestos (combinación de estimadores directos e indirectos) para áreas pequeñas en estadística regional. Comparamos tres estimadores: uno directo basado en datos muestrales de cada Comunidad Autónoma (CA), otro sintético (indirecto) que combina los datos estatales con i...

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small
samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's
(SB) scaling corrections are available in standard computer software. Often, however, the i...

Using data form the Spanish household budget survey, we investigate some aspects of household heterogeneity on several product expenditures. We adopt a latent-variable model approach to evaluate the impact of income on expenditures, controlling for the number of members in the family. Two latent factors underlying repeated measures of monetary and...

In a recent note in the Teacher's Corner of this journal, de Jong (1999) proposed a method for computing hierarchical or fixed-order regressions in the context of latent variables. The essence of this approach is to decompose the predictor variables in the regression into orthogonal components based on a Cholesky decomposition and to regress the de...

Structural equation models (SEM) are commonly used to analyze the relationship between variables some of which may be latent, such as individual ``attitude'' to and ``behavior'' concerning specific issues. A number of difficulties arise when we want to compare a large number of groups, each with large sample size, and the manifest variables are dis...

"Beauty-contest" is a game in which participants have to choose, typically, a number in [0,100], the winner being the person whose number is closest to a proportion of the average of all chosen numbers. We describe and analyze Beauty-contest experiments run in newspapers in UK, Spain, and Germany and find stable patterns of behavior across them, de...

We extend to score, Wald and difference test statistics the scaled and adjusted corrections to goodness-of-fit test statistics developed in Satorra and Bentler (1988a,b). The theory is framed in the general context of multisample analysis of moment structures, under general conditions on the distribution of observable variables. Computational issue...

Using data from the Spanish household budget survey, we investigate life- cycle effects on several product expenditures. A latent-variable model approach is adopted to evaluate the impact of income on expenditures, controlling for the number of members in the family. Two latent factors underlying repeated measures of monetary and non-monetary incom...

Asymptotic chi-squared test statistics for testing the equality of moment vectors are developed. The test statistics proposed are generalized Wald test statistics that specialize for different settings by inserting an appropriate asymptotic variance matrix of sample moments. Scaled test statistics are also considered for dealing with nonstandard co...

Least-Squares Approximation of Off-Diagonal Elements of a Variance Matrix in the Context of Factor Analysis - Volume 13 Issue 2 - Albert Satorra, Heinz Neudecker

Estimation and testing of functional or structural multivariate regression with errors in variables, with possibly unbalanced
design for replicates, and not necessarily normal data, is developed using only the sample cross-product moments of the data.
We give conditions under which normal theory standard errors and an asymptotic chi-square goodness...

In practice, several measures of association are used when analyzing structural equation models with ordinal variables: ordinary Pearson correlations (PE approach), polychoric and polyserial correlations (PO approach), and conditional polychoric correlations (CPO approach). In the case of structural equation models without latent variables, the lit...

We consider the application of normal theory methods to the estimation and testing of a general type of multivariate regression models with errors--in--variables, in the case where various data sets are merged into a single analysis and the observable variables deviate possibly from normality. The various samples to be merged can differ on the set...

We proved the algebraic equality between Jennrich's (1970) asymptotic [chi]2 test for equality of correlation matrices, and a Wald test statistic derived from the Neudecker and Wesselman (1990) expression of the asymptotic variance matrix of the sample correlation matrix.

Muthén (1984) formulated a general model and estimation procedure for structural equation modeling with a mixture of dichotomous, ordered categorical, and continuous measures of latent variables. A general three-stage procedure was developed to obtain estimates, standard errors, and a chi-square measure of fit for a given structural model. While th...

Standard methods for the analysis of linear latent variable models often rely on the assumption that the vector of observed variables is normally distributed. This normality assumption (NA) plays a crucial role in assessing optimality of estimates, in computing standard errors, and in designing an asymptotic chi-square goodness-of-fit test. The asy...

* ABSTRACT This chapter starts by reviewing the consequences of scale dependence of Structural Equation Models. Next it shows the scale dependence of the True Score MTMM model presented in Chapter 1 and its practical consequences. Finally, it suggests alternative sets of non-linear constraints for the True Score MTMM model which make the model to b...

Large-scale surveys using complex sample designs are frequently carried out by government agencies. The statistical analysis technology available for such data is, however, limited in scope. This study investigates and further develops statistical methods that could be used in software for the analysis of data collected under complex sample designs...

This chapter starts by reviewing the consequences of scale dependence of Structural Equation Models. Next it shows the scale dependence of the True Score MTMM model presented in Chapter 1 and its practical consequences. Finally, it suggests alternative sets of non-linear constraints for the True Score MTMM model which make the model to be scale inv...

In the context of linear latent-variable models, and a general type of distribution of the data, the asymptotic optimality of a subvector of minimum-distance estimators whose weight matrix uses only second-order moments is investigated. The asymptotic optimality extends to the whole vector of parameter estimators, if additional restrictions on the...

A. Satorra and P. Bentler . . . developed an approach to the asymptotic behavior of covariance structure statistics that rather naturally yields corrections to the goodness-of-fit statistic of the scaling and Satterthwaite types / present these results and . . . illustrate how they improve upon the uncorrected statistics that are now implemented in...

Standard methods in moment-structure analysis extensively use the assumption that the vector of observable variables is normally distributed. In practice, however, the data tend to deviate from this normality assumption. In contrast with parameter estimators that are consistent even under non-normality, standard errors of estimators and the asympto...

We derive an expression for the variance matrix of the vector of (uncentered) sample second-order moments under multivariate linear relations and an independence assumption. An application of the result is presented.

In moment structure analysis with nonnormal data, asymptotic valid inferences require the computation of a consistent (under general distributional assumptions) estimate of the matrix $\Gamma$ of asymptotic variances of sample second--order moments. Such a consistent estimate involves the fourth--order sample moments of the data. In practice, the u...

The paper deals with the usefulness of model modifications statistics, such as the modification index and the Wald statistic, in situations where equivalent models are involved.