Studentized concentration residuals

Studentized Concentration Residual vs. Sample Leverage Plot (Sample Diagnostic) This diagnostic is u.sed to more closeh examine individual samples. The sample ie erage for the /th sample is computed by... 

TABLE 5.16. Possible Combinations for Sample Leverage and Studentized Concentration Residuals... 

FIGURE 5.118. Studentized concentration residuals for caustic versus the sample... 

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 ... 

The last possible combination is to have a sample with large sample leverage and large studentized concentration residuals. Hus indicates a problem with the measurement vector and/or the reference concentration values. Because of the possibility of the model being forced to fit influential points, the recommendation is to reexamine the measurement and reference concentration values whenever a sample has large leverage. 

FIGURE 5.97. Sample leverage versus studentized concentration residuals for component A PLS medal. 

Studentized concentration residual vs. sample leverage plot... 

Studentized concentration residuals are concentration residuals that have been divided by the concentration standard error of estimate and VI— leverage. Sample leverage is a measure of the influence a sample measurement vector has on the model. 

RGURE 5.131. Studentized concentration residuals versus sample leverage for MCB using a six-factor PLS model. 

FlO. 18. A leverage vs studentized residual plot of same cross-validation prediction of the model in data from Fig. IS. Note that both the studentized concentration residual and the leverage of sample No. 45 are both significantly larger than the remainder of the training set. This is another confirmation that this sample is an outlier. 

The Studentized concentration residuals are frequently called just Studentized residuals . Note that these should not be confused with the X-residuals in the previous section. Now we are concerned with studying whether a particular sample cannot be predicted properly. Hence residual is interpreted in the usual way that atomic spectroscopists do, i.e. the difference between the true and the predicted concentrations of the analyte (y - preference)) where the hat (") means predicted. The simplest possibility (also used in classical univariate regression) is to plot the residuals against the sample number or the concentration and, often, the outlying samples (those which were modelled/predicted less successfully) will be obvious from the graph. 

In PLS it is common to calculate the Studentized residuals using the leverage defined above as it measures the difference between each calibrator and the average of the overall calibration set. Thus, for each calibrator, the Studentized residual can be calculated as std.res. = (y - Preference) / ( (l ) )) where 5 is the standard deviation of all concentration residuals from the calibration set (using 7— — 1 as the degrees of freedom) and h is the leverage defined above. Calibrators for which the predictions show Studentized residuals larger than 3 should be inspected carefully. 

The studSentized residual measures how well the concentration of the rth sample is fit by the model. Large positive or negative studentized residual values indicate samples that are not fit well. In the following discussion, studentized residuals greater than 2.5 are considered to be large. This is because the studentized residuals are in units of standard deviations from the mean value, and 2.5 is unusual given standard statistical assumptions. 

The student need not work up the mother liquor hut may be interested in learning the result. Concentration of the solution to a small volume is not satisfactory because of the presence of dicyclopentadiene, formed by dimerization of excess monomer the dimer has high solvent power. Hence the bulk of the solvent is evaporated on the steam bath, the flask is connected to the water pump with a rubber stopper and glass tube and heated under vacuum on the steam bath until dicyclopentadiene is removed and the residue solidifies. Crystallization from 1 1 ethyl acetate-ligroin affords 1.3 g of adduct, mp 156-158°C total yield is 95%. 

The influence of cysteine on the methionine requirement is demonstrated by the following study. The subject was a student who was awarded a Ph.U. after conducting the study on himself. The subject consumed diets that were complete except that the concentration of methionine was varied (Table 8.12), The dietary amino acids were supplied in the form of pure amino acids, rather than as protein, to allow full control over the levels of amino adds supplied by the diet. Cysteine was supplied as cy.stine. Cystine is a dimer of cysteine, in which the two cysteine residues are connected via a disulfide bond (R——R). Cystine is readily converted to cysteine in the body. The methionine requirement was assessed by determining the conditions that supported a zero or slightly positive nitrogen balance. 

When the residue is carbonized and is not greatly affected by the treatment described above, then proceed thus Place 1-2 g of solid sodium dichromate in the flask add 1-2 ml of water place in the hood with the mouth of the vessel away from you and towards the wall of the hood. Add 5-10 ml of concentrated sulfuric acid (technical grade). Clamp the vessel on a stand and heat with a free flame. If a reaction starts, remove the flame until it subsides, then heat until most of the tarry carbon has disintegtated. Allow it to cool. Pour acid mixture into a beaker and then rinse with water. The acid-mixture residue is carefully poured into the sink, unless it contains tarry matter which may cause clogging. The student is reminded that potassium permanganate is never used in place of potassium dichromate for cleaning solution. 

Problem Students often think that solubility equilibria depend on the amount of solid residue, or they succumb to the fallacy, that in the law of mass action must include the concentration of solid material. It should be shown, that it makes no difference how much solid residue is at equilibrium in the saturated salt solution. 

In this equation the confidence interval Xupper,xu,u,er) for a predicted value of jto is calculated. The parameter is the slope of the regression line, t is the Student t statistic, s is the residual mean square error, Sxx is the sum of squares of the X values from the mean of the calibration x values, and n is the total number of calibration samples. The values from these confidence limits place upper and lower bounds for the analyte concentration, jtq. at all points along the calibration range. The best confidence is found near the mean of the data, whereas lower confidence is found at the extremes. 

It is useful to plot the residuals, or the studentized residuals, against the values of the coded variables X, in turn. These should be evenly distributed, with no obvious dependence on the factor. Figure 7.1 gives the example of the response of cloud point, in the case of the formulation of an oral solution (3) already discussed in chapters 3, 5, and 6. The studentized residuals are plotted in turn against the polysorbate 80 concentration (X,), the propylene glycol concentration (Xj), and the invert sucrose medium (X,). 

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