Straight residual analysis

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

Urinary proteins were analyzed by SDS-polyacrylamide gel electrophoresis (PAGE), and a 70-kDa protein was identified as the major component of cat urine (Fig. 4.1 A). Comparative analysis of urinary proteins in several other mammals such as humans, mice, dogs, and cattle did not detect a 70-kDa protein. Therefore, the 70-kDa protein was purified from cat urine and characterized by biochemical methods (Miyazaki, Kamiie, Soeta, Taira and Yamashita 2003). Analysis of tissue distribution indicated that the 70-kDa protein is expressed in the kidney in a tissue-specific manner and secreted from the proximal straight tubular cells of the kidney into the urine (Fig. 4.IB). A full-length cDNA for a 70-kDa protein was cloned from a cat kidney cDNA library. The cDNA clone encoded a polypeptide of 545 amino acid residues. The deduced amino acid sequence shared 47% identity with cat carboxylesterase (CES, EC 3.1.1.1), and contained both the CES family protein motif (EDCLY) and a conserved active site motif (GESAG) associated with... 

Sequencing of Romalea PDF was straight forward, and it also proved to be an octadecapeptide (40). The Romalea peptide differs from Acheta PDF at a single position, the former having Leu-14 in place of Val-14. In these peptides the presence of Asp-Ala bond, positions 17-18, permitted the cleavage of the C-terminal residue by heating in dilute acid. HPLC analysis of the derivatized cleaved product enabled the identification of the C-terminus as Ala-amide. 

Assuming constant variability for the disturbances when the analysis is specified and having this assumption correct are not enough to make the long-run size of all residuals the same. If, for instance, we fit a straight line by equally weighted least squares to five equally spaced points and the model applies precisely, we find the following ... 

The four data sets in this problem were selected by Anscombe (1973) to have the same values for their slopes (= 0.50), their intercepts (= 3.00), the sums ofthe squares of their residuals S(y-ycaic)2 (= 1.53), their standard deviations ay (= 1.11), crai ( = 0.118), and crao (= 1.12), as well as their correlation coefficients r (=0.816). By all these criteria, then, they fit the same equation of a straight line equally well. However, visual examination of the graphs, or of the residual plots, yields a quite different answer only set 1 reasonably fits a straight line. In other words, the statistical analysis in terms of summary statistics does not address the validity of the assumed model, and can produce results regardless of whether or not the model is appropriate. In the present case, the first data set reasonably fits the assumed model of a linear dependence, the second set should be fitted to a parabola instead of to a line, while the third and fourth sets both contain an intentional outlier . Direct observation of the graphs before the analysis, and/or of the residuals afterwards, can often help us reject clearly inappropriate models. 

Amino Acid Analysis Results. Irradiations for these analyses were done with considerably higher doses, 0-400 kGy, to compensate for the limitations in detecting small changes in amino acid composition. All samples were desiccated and then hydrolyzed with p-toluenesulfonic acid. The number of residues determined for a particular amino acid was normalized per 1000 residues of all amino acids analyzed. Plots of this ratio for each amino acid against dose are straight lines with zero slope. Such results indicate that, within the limits of reproducibility of about 2%, none of the amino acids is discemibly modified (37,42) under these conditions. 

Thus a plot of In (R) versus t should then give a series of points falling on a straight line, as in Figure 10.50. A linear regression analysis is then performed on the residual values, with time (t) entered as the independent (x) data, and ln(i ) is entered as the dependent (y) data. The fitted slope nij represents the best value for —ka, so that... 

Outliers should not be confused with influential observations. Until now, we have used the residuals in order to find problems with a model. If we want to study the robustness of a model to perturbations, we do an influence analysis. This kind of study is done as though the model were correct. Influential observations cannot be detected by large residuals. Their removal, however, may cause major changes in subsequent use of the model. The difference can be understood from Figure 6.7. A straight-Une model that includes the influential observation will give a different slope if that observation is deleted. On the other hand, if the obvious outlier is included in the model, we will estimate larger residuals for all of the cases. 

As with univariate regression, an analysis of the residuals is important in evaluating the model. The residuals should be randomly and normally distributed. Figure 8.11 shows a plot of the residuals against the fitted values for Cl , the residuals do not show any particular pattern. Figure 8.12 plots the predicted values against the measured values. The points are reasonably close to a straight line with no obvious outliers. 

A xylan extracted from the holocellulose present in the stalks of Nicotiana tabacum has been purified by ion-exchange chromatography. The results of methylation analysis, partial hydrolysis with acid, and enzymic hydrolysis indicated that the xylan is composed of a straight chain of approximately 100 /S-(l -> 4)-linked D-xylopyranosyl residues. Curie point-g.l.c. of the xylan yielded 2-furaldehyde, which is probably formed via 3-deoxy-D- /ycero-pent-2-ulose. Another major product was identified as 3-hydroxy-2-penteno-1,5-lactone (21) by high-resolution m.s. this lactone has also been prepared by pyrolysis of a hexuronic acid-containing xylan from beech. Since the tobacco-stalk xylan contains no acidic components, the lactone (21) must arise by... 

In 1966 we noted that a log tJq - plot published by Spencer and Dillon for an ar-PS fraction consisted of two straight lines intersecting near T . Later studies by ourselves using both first derivatives and regression analysis with residuals revealed double or triple Arrhenius plots (depending on the temperature range of the data) for ar-PSs of 1.1, for a broad... 

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