Regression residual plots

For the basic evaluation of a linear calibration line, several parameters can be used, such as the relative process standard deviation value (Vxc), the Mandel-test, the Xp value [28], the plot of response factor against concentration, the residual plot, or the analysis of variance (ANOVA). The lowest concentration that has been used for the calibration curve should not be less than the value of Xp (see Fig. 4). Vxo (in units of %) and Xp values of the linear regression line Y = a + bX can be calculated using the following equations [28] ... 

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain some insight from a residual plot, which depicts the difference between the predicted and experimental values of cA using the rate constants calculated from the regression analysis. Figure 3.10 shows a random distribution of residuals for a second-order reaction, but a nonrandom distribution of residuals for a first-order reaction (consistent overprediction of concentration for the first five datapoints). Consequently, based upon this analysis, it is apparent that the reaction is second-order rather than first-order, and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is much smaller for second-order kinetics than for first-order kinetics (1.28 X 10-4 versus 5.39 xl0 4). 

FIGURE 4.8 Examples of residual plots from linear regression. In the upper left plot, the residuals are randomly scattered around 0 (eventually normally distributed) and fulfill a requirement of OLS. The upper right plot shows heteroscedasticity because the residuals increase with y (and thus they also depend on x). The lower plot indicates a nonlinear relationship between x and y. 

Examination of Data. At this point, examination of the plot of regression residuals verses transformed amount showed two conditions First, the condition of constant variance across the... 

Further analysis of linearity data typically involves inspection of residuals for fit in the linear regression form and to verify that the distribution of data points around the line is random. Random distribution of residuals is ideal however, non-random patterns may exist. Depending on the distribution of the pattern seen in a plot of residuals, the results may uncover non-ideal conditions within the separation that may then help define the range of the method or indicate areas in which further development is required. An example of residual plot is shown in Figure 36. There was no apparent trend across injection linearity range. 

Duplicate injections shown. (A) Fitting the data by nonlinear regression analysis yields a of 5.6 1.0 pM. (B) Data from A, plotted as a sigmoidal curve to better show the fit at low titrant concentrations. (C) Residuals plotted as absolute and (D) as percent of signal. 

There are no obvious features in the residuals plot to suggest that the model is unsuitable. However, some of the residuals are rather large so it is prudent to estimate how good our regression model is and what confidence we can have in data predicted from it. Estimates of the confidence in the determined slope and... 

H0 (suitability of the regression model) is examined by means of the test quantity Ftest=MSLoF/MSPE, which is compared with the corresponding value of the F-distri-bution with k-2 and n-k degrees of freedom at the significance level a. For Ftest > Fa-k-i-n-k H0 is rejected. A lack of linearity in the relation (1) can also be indicated by this test result. The LOF-test should be used in combination with the residual plot. 

Figure 2.3. Linear regression analysis with Excel. Simple linear regression analysis is performed with Excel using Tools -> Data Analysis -> Regression. The output is reorganized to show regression statistics, ANOVA residual plot and line fit plot (standard error in coefficients and a listing of the residues are not shown here).

It should be stressed that LTS regression does not throw away a certain percentage of the data. Instead, it finds a majority lit, which can then be used to detect the actual outliers. The purpose is not to delete and forget the points outside the tolerance band, but to study the residual plot in order to find out more about the data. For instance, we notice the star 7 intermediate between the main sequence and the giants, which might indicate that this star is evolving to its final stage. 

The linearity of (a part of) the range should be evaluated to check the appropriateness of the straight-line model. This can be achieved by a graphical evaluation of the residual plots or by using statistical tests. It is strongly recommended to use the residual plots in addition to the statistical tests. Mostly, the lack-of-fit test and Mandel s fitting test are used to evaluate the linearity of the regression line [8, 10]. The ISO 8466 describes in detail the statistical evaluation of the linear calibration function [11]. 

A much better way to evaluate the fitness of the regression model is by evaluating the residual plots. The residuals (e,) are plotted versus X or versus Y. Both graphs provide equivalent information for straight-line models. 

The residual plot in Figure 6.2a represents a linear relationship between the response and the independent variable. The residuals are randomly scattered around the centerline. The U-shaped residual plot of Figure 6.2b indicates that a curvilinear regression model should be fitted through the data points. Any departure from the symmetric bar shape of Figure 6.2a may indicate that the chosen regression model is inappropriate. The residual plots are also used to detect violations of other basic assumptions. This is discussed further in the text. 

The multiple linear regression models are validated using standard statistical techniques. These techniques include inspection of residual plots, standard deviation, and multiple correlation coefficient. Both regression and computational neural network models are validated using external prediction. The prediction set is not used for descriptor selection, descriptor reduction, or model development, and it therefore represents a true unknown data set. In order to ascertain the predictive power of a model the rms error is computed for the prediction set. 

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 14-26 A, A scatter plot with the Deming regression line (solid line) with an outlier (filled point). The dotted straight line is the diagonal, and the curved dashed lines demarcate the 95% confidence region. B, Standardized residuals plot with indication of the outlier.

Analysis of variance of the regression did not indicate any significant lack of fit, Lack.of-fit 4 -74 pCnt g gj (a = 5 %). The residual plots in Fig. 12.10 do not show abnormal behaviour. It was therefore assumed that the variation in yield was adequately described by the model. The isoresponse contour projections are shown in Fig.12.11... 

Residual analysis is of vital importance in any regression analysis. A residual analysis entails the careful evaluation of the differences between the observed values and the predicted values of the dependent variable after fitting a regression model to the data. Residual plots are used interalia with a view to identifying any undetected tendencies in the data, as well as outliers and fluctuation in the variance of the dependent variable (21). However, interpretation of such residual plots requires great care on account of the possible degree of subjectivity involved therein. 

Partial residuals are produced with GAM, and not the usual residual plots. Plots of residuals and functions of residuals are useful particularly for identifying patterns in the data that may suggest heterogeneity of variance or bias due to deterministic model misspecification or misspecifications of the regression variables. One particular form of bias that may exist occurs when a predictor variable is included in the model in a linear form when it actually has a curvilinear or nonlinear relationship with the response variable. A plot used by Ezekiel (23) and later referred to as a partial residual plot by Larsen and McCleary (24) is useful for this purpose. Partial residuals are defined as... 

In addition, in least-squares residual plots of S versus Xp the slope of the regression line of Si against Xj can be expected to be zero. In contrast, the regression of r against Xj should have a slope equal to the coefficient of Xj when the full model is fitted. This property of partial residuals makes these plots useful in assessing the extent of possible nonlinearity in a certain predictor (25). If the slope of the plot of r against Xj approximately equals the coefficient obtained from a ht of the full model, the specihcation of Xj in the regression model can be assumed to be correct. 

Looking at the original calibration plot there certainly seems to be well-defined curve in the data rather than a random distribution of calibration points around the regression line and hence a residual plot with a clear trend in the data is not too surprising. Note, however, it is not always easy to tell and hence a residual plot is very useful. Take the calibration plot in example 5.1. A close look at the plot suggests that there, too, may be a gentle curvature in the calibration data. The residual plot in figure 5.8 shows reasonable scatter of data and hence there is no reason to reject the linear calibration model. 

Inspect a residual plot and recalculate the regression, if necessary, after remeasuring solutions that give apparent outliers and after defining the linear range. 

In contrast, the data in the top plot of Fig. 4.2 using a constant residual variance model led to the following parameter estimates after fitting the same model volume of distribution =10.2 0.10L, clearance = 1.49 0.008 L/h, and absorption rate constant = 0.71 0.02 per h. Note that this model is the data generating model with no regression assumption violations. The residual plots from this analysis are shown in Fig. 4.4. None of the residual plots show any trend or increasing variance with increasing predicted value. Notice that the parameter estimates are less biased and have smaller standard errors than the estimates obtained from the constant variance plus proportional error model. 

Figure 4.13 Scatter plot of maximal change in albumin concentration (top) in patients dosed with XomaZyme-791 and model predicted fit (solid line). Data were fit to a Emax model using ordinary least-squares. Bottom plot is residual plot of squared residuals divided by MSE against predicted values. Solid line is least-squares regression line and dashed lines are 95% confidence interval.

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