An investigation of certain methods in the analysis of growth curves
In growth curve studies, measurements are made on individuals over a moderately large period of time and the main problem is set as the best possible evaluation of the curve which is believed to underlie the phenomenon. One important meaning of the term 'best possible evaluation' is the efficient estimation of the coefficients which, postulated by the model, characterize this curve. The important key for doing this, is the 'best' (in a certain way) estimation of the covariance matrix of observations which is supposed here to be the same for all individuals. The purpose of this thesis is to investigate certain methods which have been suggested to be appropriate for this problem both theoretically and empirically as well as to prove certain results concerning the problem of efficiency itself when an estimate of the covariance matrix of observations, irrespective of the method which calculated it, is at hand. More specifically, it is shown that REstricted Maximum Likelihood (REML) gives in general estimates of the regression coefficients whose estimated variance lies nearer to their true value quite independently from the form of the covariance matrix, than Maximum Likelihood (ML) and it is proved that the general estimated scatter of the estimated regression coefficients is greater for REML. When we measure more than one characteristics on time and for a special class of covariance models, the property that the REML estimates of the variance of estimated regression coefficients are larger than the corresponding ML, which holds for one characteristic, is lost. Asymptotically however, the two methods give identical results. Another method which is not based on the adoption of a certain parametric form for the covariance matrix is considered. A new method is suggested for its optimal choice and a comparison is made between this and three other already known methods. Empirical results suggest that it retains a very good balance between the variance of the regression coefficients and their real and optimal value for the majority of the covariance models which have been tested as possible population covariance matrices. Finally, upper and lower bounds of different types of efficiency are obtained by assuming that an estimate of the population covariance matrix has already been calculated and is not distant from the true covariance matrix more than a certain constant.