Residuals unweighted

Values V (initially set) Measurand values Y arith. means Weighting factor 1/s Residuals unweighted... 

The unweighted least squares analysis is based on the assumption that the best value of the rate constant k is the one t,hat minimizes the sum of the squares of the residuals. In the general case one should regard the zero time point as an adjustable constant in order to avoid undue weighting of the initial point. An analysis of this type gives the following expressions for first-and second-order rate constants... 

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). 

The qnantities appearing in this equation may be best nnderstood with the aid of an example. Let the unweighted residuals for a set of 10 measurements be arranged from the largest negative to the largest positive ... 

After outliers have been purged from the data and a model has been evaluated visually and/or by, e.g. residual plots, the model fit should also be tested by appropriate statistical methods [2, 6, 9, 10, 14], The fit of unweighted regression models (homoscedastic data) can be tested by the ANOVA lack-of-fit test [6, 9]. A detailed discussion of alternative statistical tests for both unweighted and weighted calibration models can be found in Ref. [16]. The widespread practice to evaluate a calibration model via its coefficients of correlation or determination is not acceptable from a statistical point of view [9]. 

Figure 3. Time-dependent anisotropy for anthracene-labeled polyisoprene in dilute hexane solution. The experimental anisotropy was obtained by setting the delay between the excitation and probe pulses to a given position and then varying the polarization of the probe beam. In the bottom portion of the figure, the smooth curve through the data is the best fit to the Hall-Helfand model(Ti=236 ps, t2=909 ps, and r(0)=0.250). Unweighted residuals for the best fit to this model are shown along with the experimental error bars in the top portion of the figure. Note that the residuals are shown on an expanded scale (lOx). The instrument response function is indicated at the left.

One method for dealing with heteroscedastic data is to ignore the variability in Y and use unweighted OLS estimates of 0. Consider the data shown in Fig. 4.2 having a constant variance plus proportional error model. The true values were volume of distribution = 10 L, clearance = 1.5 L/h, and absorption rate constant = 0.7 per/h. The OLS estimates from fitting a 1-compartment model to the data were as follows volume of distribution = 10.3 0.15L, clearance = 1.49 0.01 L/h, and absorption rate constant =0.75 0.03 per h. The parameter estimates themselves were quite well estimated, despite the fact that the assumption of constant variance was violated. Figure 4.3 presents the residual plots discussed in the previous section. The top plot, raw residuals versus predicted values, shows that as the predicted values increase so do the variance of the residuals. This is confirmed by the bottom two plots of Fig. 4.3 which indicate that both the range of the absolute value of the residuals and squared residuals increase as the predicted values increase. 

However, when the number of replicates is small, as is usually the case, the estimated variance can be quite erroneous and unstable. Nonlinear regression estimates using this approach are more variable than their unweighted least-squares counterparts, unless the number of replicates at each level is 10 or more. For this reason, this method cannot be supported and the danger of unstable variance estimates can be avoided if a parametric residual variance model can be found. 

Table 8.13 is the standard least squares model, and hence, contains exactly the same data as in Table 8.4. Notice that the MSe for the weighted values is MSe = 1 05, but for the unweighted values, is MSe = 20.2, which is a vast improvement of the model. Yet, if one plots the weighted residuals, one sees that they still show the same basic form of the unweighted residuals. This signals the need for another iteration. This time, the researcher may be better-off using a regression approach.

Albeit based on F values and hence mostly of historical value, the most popular one is the unweighted residual factor based on F R (or / 1 in SHELXL). 

We first must establish that the data are homoscedastic in order to apply the simple (unweighted) linear regression formulae. In this case simple examination of the data shows that there is no detectable trend of the absolute values Yj j — Yj j with increasing Xj, as we would expect since the range of values of Yj would increase with Xj if heteroscedasticity were present, compare Figure 8.9. (This preliminary conclusion of homoscedacity is confirmed after the event by examining the residuals calculated on the basis of this assumption (columns 7 and 8) that demonstrate no trend of rjj with Xj, the implications of these residuals for the goodness of fit are discussed later.)... 

Apply simple (unweighted) linear regression (Equations [8.19] and [8.23]) calculate and plot residuals (Yj-Yj jxed) vs Xj to search for evidence of heteroscedacity and/or nonlinearity of the data (Figure 8.9). At this point determine whether or not the cahbration obtained is fit for purpose the statistical test for goodness of fit (Equations [8.33-8.38]) wiU be helpful here, as will the calculated confidence limits for x (Equation [8.32]). 

If the tests in (2) indicate nonlinearity but no significant deviation from homoscedacity, perform an unweighted quadratic regression (Section 8.3.6). A plot of residuals can give a qualitative assurance that the situation has improved but the quantitative aspect is more complex and generally requires access to a suitable computer-based statistical package. 

Goodness of fit (correlation) for both unweighted and weighted linear least-squares regression can be measured using i) correlation coefficient (R) and/or the coefficient of determination (R ) and ii) F-test of the residuals calculated for the fit line. The F-test is preferred over R or R, since modem analytical instruments commonly generate... 

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