COMPARATIVE PRECISION IN LINEAR STRUCTURAL RELATIONSHIPS
Abstract
The problem of comparing the precisions of p((GREATERTHEQ) 2) instruments and of selecting the most precise (the best) instrument are of considerable practical interest. We assume that the measurements taken on the instruments follow a linear structural relationship model with slopes (beta)(,0),(beta)(,1),...,(beta)(,p-1) and an error covariance matrix (SIGMA)(,e) = Diag((sigma)(,0)('2),...,(sigma)(,p-1)('2)). The instrument with slope (beta)(,0) and error variance (sigma)(,0)('2) is considered to be a standard, and is called "the control". Without loss of generality, we assume (beta)(,0) = 1. The precision (pi)(,i) of instrument i is then defined to be (pi)(,i) = (beta)(,i)('2)(sigma)(,i)('-2), i = 0,...,p-1. For p = 2, the unrestricted model is not identifiable. Consequently, we examine four special cases: (1) the ratio R = (sigma)(,1)('-2)(sigma)(,0)('2) of the error variances is known; (2) the slope (beta)(,1) is equal to 1; (3) the relative precision (tau)(,0) of the control is known; (4) there exist independent consistent estimators of (sigma)(,0)('2) and (sigma)(,1)('2). The test statistics for comparing the precisions of two instruments are derived by transforming the problem of comparing the precisions between these two instruments into a problem of testing a certain correlation coefficient. The power function of the resulting tests is evaluated. For p (GREATERTHEQ) 3, the linear structural model is identifiable. We examine four cases as follows: (1) no constraints on the parameters; (2) the error variance ratios R(,1),...,R(,p-1) are known, where R(,i) = (sigma)(,i)('-2)(sigma)(,0)('2); (3) the slopes (beta)(,1),...,(beta)(,p-1) are all equal to 1; (4) the relative precision (tau)(,0) of the control is known. In each of cases (1) to (4), we apply a type of rule originally suggested by Paulson to select the most precise instrument among p instruments. In cases (2) to (4), the statistics used for comparing each instrument with the control are based on the statistics derived for p = 2, while for case (1), the statistic is based on the maximum likelihood estimators of the parameters. We also discuss a confidence interval for the ratio (psi) = (pi)(,1)(pi)(,0)('-1) of the precisions (pi)(,0) and (pi)(,1) in each of cases (1) to (4) for p = 2. For p (GREATERTHEQ) 3, we find asymptotic joint confidence regions for (psi)(,1),...,(psi)(,p-1) and for (pi)(,0),...,(pi)(,p-1), respectively, where (psi)(,i) = (pi)(,i)(pi)(,0)('-1).
Degree
Ph.D.
Subject Area
Statistics
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