Augmenting Existing Data in Multiple Regression

Abstract

Multiple regression analysis can be used to analyze data from undesigned, non-orthogonal experiments, to provide a prediction equation which may be adequate for many purposes. However, it is impossible to separate the effects for those independent variables which are highly correlated with each other. Thus, it is often desirable to augment, in an efficient manner, the existing data with a fixed number of experimental runs such that the independent variables are made more orthogonal to each other. In this study, such methods have been developed for a linear model with no interactions among the independent variables. A solution is obtained for augmenting data within a region defined by rectangular limits. A theoretical solution is obtained such that the volume of the simultaneous confidence region for the estimates of the regression coefficients is minimized and the maximum variance of a predicted response in the region is minimized under the restriction that the correlations among the independent variables are eliminated, in order that their effects can be estimated and tested separately. With a moderate number of additional data points, these augmented designs can be constructed such that the desired conditions are approximately fulfilled with a considerable gain over the information contained in the original data.

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