Residual plot

When the residuals versus jc are plotted, these values should distribute 

Residual Plots A more formal way to test fire randomness of file oror distribution is by means of a residual plot. A residual is the difference between an actual data point (an expramental value of —rp in this case) and the value predicted by a model. For example, the residual for the data point at —rp = 6.21 (Jimol/min-mg, [F] = lOOmmol/1 is 

This figure shows that the first-order rate equation is not adequate. First, most of the residuals are very large compared to the values of the experimental rates. Second, the residuals are not randomly distributed around zero. The residuals vary in a very systematic fashion with fructose concentration. All of the positive residuals occur at low fmctose concentrations, 500 nunoI/1. The only two negative residuals occur at fructose concentrations above this value. 

We could construct residual plots for variables other than the fructose concentration. For example, it is common to plot the residuals against the measured values of the dependent variable, in this case the rate of fructose disappearance. We shall construct and discuss such a plot shortly, when we test a Michaelis-Menten rate equation against the data. 

We might also construct a plot of the residuals against the technician who ran each experiment, to look for systematic operator error. Another possibility is to examine the residuals against the source of a key raw material, e.g., fructose, if the material was obtained from more than one source. 

It was shown in Chapter 7 how different, subarrays of the residual E can be made and how statistics can be calculated for them. The goals of the analysis often dictate how residuals should be viewed and treated. There are many possibilities and not all of them fit each specific three-way data array. 

In multi-way regression models there are the residuals Ey and Ex(see Chapters 4). These residuals can be used as described in Chapter 7 for calculating a number of different sums of squares. These can also be visualized as described here for component models. 

Let us discuss more about residual plots. Important plots to generate in terms of residuals include  

The residual values, y, — y, = e, plotted against the fitted values, %. This residual scatter graph is useful in  

FIGURE 8.5 Logio colony counts with averages for males. Example 8.1. 

Residuals can be useful in model diagnostics in multiple regression by plotting interaction terms. 

After the model has been constructed and the parameters estimated, it is crucial to evaluate it. In this book, we describe two methods for evaluating models residual plots and lack of fit. Residual plots are discussed in Section 7.8.1 and lack of fit is described in the context of the analysis of variance (ANOVA) table. The error is defined as 

Different types of residual plots are constructed in order to check that there are no trends pertaining to the errors. The error can be plotted versus time, independent variable (X), predicted response variable ( ), in addition to other parameters, and are discnssed below. 

The residual should be normally distributed as described (e N(0, a)). This means that if we divide by the variance, the following normal distribntion would result  

95% of the standardized errors should be in the — 1.96 to 1.96 range and approximately 99% in the —2.576 to 2.576 range. For other degrees of freedom, the t-values should be taken from the t-distribution (see Appendix C). However, because the variance is not constant for each error, this is not exact. Therefore, it is better to use the studentized residuals, 

It is usefule to construct Residual Plots based on the error, e, or the standardized residuals, for many applications. 

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Pigure 8.8a and b, respectively, show fluorescence autocorrelation curves of R6G in ethylene glycol and R123 in water at 294.4 K. The solid lines in these traces are curves analyzed by the nonlinear least square method with Eq. (8.1). Residuals plotted on top of the traces clearly indicate that the experimental results were well reproduced by the... 

With the Nelder-Mead search, 58 full integrations and 0.6 s computer time on a VAX 6000 computer were needed to get the following result with initial and final values, standard deviation, residual plot and statistical information. 

TIME OBSERVED PREDICTED % ERROR RESIDUAL PLOT ... 

Figure 12 Residuals plot for the nonlinear model 5.2.1 Hamaker equation...

Figure 5. L-Alanine. Fit to noisy data. Calculation A. Distribution of residual structure factor amplitudes at the end of the MaxEnt calculation on 2532 noisy data up to 0.463A. Residuals plotted ... 

Homoscedasticity. Unequal variances are recognizable from residual plots as in Fig. 6.8c where frequently ey is a function of x in the given trumpet-like form. In such a case, the test of homoscedasticity can be carried out in a simple way by means of the Hartley test (Fmax test), Fmax = smax/5min> see Sect. 4.3.4 (1). 

The residual plot of the difference between the best computed curve and the experimental data is shown in Figure 8. The largest variance is observed in the vicinity of the maximum isocyanate absorbance and probably arises because of the large changes in concentration which arise during the early stages of cure. 

The weighted residuals plots in Fig. 3.4 are the results of the analyses, where the shape of the reference spectra are matched to those of acceptor and donor spectra by least-squares refinement. Poor shape-analysis leads to high weighted residuals, which can reveal impurities, decomposition, or other artifacts. In the present cases, no difficulties were encountered. 

Fig. 3.4. Microtiter plate UV spectra taken in in acceptor wells. The weighted residuals plots triplicate of propranolol reference, donor, and indicate that the shapes of spectra in donor acceptor solutions, at iso-pH 7.4 in 20% wt/vol and acceptor wells are in agreement with those soy lecithin in dodecane. (a) After 15 h per- in the reference wells, confirming that neither meation time, surfactant-free (b) 3 h, 35 mM decomposition nor impurities were detectable.

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] ... 

Linearity should always be assessed initially by visual inspection of the plotted data and then by statistical evaluation. The linearity of the instrument response needs to be established because without this information it is difficult to attribute causes of non-linearity. The supporting statistical measures include correlation coefficient (r, r2, etc), residual plot, residual standard deviation and significance... 

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). 

Probability plot Q-Q plot P-P Plot Hanging histogram Rootagram Poissonness plot Average versus standard deviation Component-plus-residual plot Partial-residual plot Residual plots Control chart Cusum chart Half-normal plot Ridge trace Youden plot... 

Of particular value in kinetic studies are residual plots using the linearized form of the Hougen-Watson equation. For the model of Eq. (18), for example, we obtain... 

The plot of the residuals of this adsorption constant, using the data of Fig. 17, is shown in Fig. 18. A substantial trending effect is evident (it is also evident from Fig. 17). This is of interest in itself, but it is especially desirable to utilize the information of Fig. 18 to learn how the model must be modified to remove the observed defect. The other residual plots of the section can also assist this objective. This topic, called model-building, is treated in Section V. 

Figure 30 portrays the grid of values of the independent variables over which values of D were calculated to choose experimental points after the initial nine. The additional five points chosen are also shown in Fig. 30. Note that points at high hydrogen and low propylene partial pressures are required. Figure 31 shows the posterior probabilities associated with each model. The acceptability of model 2 declines rapidly as data are taken according to the model-discrimination design. If, in addition, model 2 cannot pass standard lack-of-fit tests, residual plots, and other tests of model adequacy, then it should be rejected. Similarly, model 1 should be shown to remain adequate after these tests. Many more data points than these 14 have shown less conclusive results, when this procedure is not used for this experimental system.

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. 

Systematic deviations from the assumed model yield systematic patterns in the residuals and can, therefore, be detected by checking independence of the residuals. Residuals plot, particularly the residuals e, versus expected values y plot, is also well suited to detect non-linearities since the plot will show curved patterns instead of randomness. ... 

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. 

Measurement Residual Plot (Model, Sample, and Variable Diagnostic) The residue are the portion of the original data that is not described by a given number of PCs. Tlie number of elements in the residuals is the same as the number of measurement variables in the original matrix. To understand the residue, consider appl> ing PCA to a data set. Wlien the first PC is estimated, it is jKissible to determine the variation that is described by this first dimension or factor. The contribution of this PC can be subtracted from the original data set, resulting in a residual for each sample. This process can be... 

The residual plots can also be used to identify unusual samples. When one or more samples have a residual vector that is significantly different in magnitude or shape from the majority, they should be examined more closely. 

Measunment Residual Plot (Model, Sample, and Variable Diagnostic) Figure 4.3 c shows the residuals after using 1, 3, and 4 principal components, respctively. The residuals have been convened back from the autoscaled ui to the original measurement imits to facilitate comparisons. In this exampfe these plots do not definitively indicate the inherent dimensional-it> of the dan set. 

Measurement Residual Plot (Model, Sample, and Variable Diagnostic) The residuals plots (not shown) after two principal components showed structure, but after three principal components no significant structure remained. 

Measurment Residual Plot There are residual plots for each unknown sample for every SIMCA model. Tlie residual spectra for samples that belong to a class are expected to resemble in magnitude and shape normally distributed noise as fotsrd in the training set Depending on the structure of the residuals, it may be possible to identify failures in the instrument (e.g., excessive noise) or chemical differences between tlie calibration and unknown samples (e.g., peaks in the residuals). The residual plot may help identify why a sample is not classified iiso any given class. 

Measmsment Residual Plot The PCA residuals for the four unknowns tis-ing the TB. model are shown in Figure 4.90 with the range of the calibration residuals own as solid horizontal lines. Unknowns 1, 2, and 4 have large residuals, which is reflected in the values. Unknown 3 has a small residual... 

Concentration Residuals vs. Predicted Concentration Plot (Model and Sample Diagnostic) The plot of concentration residuals (c — c) versus predicted concentration gives similar infonnation to tlie predicted vs. known plots. Ideally this plot has a scatter of points about a line with slope and intercept equal to zero. Figure 5-14 shows the concentration residual versus predicted plots that correspond to the scenarios presented in Figure 5.12. The residual plot enhances features that can go unnoticed in the actual versus predicted plots. 

CalibraJwn Measurement Residual Plot (Model Diagnostic) After the pure specta are estimated (S), they are used with the original C matrix to generate esti es of the mixture spectra (R CS). These are then used to calculate a caUbration residual matrix which contains the portion of the mixture spectra that are not fit by the estimated pures (Equation 5.18). 

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