Mplus Users Guide v6.. - Muthén & Muthén

CHAPTER 7algorithm will be used. Note that numerical integration becomesincreasingly more computationally demanding as the number of factorsand the sample size increase. In this example, two dimensions ofintegration are used with 225 integration points. The ESTIMATORoption can be used to select a different estimator. The LINK option isused with maximum likelihood estimation to select a logit or a probitlink for models with categorical outcomes. The default is a logit link. Inthis example, the probit link is used because the threshold model forliabilities uses normally-distributed latent response variables.In the overall model, the (1) following the first bracket statementspecifies that the thresholds of u1 and u2 are held equal across twins.The two BY statements define a factor behind each outcome. This isdone because covariances of categorical outcomes are not part of themodel when maximum likelihood estimation is used. The covariances ofthe factors become the covariances of the categorical outcomes or moreprecisely the covariances of the latent response variables underlying thecategorical outcomes. The means of the factors are fixed at zero andtheir variances are held equal across twins. The variance of eachunderlying response variable is obtained as the sum of the factorvariance plus one where one is the residual variance in the probitregression of the categorical outcome on the factor.In the MODEL command, labels are defined for three parameters. Thelabel varf is assigned to the variances of f1 and f2. Because they aregiven the same label, these parameters are held equal. The label covmzis assigned to the covariance between f1 and f2 for the monozygotictwins and the label covdz is assigned to the covariance between f1 and f2for the dizygotic twins. In the MODEL CONSTRAINT command, theNEW option is used to assign labels to three parameters that are not inthe analysis model: a, c, and h. The two parameters a and c are used todecompose the covariances of u1 and u2 into genetic and environmentalcomponents. The value .001 is added to the variance of the factors toavoid a singular factor covariance matrix which comes about because thefactor variances and covariances are the same. The parameter h does notimpose restrictions on the model parameters but is used to compute theheritability estimate and its standard error. This heritability estimateuses the residual variances for the latent response variables which arefixed at one. An explanation of the other commands can be found inExample 7.1.190