The objectives of this proposal are to develop new methodology in the area of longitudinal data analyses with multivariate responses, and to concentrate on models that have biological interpretations. Multivariate longitudinal data models are models where more than one response variable is observed on each subject at each observation time. The data structure may be unbalanced where different subjeCts are observed at different times and some of the response variables may be missing at an observation time. The previous research of the Principal Investigator on multivariate unequally spaced Iongitudinal data models will be extended to include multivariate continuous time AR(2) structures. The reason for this extension arose from the application of the multivariate AR(1) structure to a study of the effects of diet and weight on blood lipids and insulin in Hispanics and non-Hispanics males and females. After preliminary studies which indicated that the weight of a subject may influence the rate of change of another variable such as blood insulin level, there were hypotheses that suggest that the rate of weight gain or loss may also be important. Since the Continuous time multivariate AR(2) model includes the rate of change of a predictor variable as part of the state equation, it would be possible to test whether the rate of change of a response variable affects the rate of Change of another response variable. Another area of interpreting longitudinal data analyses is in interpreting the effects of time varying covariates. It is possible to decompose a time varying covariate into two covariates. The first is a fixed covariate that is the average of the time varying covariate for a given subject, and the second is a time varying covariate that is the deviation from this mean. The interpretation of the first coefficient would be the effect of this covariate across subjects. The interpretation of the time varying covariate. with the mean subtracted. would be the effect of this covariate within subjects. The difference is """"""""heavier people have higher insulin levels"""""""" vs. """"""""if a person gains weight, his or her insulin level will have a tendency to increase"""""""". The new methodology will be developed into computer software that will be distributed on request at no Cost, as has been The policy in the past. The multivariate continuous time software that handles unequally spaced and unbalanced data that is now available from the Principal Investigator is not available from any commercial source.
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