Diagnostic Plots

In line with the analysis above, diagnostic plots were computer generated to depict the various scenarios in terms of N = f(C), M = f(Q, — ln(l — Q =f(time), and C = f(time). 

Number of polymer chains vs conversion. B Number average molecular weight vs conversion. C Rate of polymerization vs time. D Conversion vs time. Diagnostic plots for instantaneous initiation (At), with various chain transfers (A2a, A2b, A2c) and for slow initiations (A3a, A3b) 

As shown in Fig. IB, slow initiation leads to higher than theoretical mol. wts. at low conversions, but the effect diminishes with increasing conversion. Chain transfer to monomer is difficult to discern at low molecular weights (or conversions), but it becomes more pronounced at higher molecular weights, particularly with zero order chain transfer, as pointed out for example in [1, 20, 51]. 

Since chain transfer to monomer does not change the number of growing chains, the conversion vs time (Fig. ID) and rate vs time (Fig. 1C) plots run very close to the theoretical one. The only exception is the scenario with slow initiation (A3a, Fig. 1C) which shows a noticeable downward shift after a brief induction period. The magnitude of this shift depends on the extent of slow initiation. 

The scenarios in series B concern initiation only by impurity ( HX ). Slow initiation was not considered because according to the experimental data 

Since in scenario Cl, N, is larger than I , the diagnostic Ni vs C plot will be below the ideal scenario of A1 just after the onset of polymerization, and the rate of polymerization will be higher than that of A1 (see Fig. 5Q. Chain transfer has the same effect as in scenarios of e.g., B2a, B2b, and B2c (see Fig. 5A-D). If initiation due to added initiator is slow, but initiation by H2O is relatively fast, the diagnostic plots will run below that of A1 in the N vs C, — ln(l — C) vs t and C vs t plots (Fig. 6A, C, D, scenarios C3a and C3b) but M will be higher than theoretical (Fig. 6B, scenarios C3a and C3b) at low conversions. After N,o,ai becomes larger than Iq, the curves that run under the theoretical line will cross above it (see Fig. 6A) and those running over the theoretical line will cross under it (see Fig. 6B, C, D). If the effects of slow 

---

Outlier identification is best done with a diagnostic plot based on robust PCA (Section 3.7.3) classical PCA indicates only extreme outliers. 

Outliers may heavily influence the result of PCA. Diagnostic plots help to find outliers (leverage points and orthogonal outliers) falling outside the hyper-ellipsoid which defines the PCA model. Essential is the use of robust methods that are tolerant against deviations from multivariate normal distributions. 

The remaining diagnostic plots shown in Figure 4.17 are the QQ-plot for checking the assumption of normal distribution of the residuals (upper right), the values of the y-variable (response) versus the fitted y values (lower left), and the residuals versus the fitted y values (lower right). The symbols + for outliers were used for the same objects as in the upper left plot. Thus it can be seen that the... 

FIGURE 4.17 Diagnostic plots from robust regression on the ash data. 

Figure 6.17 shows some diagnostic plots for the optimal model resulting from the analysis of the BIC values in Figure 6.16. The left picture visualizes the result of the classification, the middle picture shows the uncertainty of the classification by symbol size, and the right picture shows contour lines for the clusters. The result of model-based clustering fits much better to the visually evident groups than the result from /t-means clustering (shown in Figure 6.9).

Figure 2.28 Diagnostic plot of Equation 2.32 taken from Ref [147].

The other axis of the diagnostic plot is the studentized concentration residuals. These residuals are similar to the concentration residuals in Figure 5.95, but have been converted to standard deviation units as shown in Equation 5-38 ... 

As mentioned above, PLS can render very useful plots to diagnose whether the model seems adequate or not. The diagnostic plots that one can use depend somewhat on the commercial software available, but probably the following sections take account of the most common and typical ones. 

Figure 4.19 Diagnostic plots t va t scores to inspect for anomalous samples on the spectral space (X-block).

Diagnostic plots for heterogeneous catalytic electrode reactions at the RRDE have many features in common with those for simple parallel reactions [178]. This type of analysis is important in the investigation of the oxygen electrode reaction where non-electrochemical surface processes can occur. 

Fig. 13. Diagnostic plots for electrode reactions with coupled homogeneous reactions, illustrated for the RDE. (a) CE mechanism. Curve A, no effect from chemical reaction (5k = 0) curve B, effect of preceding chemical reaction (5k >0). (b) Catalytic mechanism. Curve A, in the absence of parallel chemical reaction curve B, experimental dependence predicted from eqn. (175).

PLS has been used mainly for calibration purposes in analytical chemistry. In this case the determination of unknown concentrations is the most important demand. In spectroscopic research, there is also the interpretation of diagnostic plots such as the score plots and loading plots as a function of reaction mechanisms and spectroscopic background knowledge. Also the interpretation of rank as complexity of a mechanism is a valuable tool. A nice property of latent variable methods is that they do not demand advanced knowledge of the system studied, but that the measurements... 

For the artificial data of Figure 6.4a, the corresponding diagnostic plot is shown in Figure 6.4b. It exposes the robust residuals / clts vs. the robust... 

The NONMEM (nonlinear mixed-effects modeling) software (Beal et al. 1992), mostly used in population pharmacokinetics, was developed at the University of California and is presently distributed by Globomax. For data management, post processing and diagnostic plots, the software S-plus (Mathsoft) is frequently used. 

Criticism seeks to determine if a fitted model is faulty. This is done by examining the residuals (departures of the data from the fitted model) for any evidence of unusual or systematic errors. Sampling theory and diagnostic plots of the residuals are the natural tools for statistical criticism their use is demonstrated in Chapters 6 and 7 and in Appendix C. 

Diagnostic plots of one parameter as a function of the dimensionless rate are common in numerical analyses of rate constants. Manipulation of the scan rate is one way to experimentally change the dimensionless rate constants. Having more than one adjustable rate constants which depend on the scanrate make the determination of rate constants less straightforward. Changes in v can however, be used to show the qualitative changes in the i-E scan that are characteristic of the ec mechanism. This will be demonstrated later. 

Try to fit a response model using the significant variables only. Make diagnostic plots of the residuals to analyze the fit. If an independent estimate of the experimental error is available, use analysis of variance of the reduced model to find out if there is a significant lack of fit. If the reduced model withstands the diagnostic tests, it is a good indication that the important variables have been found. 

Diagnostic plots can only tell us about the deficiencies of the model and not about its adequacy unless we compare it to the same type of plots based on other contending models. In practice, if we cannot see any problems in our battery of diagnostic plots, then we assume the model is without substantive error. Therefore, it is important to use a multitude of graphs to inspect as many aspects as possible of the model. However, we need to be careful because a diagnostic plot may appear suboptimal even if the model is adequate. There are basically two reasons for this phenomenon. 

When a model is used for descriptive purposes, goodness-of-ht, reliability, and stability, the components of model evaluation must be assessed. Model evaluation should be done in a manner consistent with the intended application of the PM model. The reliability of the analysis results can be checked by carefully examining diagnostic plots, key parameter estimates, standard errors, case deletion diagnostics (7-9), and/or sensitivity analysis as may seem appropriate. Conhdence intervals (standard errors) for parameters may be checked using nonparametric techniques, such as the jackknife and bootstrapping, or the prohle likelihood method. Model stability to determine whether the covariates in the PM model are those that should be tested for inclusion in the model can be checked using the bootstrap (9). 

Goodness-of-Fit. It is implied in steps 2 to 6 above that diagnostic plots (e.g., weighted residual versus time, weighted residual versus predicted observations, population observed versus predicted concentrations, individual observed versus predicted concentrations) and a test statistic such as the likelihood ratio test would be used in arriving at the base model (see Section 8.6.1.1 for goodness of fit). Once the base model (with optimized structural and variance models) has been obtained, the next step in the PM model identification process is the development of the population model. 

In most cases the choice between two competing nonheirarchical models boils down to choosing the model with a more stable formulation. This is because an examination of the diagnostic plots may not yield any difference in how the models characterize the data, and just choosing a model with a lower objective function may not necessarily indicate that it is a better model (e.g., see Ref. 18). 

---