Singularities and Constructive Methods for Their Treatment by Pierre Grisvard, Wolfgang L. Wendland, John R. Whiteman

By Pierre Grisvard, Wolfgang L. Wendland, John R. Whiteman

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Although it may happen that multivariate normality holds when all observed variables are individually normally distributed, it is desirable to also examine bivariate normality that is generally not a consequence of univariate normality. In fact, if the observations are from a multivariate normal distribution, each bivariate distribution should also be normal, like each univariate distribution. , Tabachnik & Fidell, 2001). , Johnson & Wichern, 2002; Khattree & Naik, 1999; Marcoulides & Hershberger, 1997).

In fact, unless its metric is defined, the scale of the latent variable will remain indeterminate. Subsequently, this will lead to model-estimation problems and unidentified parameters and models (discussed later in this chapter). For any independent latent variable included in a given model, the metric can be fixed in one of two ways that are equivalent for this purpose. Either its variance is set equal to a constant, usually 1, or a path going out of the latent variable is set to a constant (typically 1).

Conversely, a researcher may elect to free some initially fixed parameters, rendering them free parameters, after making sure of course that the model remains identified (see below). The third type of parameters are called constrained parameters, also sometimes referred to as restricted or restrained parameters. Constrained parameters are those that are postulated to be equal to one another—but their value is not specified in advance as is that of fixed parameters—or involved in a more complex relationship among themselves.

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