Variance nonconstant

Garden, J. S., Mitchell, D. G., and Mills, W. N., Nonconstant Variance Regression Techniques for Calibration-Curve-Based Analysis, Anal. Chem. 52, 1980, 2310-2315. 

Garden JS, Mitchell DG, Mills WH (1980) Nonconstant variances regression techniques for calibration-curve-based analysis. Anal Chem 52 2310... 

Hypothetical calibration data showing replicate standard analyses with (a) constant and (b) nonconstant variance. 

Correction for nonconstant variance. To correct for nonconstant variance, it is necessary to weight standard measurements according to their local variance, S. For each standard concentration the variance is determined by repetitive analysis at that level, and a weighting factor, w = 1/s, is calculated. 

When dealing with pharmacokinetic data, it is actually quite rare for the assumption of constant variance to be met. When the observations exhibit nonconstant variance, the data are heteroscedastic and the basic model needs to be modified to... 

We will consider Xi (months) as the main predictor value with the greatest value range, 1 through 12. Note, by a t-test, each independent predictor variable is highly significant in the model (p < 0.01). A plot of the e,s vs. xi, presented in Figure 8.11, demonstrates, by itself, a nonconstant variance. Often, this pattern is masked by extraneous outlier values. The data should be cleaned of these values to better see a nonconstant variance situation, but often the Modified Levene will identify a nonconstant variance, even in the presence of the noise of outlier values. 

The method of least squares is a very popular procedure wherein a curve can be drawn through a set of data in an attempt to extract the essential nonrandom variation of data responding to various levels of an inpnt variant (Fignre 4.2.7). Although a very popular technique, the method of least squares is only valid for data that is linear and has a constant variance. For nonlinear relationships or nonconstant variances, modifications of the technique are available, but they, too, have limitations. 

Lavagnini, L Favaro, G. and Magno, F. Nonlinear and nonconstant variance calibration curves in analysis of volatile organic compounds for testing of water by the purge-and-trap method coupled with gas chromatography/mass spectrometry. Rapid Communications in Mass Spectrometry 2004,18 (12), 1383-1391. 

These plots can also provide information about the assumption of constant error variance (Section III) made in the unweighted linear or nonlinear least-squares analyses. If the residuals continually increase or continually decrease in such plots, a nonconstant error variance would be evident. Here, either a weighted least-squares analysis should be conducted (Section III,A,2) or a transformation should be found to stabilize the error variance (Section VI). 

Davidian, M. and Haaland, P.D. Regression and calibration with nonconstant error variance. Chemometrics and Intelligent Laboratory Systems 1990 9 231-248. 

The topic of nonlinear calibration for LBAs, such as immunoassays, has been reviewed in detail in a number of publications [4,8,9,15 17]. Typically, immunoassay calibration curves are inherently nonlinear [9]. Because the response error relationship is a nonconstant function of the mean response, weighting is needed to account for the heterogeneity in response variances. The four- or five-parameter logistic models are accepted widely as the standard models for fitting nonlinear sigmoidal calibration data [3 5,8,9,16,17], This model can be described... 

We have discussed transforming y and x values to linearize them, as well as removing effects of serial correlation. But transformations can also be valuable in eliminating nonconstant error variances. Unequal error variances are often easily determined by a residual plot. For a simple linear regression, y = ho + hi JCi e, the residual plot will appear similar to Figure 8.8, if a constant variance is present. 

Once a transformation is determined for the regression, substitute y for y and plot the residuals. The process is an iterative one. It is particularly important to correct a nonconstant when providing confidence intervals for prediction. The least squares estimator will still be unbiased, but no Imiger for a minimum variance probability. 

Equation (5.41) can therefore be considered as the ANOVA (Analysis of Variances) decomposition of T(x) and has several important properties (Sobol 2001). The expected value of all nonconstant component functions in Eq. (5.41) is zero and the terms in (5.41) are orthogonal (SoboT 2001). The notation of the zeroth, first, second order, etc. in the HDMR expansion should not be confused with the terminology of a Taylor series (see Eq. (5.3)) since the HDMR expansion is always of finite order (Rabitz and Ah 2000). The higher-order terms reflect the cooperative effects of increasing numbers of input variables acting together to influence the output 7(x). The HDMR expansion is computationally very efficient if higher-order... 

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