Variance quotient

Fig. 2. Knowing from the/test (variance quotient) that the assumption of the same standard deviation does not apply, we have to balance the disequilibrium by a factor-the weighting factor. Otherwise the calculation will be wrong. The 2-fold standard deviation...

The detectability of critical defects with CT depends on the final image quality and the skill of the operator, see figure 2. The basic concepts of image quality are resolution, contrast, and noise. Image quality are generally described by the signal-to-noise ratio SNR), the modulation transfer function (MTF) and the noise power spectrum (NFS). SNR is the quotient of a signal and its variance, MTF describes the contrast as a function of spatial frequency and NFS in turn describes the noise power at various spatial frequencies [1, 3]. 

Finally, a measure of lack of fit using a PCs can be defined using the sum of the squared errors (SSE) from the test set, flSSETEST = Latest 2 (prediction sum of squares). Here, 2 stands for the sum of squared matrix elements. This measure can be related to the overall sum of squares of the data from the test set, SStest = -Xtest 2- The quotient of both measures is between 0 and 1. Subtraction from 1 gives a measure of the quality of fit or explained variance for a fixed number of a PCs ... 

The condition of variance homogeneity can be proven with the help of statistical tests (/- -test for quotient of variances at the lower and upper end of the calibration range, or better, in the case of rij>5 by using the Bartlett-test, which includes all of the variances in the calibration range). If variance homogeneity is violated, a weighted least squares regression (WLS) should be used. 

Such threshold values are often estimated using no-observed-effect concentrations or levels (NOECs or NOELs). It might be tempting to substitute the individual ECx values in the CA equation (Equation 4.2) with NOELs in order to calculate a mixture NOEL. But this would imply that all NOELs provoke the same, statistically insignificant effect that is, all of them must have been determined in an identical experimental setup (in terms of number of replicates, spacing of test concentrations, variance structure), which is hardly ever the case. Nevertheless, a range of methods, such as TEFs or TEQs (see Chapters 1 and 5), makes use of a CA-like approach and sums up NOEL-based hazard quotients. This introduces an additional source of uncertainty in the risk assessment, which is fundamentally different from the question of whether CA is an appropriate concept for the mixture of interest. 

Different safety factors may have been used in the derivation of the reference values of the individual substances (RfDA deterministic HI thus sums risk ratios that may reflect different percentile values of a risk probability distribution. Assessment and interpretation of the uncertainty in the HI may be severely hampered by this summation of dissimilar distribution parameters. In a probabilistic risk assessment, the uncertainty in the exposure and reference values is often characterized by lognormal distributions. The ratio of 2 lognormal distributions also is a lognormal distribution. The variance in a quotient of 2 random variables can be approximated as follows (Mood et al. 1974, p 181) ... 

The F-test is based on the F-probability distribution curve and is used to test the equality of variances obtained by statistical sampling. The distribution describes the probabilities of obtaining specified ratios of sample variance from the same parent population. Starting with a normal distribution with variance cr, if two random samples of sizes ni and //2 are taken from this population and the sample variances, si and Sz, calculated then the quotient Si /s2 will be close to imity if the sample sizes are large. By taking repeated pairs of samples and plotting the ratio, F= s /s-i, the F-distribution curve is obtained. ... 

K C/Kv°, is typically reported in terms of 95-percent confidence limits on the average value. For high volatility compounds, the 95-percent confidence limits are reported to be 5 to 10 percent of the magnitude of (K C/K °). The regression approach typically used to determine (K /Ky0 and the specification of confidence limits based on the t-distribution imply that the error distributions are assumed to be normal. However, even if the errors of Kvc and Kv° estimates are each normally distributed, the error distribution associated with the quotient of the two is not necessarily normally distributed. The regression approach assumed may produce an estimated value of (Kvc/Kv°) lower than the true value and underestimate the variance. Monte Carlo simulation of the regression approach did not confirm underestimation of (Kvc/Kv°) however, the variances appeared to be underestimated by a factor of 2.3. 

Differences in variance are tested by using an F-distribution, representing the quotient of the mean square between sampies and the mean square within samples. This is called the sample variance ratio. This material is comprehensively treated in the statistical handbooks (Bennett and Franklin. 1954). 

With any functional form beyond a sum or difference, the variance of Y is influenced by the value for the differentials. For a ratio or quotient. 

The quotient of the variances is used for this calculation. Therefore, the method might be called weighting by variance ratio. 

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