Dietrich von Rosen

The growth curve model: a review

A survey is given of papers which have influenced or have been influenced by the Growth Curve Model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical version of the model, tests, extensions, incomplete data, Bayesian approaches and covariance structures.

Local influence assessment in the growth curve model with unstructured covariance

Abstract In this paper, a local influence approach is employed to assess adequacy of the growth curve model with an unstructured covariance, based on likelihood displacement. The Hessian matrix of the model is investigated in detail under an abstract perturbation scheme. For illustration, covariance-weighted perturbation is discussed and used to analyze two real-life biological data sets, which show that the criteria presented in this article are useful in practice.

Approximating by the Wishart distribution

Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.

RESIDUALS IN THE GROWTH CURVE MODEL

Residuals for the Growth Curve model will be discussed. In univariate linear models as well as the ordinary multivariate analysis of variance model residuals are based on the difference between the observations and the mean whereas for the Growth Curve model we have three different residuals all showing various aspects useful for validating analysis. For these residuals some basic properties are established.

Uniqueness conditions for maximum likelihood estimators in a multivariate linear model

Abstract Necessary and sufficient uniqueness conditions are given for the maximum likelihood estimators of the parameters in the mean in a multivariate linear model. These are applied to the Growth Curve model when linear restrictions exist on the parameter describing the mean.

A family of bivariate distributions with non-elliptical contours

In this paper, a family of copulas with two parameters is proposed and its dependence analysis is performed. The corresponding family of bivariate distributions with specified marginals is constructed. For normal marginals, the new distributions are non-elliptical and can be applied in data analysis. They provide various alternative hypotheses for testing normality. Finally, an example is given.

On the posterior distribution of the covariance matrix of the growth curve model

For the growth curve model with an unstructured covariance matrix, the posterior distributions of the dispersion matrix is derived under a non-informative prior distribution. The results are especially useful for Bayesian inference as well as Bayesian diagnostics of the model.

Distribution and Density Approximations of a Maximum Likelihood Estimator in the Growth Curve Model

Moments of arbitrary order are obtained and expansions of the distribution and density functions of the maximum likelihood estimator of the mean structure in the Growth Curve model are considered. Results are given for expansions of arbitrary order. The results are obtained with the help of normally distributed variables and inverted Wishart variables.

Homogeneous matrix equations and multivariate linear models

Abstract The growth curve model (Potthoff and Roy, 1964) and an extension (von Rosen, 1989) are considered. The mean structures for the models are given by ABC and ∑ i A i B i C i , respectively, where the A and C matrices are known and the B matrices unknown. The purpose is to discuss homogeneous matrix equations DBE =0, D i BE i =0, i = 1,2,…, s and D 1 B 1 E 1 + D 2 B 2 E 2 =0 regarded as restrictions on the parameter space in statistical models. Among other results, we will see what kind of restrictions have to be imposed on the D and E matrices in order to obtain interpretable maximum likelihood estimators.

Partial Least Squares and a Linear Model

Partial least squares (PLS) is considered from the perspective of a linear model. It is shown that the PLS-predictor is identical to the best linear predictor in a linear model with random coefficients.