Williams plot

For an analysis of the AD of regression models, the author has always used the Williams plot, which is now widely applied by other authors and commercial software. The Williams plot is the plot of standardized cross-validated residuals (R) versus leverage (Hat diagonal) values (h from the HAT matrix). It allows an immediate and simple graphical detection of both the response outliers i.e., compounds with cross-validated standardized residuals greater than 2-3 standard deviation units) and structurally anomalous chemicals in a model (h>h, the critical value being h = 3p /n, where p is the number of model variables plus one, and n is the number of the objects used to calculate the model).40,62,66... 

Modified Bellamy-Hallam-Williams plot of shift in OH frequency of acidic hydroxyl groups of different zeolites against the shift in frequency of framework silanol hydroxyls as probe molecules of different basic strengths (represented by numbers 1-17) are adsorbed. See reference for details. [Figure reproduced from reference 23 with permission. Copyright 1997 American Chemical Society.]... 

Williams plotted the film-thinning time for deformable and nondeformable droplets against droplet radius. While an increase in droplet size increases the time required for thinning of a deformable droplet, nondeformable droplets experience a reduction in film thinning time as their size increases. It is interesting also to note the square relationship on thinning rate wifii nondeformable droplets and an inverse square relationship for deformable droplets. Clearly, increasing the apphed field across a system with deformable droplets could result in a reduction in coalescence efficiency. 

A systematic way of carrying out such a comparison makes use of the so-called Bellamy-Hallam-William plot, a procedure borrowed from solution chemistry, which is as follows. A set of molecules of basic nature is considered. The shift suffered by H-bonding to each molecule in this set is measured by the O - H stretching mode of an acidic hydroxyl (e.g., in H-ZSM-5) and successively plotted against the same type of data for a reference O - H species (e.g., the isolated silanol in H-MCM-41). A proportionality is seen, the slope of the straight line constituting a measure of relative acidity of the two hydroxyl species. 

The effect of substituents on the rate of the reaction catalysed by different metal ions has also been studied Correlation with resulted in perfectly linear Hammett plots. Now the p-values for the four Lewis-acids are of comparable magnitude and do not follow the Irving-Williams order. Note tlrat the substituents have opposing effects on complexation, which is favoured by electron donating substituents, and reactivity, which is increased by electron withdrawirg substituents. The effect on the reactivity is clearly more pronounced than the effect on the complexation equilibrium. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Figure 4.4 Data in Fig. 4.3 when plotted di (h , - hfi versus time. The slope of the lines represents the internuclear distance r that corresponds to the rate of nOe buildup, which is directly proportional to r . (Reprinted from D. H. Williams et al.,J. Am. Chem. Soc. 105,1332, copyright (1983), with permission from The American Chemical Society, 1155 16th Street, N.W. Washington, D.C. 20036, U.S.A.). 

Williams et al. (2002) have plotted the relationship between the spatial resolution and the MMF in several electron-probe instruments, and this is reproduced in Figure 5.27. 

Robert Williams Yes, but where does the line intersect the axes Where is the intercept, because if you plot intercepts, do you get any result at all about mortality It would be very strange if you didn t. 

Fig. 2.8. The variation of stability constants, K, for the complexes of M2+ ions of the Irving-Williams series, ox, oxalate en, ethylenediamine (note that the constants plotted are absolute, not effective constants at pH = 7).

When a graph is made of log fCj versus the sum of the first and second ionization potentials for the metals, the result is shown in Figure 19.5. Clearly, the relationship is linear for the metal ions except Zn2+, for which the complex is considerably less stable than that with Cu2+. These results are in accord with the Irving-Williams series described earlier. The graph obtained when log K2 is plotted against the total ionization potential has exactly the same characteristics, as is shown in Figure 19.6. 

The results (Fig. 10) clearly show that the magnitude of the KIE is related to the looseness of the transition state. Thus, the results of this study are in agreement with the conclusions based on the theoretical calculations by Barnes and Williams (1993) and Poirier et al. (1994). Another interesting observation is that the slopes of the plots of KIE versus elongation of the C—X bond on going to the transition state are almost identical for the three identity reactions. 

Fig. 27. Semilogarithmic plot of the nonradiative triplet rate constant against (E— o)/> for the normal and deuterated hydrocarbons listed in Ref. t)). The broken line, derived from phosphorescence spectra, is taken from Ref. t). The slopes of the two solid lines differ by a factor 1.35. (O.Ci-jjH, E = 4000 cm l 0 Ci fl Z>u, =5500 cm t). The following totally deuterated hydrocarbons are included benzene, triphenylene, acenaphtene, naphthalene, phenanthrene, chrysene, biphenyl, p-terphenyl, pyrene, 1,2-benzanthracene, anthracene (in the order of increasing /S). (From Siebrand and Williams, Ref. l)...

Fig. 1.9 Time dependence of absorbance obtained after mixing Con A (200 pM) and 10 (a) (20 pM). The semi-log plot of the data (inset) shows even clearer the hiphasic nature of the reaction. Reprinted with permission from T. J. Williams, J. A. Shafer, I. T. Goldstein and T. Adamson, J. Biol. Chem. 253, 8538 (1978).

Fig. 4-3 Plot of percent conversion versus time for emulsion polymerizations of styrene with different concentrations of potassium laurate at 60° C. The moles of emulsifier per polymerization charge (containing 180 g H2O, 100 g styrene, 0.5 g K2S2O8) are 0.0035 (plot 1), 0.007 (plot 2), and 0.014 (plot 3). After Williams and Bobalek [1966] (by permission of Wiley-Interscience, New York).

The second study is relevant to a discussion in the paper by Dr. Wilkins of instances in which the relative rates are related to stabilities. We have come across a very striking example of such a relationship in the activation of the enzyme ribo-nuclease by metal ions. In Figure E the activity is plotted vs. concentration of metal ion added to the enzymatic reaction mixture. In the absence of metal the activity is as indicated by the straight line in the center of the figure. It can be seen that a list of the transition metals in the order of concentrations giving maximum activity corresponds rather neatly with the Irving-Williams series. 

Transmission electron micrographs (TEM) of submicrometer-size particles show faceted particles, and selected area electron diffraction (SAED) patterns of isolated particles show that they are formed by a small number of crystallites (Fig. 9.2.14a), This result is consistent with the mean size of the crystallites, which can be inferred from the x-ray diffraction lines broadening analysis using a William-son-Hall plot (35) in order to take into account the contribution of microstrains to the line broadening. Over the whole composition range, the mean crystallite size is in the range 40-60 nm for particles with a mean diameter in the range 200-300 nm (Table 9.2.5) (33). 

YBa2Cu307 is similar to the vs. S plot the plot of against V+ is linear (Tallon Williams, 1990). 

An industrial oil-extended synthetic cis-polyisoprene was investigated by von Meerwall and Ferguson30). Following Boss, et al. 29), they substracted the unattenu-atable spin echo arising from the rubber, obtaining the diffusivity of the extender oil from the remainder. They demonstrated that no departures from Fickian diffusion occur, and measured the diffusion of the oil, both in the rubber and in the pure liquid, between —10 °C and 130 °C. Since the plot of log D vs. 1/T was not a straight line it was necessary to invoke the Williams-Landel-Ferry temperature dependence,... 

Thus a plot of 1/Mw ipp(cell) vs. c0, could yield B from the limiting slope, and BLs could be estimated from Equation 90, provided one had an estimate of Mz, the z-average molecular weight. This method was restricted to lower speeds (to avoid solute redistribution problems) and to solutions that were not too nonideal. More recently, Deonier and Williams (37) showed that... 

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

An alternative to constructing the Arrhenius plot log(K) against 1/T is to shift the plots of parameter against time along the time axis to construct a master curve. Use can be made of the Williams, Landel, Ferry (WLF) equation -... 

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