Variance weighting

Fig. 8. Plot of data from patients having Hver diseases A or B or unknown X (a) on two blood enzymes (b) scores of points on the first two eigenvectors obtained from an eight-dimensional enzyme space and (c) eigenvector plot of the variance weighted data. Variance weights ranged from 3.5 to 1.2 for the eight blood enzymes measured. A weight of 1.0 indicates no discrimination information (22).

This GLS estimator is akin to inverse variance-weighted regression discussed in Section 8.2.3. Again there is a limitation V can be inverted only when the number of calibration samples is larger than the number of predictor variables, i.e. spectral wavelengths. Thus, one either has to work with a limited set of selected wavelengths or one must apply other solutions which have been proposed for tackling this problem [5]. 

From Eq. (6.14) it suggests itself that squared variance weighting wyi 1 /s2 should be applied. In analytical practice, frequently relative weights 1... 

The effective variance weighting scheme (Equation 12) can be used also in this model to down-weight the chemical elements with high uncertainties. 

This solution provides two benefits. First, it propagates a confidence interval around the calculated source contributions which reflects the cumulative uncertainty of the input observables. The second benefit provided by this "effective variance" weighting is to give those chemical properties with larger uncertainties, or chemical properties which are not as unique to a source type, less weight in the fitting procedure than those properties having more precise measurements or a truly unique source character. 

When feature selection is used to simplify, because of the large number of variables, methods must be simple. The univariate criterion of interclass variance/intraclass variance ratio (in the different variants called Fisher weights variance weights or Coomans weights is simple, but can lead to the elimination of variables with some discriminant power, either separately or, more important, in connection with other variables (Fig. 36). 

If there is no reason to choose one specific measure, the separate (reliable) measures are combined [10] into a single assigned value, being the inverse-variance weighted (for uncertainty) average of the independent measures of analyte concentration. 

Now we can combine several measurements of 0. Ej) to form the variance-weighted least-squares estimate... 

Note 6. Combining the inequality constraint 0] with known Internal Poisson variance has been used by Guinn for total variance estimation in activation analysis ( 2), by Donahue (88) for the same purpose in accelerator mass spectrometry, by Heydorn (18.89) for the analysis of precision of gamma ray spectrum analysis, and by Currie ( ) for optimal weighting in counting experiments. The variance weighted t approximation derives from work by Cochran (90). Cochran s work, however, applied to a somewhat different model than used here. The statistical properties of this extension of his technique have not been studied. 

Several methods have been found adequate for low event count situations. For odds ratio summary measures, the Mantel-Haenszel, Peto, and exact methods appear to work well. The commonly used method in meta-analysis based on inverse variance weights does not perform well in low event count situations because the weights are not stable with low event counts. The Mantel-Haenszel risk difference appears to work well as well. This method has the added benefit in that unlike the methods for the odds ratio, this... 

Fig. 4.19. Scores plot for a set of NOA samples described by sensory QDA data. The QDA data was autoscaled and variance-weighted (see reference for details). Symbols are the same as those used in Fig. 4.14 (from Lin ef a/. 1993, with permission of Elsevier Science).

Fisher-weighting and variance-weighting are different procedures for weighting variables according to their ability to classify samples (see Varmuza 1980). 

Schwartz (1994) did a meta-analysis on three longitudinal and four cross-sectional studies relating PbB to full scale IQ changes in schoolchildren. The methodology involved inverse variance weighting in a random effects model. Schwartz reported that an increase of PbB of 10—20 ng/dl produced a decline of 2.6 (rounding) IQ points for all seven original data sets. The effect... 

The goodness-of-fit S is a measure of the extent to which the calculated model values Cy agree with the observations Oy. Eor miminum-variance weights, S is calculated as... 

In the final cycle, the best estimate of minimum-variance weights, w = should be used, where u = Uy2 refinements against f or f. ... 

A modified approach was used to characterize the performance of a fuzzy similarity measure in comparison with the estimation of the correlation coefficient. Here the variance weighted relative cardinality was superior to the classical method for identification of UV spectra. ... 

There is good consistency in the estimated effect of PM,o across these studies. Effect estimates range between 0.5 and 1.6% increase in daily mortality for each 10- j,g/m increase in PMio concentration. A weighted mean of the study-specific effect estimates, with inverse variance weights (36) gives a combined effect estimate for these studies of 0.7% (95% Cl 0.6-0.9%) increase in daily mortality for each 10- j,g/m increase in PMio. 

Variance Weighted Distance Between Cluster Centers... 

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