Residual studentized

Decision command Use the maximum absolute studentized residual method to detect inconsistent values. 

VHien this method is used, Table II shows the results when the regression model is the normal first order linear model. Since the maximum absolute studentized residual (Max ASR) found, 2.29, was less than the critical value relative to this model, 2.78, the conclusion is that there are no inconsistent values. 

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. 

Laige Studentized Residual Large Studentized Residual... 

All other sastplcs have reasonable leverages and acceptable studentized residuals. 

FIGURE 5.104. Studentized residuals versus sample leverage for three-factor PLS model for component A, corrected data.. ... 

One method for identifying influential cases is to examine plots of the residuals ei = yi-yi. Here, the problem is that residuals for calibration samples near the mean of the x-values have greater variance than residuals for cases at the extreme x-values. A common method for solving this scaling problem is to standardize the residuals to give the so-called studentized residuals, r defined as... 

Calibration samples having large studentized residuals should be carefully scrutinized as possible outliers. 

Figure 4. Normal probability plot of the studentized residuals from the plasma etching experiment.

Compound Number Leverage Residual Standardized Residual Studentized Residual Dfits... 

Studentized Residuals Studentized Residuals Studentized Residuals... 

Fig. 19.7 Plots of normal probability vs. studentized residuals for models describing solvent resistance of BPA-HQ-RS copolymer compositions in (a) THF, (b) chloroform, (c) MEK. Reprinted with permission from Potyrailo et al.23 Copyright 2006 American Chemical Society...

Many of the plots just suggested are not limited to ordinary residuals. Weighted residuals, partial residuals, studentized residuals, and others can all be used to aid in model diagnostics. Beyond residual plots, other plots are also informative and can help in detecting model inadequacies. One notable plot is a scatter plot of observed versus predicted values usually with the line of unity overlaid on the plot (Fig. 1.8). The model should show random variation around the line of unity. Systematic deviations from the line indicate model misspecification whereas if the variance of the predicted values increases as the observed values increase then the variance model may be inappropriate. 

Under the assumption that the residuals are independent, normally distributed with mean 0 and constant variance, when the sample size is large, standardized residuals greater than 2 are often identified as suspect observations. Since asymptotically standardized residuals are normally distributed, one might think that they are bounded by — oo and +00, but in fact, a stand-ardized residual can never exceed y/(n - p)(n - l)n-1 (Gray and Woodall, 1994). For a simple linear model with 19 observations, it is impossible for any standardized residual to exceed 4. Standardized residuals suffer from the fact that they a prone to ballooning in which extreme cases of x tend to have smaller residuals than cases of x near the centroid of the data. To account for this, a more commonly used statistic, called studentized or internally studentized residuals, was developed... 

Upper bounds for externally studentized residuals have not been developed. Externally studentized residuals are distributed as a Student s t-distribution with n—p—1 degrees of freedom. Thus in the case of a single outlier observation, a quick test would be to compare the value of the external studentized residual to the appropriate t-distribution value, although as Cook and Weisberg (1999) point out, because of issues with multiplicity a more appropriate comparison would be Student s t-distribution with a/n critical value and n—p—1 degrees of freedom. In general, however, a yardstick of 2 or 2.5 is usually used as a critical value to flag suspect observations. 

Examination of the collinearity diagnostics indicated that two of the observations had HAT values greater than the yardstick of 2 x 3/26 or 0.23. One studentized residual was greater than + 2 (Subject 3). This subject also had a DFBETA of 1.023 for the intercept and — 1.084 for the parameter associated with BSA, indicating that these parameters would change by more than 1 standard error should this subject be removed... 

Another quick and dirty test for heteroscedasticity suggested by Carroll and Ruppert (1988) is to compute the Spearman rank correlation coefficient between absolute studentized residuals and predicted values. If Spearman s correlation coefficient is statistically significant, this is indicative of increasing variance, but in no manner should the degree of correlation be taken as a measure of the degree of heteroscedasticity. They also suggest that a further refinement to any residual plot would be to overlay a nonparametric smoothed curve, such as a LOESS or kernel fit. 

Hence, raw residuals are often standardized by some quantity, usually its standard deviation, to account for the unequal variance of the residuals. If the estimate of the standard deviation is based on the observation in question the method is called internal studentization, but if the observation is not included in the estimate of the standard deviation, then the method is called external studentization. Studentized residuals can be estimated... 

Gray, J.B., and Woodall, W.H. The maximum size of standardized and internally studentized residuals in regression analysis. American Statistician 1994 48 111-113. 

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,). 

The applied researcher, then, needs to remove extreme values that are truly nonrepresentational and include extreme values that are representational. The researcher must also discover the phenomena contributing to these values. Rescaling the residuals can be very valuable in helping to identify outliers. Rescaling procedures include standardizing residuals, studentizing residuals, and jackknife residuals. 

For smaller sample sizes (n < 30), the use of the Studentized approach is recommended, as it follows the Student s f-distribution with n — k — df. The Studentized residual Sr,) is computed as... 

The standardized and Studentized residuals generally convey the same information, except when specific e, residuals are large, the /r values are large, and/or the sample size is small. Then use the Studentized approach to the residuals. 

If the standard regression assumptions are met, and the same number of replicates is taken at each x, value, the standardized, the Studentized, and jackknife residuals look the same. Outliers are often best identified by the jackknife residual, for it makes suspect data more obvious. For example, if the /th residual observation is extreme (hes outside the data pool), the, ) value will tend to be much smaller than which will make the r(, ) value larger in comparison to Sr the Studentized residual. Hence, the r(, ) value will stand out for detection. 

A Cook s distance value (Cd,) may be large because an observation is large or because it has large Studentized residuals, Sr,. The Sr, value is not seen in Equation 8.26, but Cd,) can also be written as... 

Studentized residual = Sr. Jackknife residual = (-o-Leverage value = hi. 

Because this author prefers the jackknife procedure, we will use it for an example of a complete analysis. The same procedure would be done for calculating standardized and Studentized residuals. First, a Stem-Leaf display was computed of the T(, ) values (Table 8.19). 

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