Epsilon Open Archive

Abstract

New residuals taking the bilinear structure into account are defined for the Extended Growth Curve Model. It is shown that the ordinary residuals are defined by projecting the observation matrix on the space orthogonal to the one generated by the design matrices which turn out to be the sum of two tensor product spaces. The space on which the ordinary residuals are defined is then decomposed into four orthogonal spaces and new residuals are defined by projecting the observation matrix on the resulting four spaces. The information contained in them is used to check the adequacy of the model and to check if there are extreme observations. Tests are proposed for the Growth and Extended Growth Curve models which turn out to be functions of appropriate residuals. It is shown that the distributions of the tests under the null hypotheses are independent of the unknown covariance matrix. The distributions are difficult to find, however, two suggestions are made to tackle this problem. We consider a conditional approach and discuss why it is appropriate in our situation. Moreover, it is shown that the distribution of the conditional test under the null and alternative hypotheses can be written as sums of independent central and non-central chi-square random variables, respectively. However, the exact distributions, which are available as an infinite series, are too complicated to be used in practice and approximations are needed. We use Satterthwaite's approximation to find the critical point. Under the alternative, an approximation similar to that of Satterthwaite is provided for obtaining an approximate power. However, our approach is different and new ideas are utilized to get the approximation. Numerical examples are given to illustrate the results.