Optimum designs under experimental constraints for a covariate model and an intra-class regression model
Consider the basic covariate model with one treatment at I levels and additional first-order regression on a cofactor with values in [-1,1]. Mathematically, we are dealing with random variables \(X(i,t)=\beta_ i+\alpha t+\epsilon (i,t),\) \(i=1,...,I\) and \(t\in [-1,1]\). Here \(\beta_ i\) denotes the fixed effect of treatment level i, \(\alpha\) the regression parameter and \(\epsilon\) (i,t) the error random variable which is assumed to have mean 0 and variance \(\sigma^ 2\) for all i and t. Also, we consider the corresponding intra-class regression model where the slope of the regression line depends on the treatment level, i.e., \(X(i,t)=\beta_ i+\alpha_ it+\epsilon (i,t).\) In applied design problems often upper and lower bounds on the number of replications for each treatment level are given reflecting the limited availability of some levels and the need for accurate estimators, respectively. With respect to these constraints, the author has derived necessary and sufficient conditions for D-optimum designs for inference on (i) the treatment effects, (ii) the regression parameters and (iii) all parameters. Also, under these conditions, simple finite algorithms generating optimum designs have been presented.
finite algorithms generating optimum designs
scientific article; zbMATH DE number 3923906