Residual, variance

If the calibration is repeated and a number of linear regression slopes b are available, these can be compared as are means. (See Section 1.5.1, but also Section 2.2.4.) 

Item Value Equation Item Value Equation 

18) Vmisc contains all other variance components, e.g., that stemming from the v-dependent Sy (heteroscedacity). (See Fig. 2.5.) 

Taking the square root of Vres. one obtains the residual standard deviation, i res, a most useful measure  

5res has the same dimension as the reproducibility and the repeatability, namely the dimension of the measurement those are the units the analyst is most familiar with, such as absorbance units, milligrams, etc. 

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Since a series of t-tests is cumbersome to carry out, and does not answer all questions, all measurements will be simultaneously evaluated to find differences between means. The total variance (relative to the grand mean xqm) is broken down into a component Vi variance within groups, which corresponds to the residual variance, and a component V2 variance between groups. If Hq is true, Vi and V2 should be similar, and all values can be pooled because they belong to the same population. When one or more means deviate from the rest, Vj must be significantly larger than Vi. 

The residual variance Vres summarizes the vertical residuals from Figure 2.4 it is composed of... 

Figure 2.5. Three important components of residual variance. The residuals are graphed versus the independent variable x.

This empirical test is based on the so-called Scree-plot which represents the residual variance as a function of the number of eigenvectors that have been extracted [42]. The residual variance V of the r -th eigenvector is defined by ... 

Fig. 31.15. Scree-plot, representing the residual variance V as a function of the number of factors r that has been extracted. The diagram is based on a factor analysis of Table 31.2 after log double-centering. A break point occurs after the second factor, which suggests the presence of only two structural factors, the residual factors being attributed to noise and artefacts in the data.

It is assumed that the structural eigenvectors explain successively less variance in the data. The error eigenvalues, however, when they account for random errors in the data, should be equal. In practice, one expects that the curve on the Scree-plot levels off at a point r when the structural information in the data is nearly exhausted. This point determines the number of structural eigenvectors. In Fig. 31.15 we present the Scree-plot for the 23x8 table of transformed chromatographic retention times. From the plot we observe that the residual variance levels off after the second eigenvector. Hence, we conclude from this evidence that the structural pattern in the data is two-dimensional and that the five residual dimensions contribute mostly noise. 

As in example 1, the explained variance (the total variance minus the residual variance) is calculated by comparing the true process data with estimates computed from a reference model. This explained variance can be computed as a function of the batch number, time, or variable number. A large explained variance indicates that the variability in the data is captured by the reference model and that correlations exist among the variables. The explained variance as a function of time can be very useful in differentiating among phenomena that occur in different stages of the process operations. 

By means of this reduction of dimensions the information in the form of variance is subdivided into essential contributions (common and specific variance) on one hand and residual variance on the other ... 

Both assumptions are mainly needed for constructing confidence intervals and tests for the regression parameters, as well as for prediction intervals for new observations in x. The assumption of normal distribution additionally helps avoid skewness and outliers, mean 0 guarantees a linear relationship. The constant variance, also called homoscedasticity, is also needed for inference (confidence intervals and tests). This assumption would be violated if the variance of y (which is equal to the residual variance a2, see below) is dependent on the value of x, a situation called heteroscedasticity, see Figure 4.8. 

Besides estimating the regression coefficients, it is also of interest to estimate the variation of the measurements around the fitted regression line. This means that the residual variance (j2 has to be estimated which can be done by the classical estimator... 

The denominator n 2 is used here because two parameters are necessary for a fitted straight line, and this makes s2 an unbiased estimator for a2. The estimated residual variance is necessary for constructing confidence intervals and tests. Here the above model assumptions are required, and confidence intervals for intercept, b0, and slope, b, can be derived as follows ... 

Similar to Equation 4.26, an unbiased estimator for the residual variance cr2 is... 

The applicability of Eq. (45) to a broad range of biological (i.e., toxic, geno-toxic) structure-activity relationships has been demonstrated convincingly by Hansch and associates and many others in the years since 1964 [60-62, 80, 120-122, 160, 161, 195, 204-208, 281-285, 289, 296-298]. The success of this model led to its generalization to include additional parameters in attempts to minimize residual variance in such correlations, a wide variety of physicochemical parameters and properties, structural and topological features, molecular orbital indices, and for constant but for theoretically unaccountable features, indicator or dummy variables (1 or 0) have been employed. A widespread use of Eq. (45) has provided an important stimulus for the review and extension of established scales of substituent effects, and even for the development of new ones. It should be cautioned here, however, that the general validity or indeed the need for these latter scales has not been established. 

Once the general library has been constructed, those products requiring the second identification step are pinpointed and the most suitable method for constructing each sublibrary required is chosen. Sublibraries can be constructed using various algorithms including the Mahalanobis distance or the residual variance. The two are complementary, so which is better for the intended purpose should be determined on an individual basis. 

Constructing the library. First, one must choose the PRM to be used (e.g. correlation, wavelength distance, Mahalanobis distance, residual variance, etc.), the choice being dictated by the specific purpose of the library. Then, one must choose constraction parameters such as the math pretreatment (standard... 

Distance-based methods possess a superior discriminating power and allow highly similar compounds (e.g. substances with different particle sizes or purity grades, products from different manufacturers) to be distinguished. One other choice for classification purposes is the residual variance, which is a variant of soft independent modeling of class analogy (SIMCA). 

Raw materials Mahalanobis distance and residual variance Classification models development 24... 

Raw materials Correlation coefficient, Mahalanobis distance, residual variance Libraries and sublibraries 63... 

Herbal medicines Wavelength distance, residual variance, Mahalanobis distance and SIMCA PRMs comparison 70... 

N.K. Shah and P.J. Gemperhne, Comhination of the Mahalanobis distance and residual variance pattern recognition techniques for classification of near-infrared reflectance spectra, Anal. Chem., 62, 465-470 (1990). 

If there is no theory available to determine a suitable transformation, statistical methods can be used to determine a transformation. The Box-Cox transformation [18] is a common approach to determine if a transformation of a response is needed. With the Box-Cox transformation the response, y, is taken to different powers A, (e.g. -2transformed response can be fitted by a predefined (simple) model. Both an optimal value and a confidence interval for A can be estimated. The transformation which results in the lowest value for the residual variance is the optimal value and should give a combination of a homoscedastical error structure and be suitable for the predefined model. When A=0 the trans-... 

The appropriate test when comparing more than two means is analysis of variance (ANOVA). The essential process in ANOVA is to split up, or decompose, the overall variance in the data. This variability is due to differences between the means due to the treatment effect (between-group variance) and that due to random variability between individuals within each group (within-group variance, sometimes called unexplained or residual variance), hence the name analysis of variance. ... 

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