Covariance structure analysis of partially additive ipsative data using restricted maximum likelihood estimation

A data matrix is said to be ipsative when the sum of the scores obtained over the variables for each subject is a constant. In this article, a general type of ipsative data known as partially additive ipsative data (PAID) is defined. Ordinary additive ipsative data (AID) is a special case. Due to the specific nature of the research design or measurement process, the observed vector (x) under bar is PAID with an underlying nonipsative vector (y) under bar. It is shown that if the underlying distribution of (y) under bar is multivariate normal with structured covariance matrix Sigma = Sigma(<(theta)under bar>), the observed (x) under bar will have a degenerate normal distribution. As a result, ordinary maximum likelihood estimation of <(theta)under bar> cannot be carried out directly. A transformation of (x) under bar is suggested so that the transformed vector (x) under bar* = B (x) under bar will have a nonsingular density and restricted maximum likelihood (REML) estimation can be applied. A simulation study is conducted to investigate the effect of sample size and other model characteristics on the performance of the REML estimators and the sampling behavior of the goodness of fit statistic. It is found that the REML estimates are in general close to the true parameter values, but they have larger standard errors as compared with the ordinary MLE based on (y) under bar. The test statistic is well behaved when sample size is large enough. Moreover, the likelihood of obtaining a convergent solution depends on a number of factors such as sample size, number of indicators per latent factor, and degree of ipsativity. Finally, statistical decisions (reject or not reject the hypothesized model) based on (x) under bar* are in general consistent with that based on (y) under bar.