Testing sphericity and intraclass covariance structures under a growth curve model in high dimension

In this article, we consider the problem of testing (a) sphericity and (b) intraclass covariance structure under a growth curve model. The maximum likelihood estimator (MLE) for the mean in a growth curve model is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. The MLE for the covariance matrix is based on the MLE for the mean, which can be very poor for p close to N. For both structures (a) and (b), we modify the MLE for the mean to an unweighted estimator and based on this estimator we propose a new estimator for the covariance matrix. This new estimator leads to new tests for (a) and (b). We also propose two other tests for each structure, which are just based on the sample covariance matrix.

To compare the performance of all four tests we compute for each structure (a) and (b) the attained significance level and the empirical power. We show that one of the tests based on the sample covariance matrix is better than the likelihood ratio test based on the MLE.     

General multivariate linear models for longitudional studies

Longitudinal studies occcur frequently in many different disciplines. To fully utilize the potential value of the information contained in a longitudinal data, various multivariate linear models have been proposed. The methodology and analysis are somewhat unique in their own ways and their relationships are not well understood and presented. This article describes a general multivaritate linear model for longitudinal data and attempts to provide a constructive formulation of the components in the mean response profile. The objective is to point out the extension and connections of some well-known models that have been obscured by different areas of application. More imporiantly, the model is expressed in a unified regression form from the subject matter considerations. Such an approach is simpler and more intuitive than other ways to modeling and parameter estimation. As a cmsequeace the analyses of the general class cf models for longitudional data can be casily implemented with standard software.     

Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

On a gmanova model likelihood ratio test criterion

For the generalized MANOVA (GMANOVA) model of Potthoff and Roy (1964), X = BξA + E, Khatri (1966) derives the likelihood ratio test criterion for test-ing the composite double linear null hypothesis CξV = 0, C,V known. This criterion plays an important role in statistics, and several authors have recently studied its further properties. However, Khatri's (1966) de-reviation of the distribution of this criterion is involved. By noting that the GMANOVA model is re-stricted MANOVA model, this paper presents an alter-native simple derivation of the distribution of this criterion. The derivation is based on the generalized Sverdrup's lemma, Kabe (1965).     

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.     

On Comparisons of Exact Powers of Bivariate GMANOVA Tests

Exact powers of four classical tests in a GMANOVA model are compared numerically when the order of the error sum of square matrix is 2. The four tests are likelihood ratio (=LR), Pillai's V, Hotelling's T ², and Roy's largest root tests. It turns out that for small sizes, there are a few cases in which Rothenberg's condition for the relative magnitude of asymptotic powers of three standard tests does not hold.     

Test for the mean matrix in a Growth Curve model for high dimensions

We consider the problem of estimating and testing a general linear hypothesis in a general multivariate linear model, the so-called Growth Curve model, when the p × N observation matrix is normally distributed.

The maximum likelihood estimator (MLE) for the mean is a weighted estimator with the inverse of the sample covariance matrix which is unstable for large p close to N and singular for p larger than N. We modify the MLE to an unweighted estimator and propose new tests which we compare with the previous likelihood ratio test (LRT) based on the weighted estimator, i.e., the MLE. We show that the performance of these new tests based on the unweighted estimator is better than the LRT based on the MLE.     

On the difference in inference and prediction between the joint and independent f-error models for seemingly unrelated regressions

We consider likelihood and Bayesian inferences for seemingly unrelated (linear) regressions for the joint niultivariate terror (e.g. Zellner, 1976) and the independent t-error (e.g. Maronna, 1976) models. For likelihood inference, the scale matrix and the shape parameter for the joint terror model cannot be consistently estimated because of the lack of adequate information to identify the latter. The joint terror model also yields the same MLEs for the regression coefficients and the scale matrix as for the independent normal error model. which are not robust against outliers. Further, linear hypotheses with respect

to the regression coefficients also give rise to the same mill distributions AS for the independent normal error model, though the MLE has a non-normal limiting distribution. In contrast to the striking similarities between the joint t-error and the independent normal error models, the independent f-error model yields AiLEs that are lubust against uuthers. Since the MLE of the shape parameter reflects the tails of the data distributions, this model extends the independent normal error model for modeling data distributions with relatively t hicker tails. These differences are also discussed with respect to the posterior and predictive distributions for Bayesian inference.     

A note on the manova model with double linear restrictions

In this note we show how one may construct goodness-of-fit tests to test hypotheses for the restricted MANOVA and GMANOVA models using the multivariate seemingly unrelated regression (MSUR) model.