Least-squares procedures, weighted analysis

Two procedures for improving precision in calibration curve-based-analysis are described. A multiple curve procedure is used to compensate for poor mathematical models. A weighted least squares procedure is used to compensate for non-constant variance. Confidence band statistics are used to choose between alternative calibration strategies and to measure precision and dynamic range. 

As part of a study of biscyclic dibenzacridines, the crystal structure of 1,2-8,9-dibenzacridine (72) has been investigated by Mason (1957, 1960). The space group was shown to be Pna2v thus, with four molecules in the unit cell, no molecular symmetry is required. The structure was determined from an examination of the weighted reciprocal lattice and trial and error methods, refinement being by two-dimensional least-squares procedures. At the conclusion of the analysis,... 

Correct weighting procedures for least-squares analysis have been discussed in... 

Procedure Briefly, the procedure developed consists of the transformation of the positional parameters of the M4Y4 atoms, together with their standard deviations, to an orthonormal coordinate system. One then performs a least squares analysis in order to adjust rotations about the orthonormal axes to obtain the best fit to an assumed symmetry. One can then test the variance of an observation of unit weight by an appropriate chi square test. Further details are given by Averill et al.2 ... 

D Rodbard, DM Hutt. Statistical analysis of radioimmunoassays and immunoradio-metric (labelled antibody) assays A generalized weighted, iterative, least-squares method for logistic curve fitting. In Radioimmunoassay and Related Procedures in Medicine, Vol I. Vienna International Atomic Energy Agency, 1974, p 165. 

I) Carry out a weighted least-squares analysis of the data u,sing a procedure similar to the one described in Problem 10-14 that allows for uncertainties in both the, v and y variables. Compare your results to those obtained in (k). 

Before the advent of computer technology and the computerized statistical methods for data analysis, a procedure that was employed extensively in the analysis of enzyme kinetic data was linear regression. It is important to point out that linear regression performed by the least squares method should not be used unless the values are weighted. If it is used without proper weighting one can get bad results. 

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