Simulating multivariate nonnormal distributions

Vale, C. David; Maurelli, Vincent A.

doi:10.1007/bf02293687

Cited by 393 publications

(390 citation statements)

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“…10 For each replication, we fit the relevant population factor model using both full and robust WLS estimation as implemented with Mplus (Version 2.01; L. K. Muthén & Muthén, 1998). 11 10 We generated the raw data with the EQS implementation of the methods of Fleishman (1978), who presented formulae for simulating univariate data with nonzero skewness and kurtosis, and Vale and Maurelli (1983), who extended Fleishman's method to allow simulation of multivariate data with known levels of univariate skewness and kurtosis and a known correlational structure. To ensure that EQS properly created the data according to the desired levels of skewness and kurtosis for the y* variables, we generated a set of continuous data conforming to Model 1 with N = 50,000 for each level of our manipulation of the y* distributions.…”

Section: Data Generation and Analysismentioning

confidence: 99%

An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis With Ordinal Data.

Flora¹,

Curran²

2004

Psychological Methods

2,408

1,866

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Confirmatory factor analysis (CFA) is widely used for examining hypothesized relations among ordinal variables (e.g., Likert-type items). A theoretically appropriate method fits the CFA model to polychoric correlations using either weighted least squares (WLS) or robust WLS. Importantly, this approach assumes that a continuous, normal latent process determines each observed variable. The extent to which violations of this assumption undermine CFA estimation is not well-known. In this article, the authors empirically study this issue using a computer simulation study. The results suggest that estimation of polychoric correlations is robust to modest violations of underlying normality. Further, WLS performed adequately only at the largest sample size but led to substantial estimation difficulties with smaller samples. Finally, robust WLS performed well across all conditions. Variables characterized by an ordinal level of measurement are common in many empirical investigations within the social and behavioral sciences. A typical situation involves the development or refinement of a psychometric test or survey in which a set of ordinally scaled items (e.g., 0 = strongly disagree, 1 = neither agree nor disagree, 2 = strongly agree) is used to assess one or more psychological constructs. Although the individual items are designed to measure a theoretically continuous construct, the observed responses are discrete realizations of a small number of categories. Statistical methods that assume continuous distributions are often applied to observed measures that are ordinally scaled. In circumstances such as these, there is the potential for a critical mismatch between the assumptions underlying the statistical model and the empirical characteristics of the data to be analyzed. This mismatch in turn undermines confidence in the validity of the conclusions that are drawn from empirical data with respect to a theoretical model of interest (e.g., Shadish, Cook, & Campbell, 2002).This problem often arises in confirmatory factor analysis (CFA), a statistical modeling method commonly used in many social science disciplines. CFA is a member of the more general family of structural equation models (SEMs) and provides a powerful method for testing a variety of hypotheses about a set of measured variables. By far the most common method of estimation within CFA is maximum likelihood (ML), a technique which assumes that the observed variables are continuous and normally distributed (e.g., Bollen, 1989, pp. 131-134). These assumptions are not met when the observed data are discrete (as occurs when using ordinal scales), thus significant problems can result when fitting CFA models Correspondence should be addressed to Patrick J. Curran, Department of Psychology, University of North Carolina, Chapel Hill, NC 27599-3270. curran@unc.edu. Additional materials are on the web at http://dx.doi.org/10.1037/1082-989X.9.4.466.supp. NIH Public Access Author ManuscriptPsychol Methods. Author manuscript; available in PMC 2011 August 9. NIH...

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Section: Data Generation and Analysismentioning

confidence: 99%

An Empirical Evaluation of Alternative Methods of Estimation for Confirmatory Factor Analysis With Ordinal Data.

Flora¹,

Curran²

2004

Psychological Methods

2,408

1,866

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“…9 In the simulation model, the producer's final wealth is a function of four random variables: farm yield (y), area yield (q), farm cash price (p) and futures price (f). Given the small size of our samples, we chose to apply the method proposed by Fleishman (1978) and extended by Vale and Maurelli (1983) to generate multivariate non-normal distributions. Fleishman's technique to simulate non-normal random numbers in the univariate case consists of defining a random variable as a linear combination of the first three powers of a standard normal random variable.…”

Section: Selected Farmsmentioning

confidence: 99%

Hedging price risk in the presence of crop yield and revenue insurance

Mahul¹

2003

European Review of Agriculture Economics

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“…It finally computes the average of the differences between the ordinary weight matrix Wand the weight matrix W' generated from the bootstrap procedure. Procedure that, although not necessary for realization of the two corrections proposed by Yung and Bentler (1994), could be of great value in carrying out Monte Carlo simulations, generating random samples of size n in p variables, with skewness and kurtosis previously selected by the user, according to Vale and Maurelli (1983) algorithms. The Fleishman constants can be estimated using the following FLEfS procedure.…”

Section: W~(equation 10)mentioning

confidence: 99%

“…Once the correct operation of the application was checked, a Monte Carlo simulation procedure was carried out using the model shown in Figure 2 and manipulating the sample size (300 and 500 cases) and nonnormality by observing variable skewness and kurtosis through three levels (Skew 0 -Kurt 6, Skew 1.5 -Kurt 3, Skew 3 -Kurt 15). For each one of the six 3 X 2 conditions, 300 independent samples were generated from the variance and covariance matrix in Joreskog and Sorborn (1986, p. III98) by means of the Fleishman (1978) and Vale and Maurelli (1983) algorithms, implemented in the VALE procedure.…”

Section: W~(equation 10)mentioning

confidence: 99%