Cook s distance

Cook s Distance Plot (Model Diagnostic) A statistic known as Cook s distance can be used to detect calibration data outliers by identilying which samples are most influential on the model. Now that the selected variables have been finalized, it is good practice to examine the calibration data for influential samples. These samples should be investigated and removed if it is determined that they have an unusual effect on the model. 

FIGURE 5.69. Cook s distance versus sample number for component A. 

Cook s distance plot (Cook s distance vs. sample number)... 

Cook s Distance Plot (Model Diagnostic) In Figure 5.80, the Cook s distance is plotted versus sample number for tlie three-variable caustic model. All values are much less than 1 (the maximum is 0.14), which is an indication that no samples are potentially over ) influential on the model parameters. 

Distance measures, such as Cook s distance, combine the concept of leverage and residuals to compute an overall measure of a calibration sample s influence on the calibration model. Cook s distance is computed as... 

To find the outlying observations, two strategies can be followed. The first approach is to apply a classical method, followed by the computation of several diagnostics that are based on the resulting residuals. Consider, for example, the Cook s distance in regression. For each observation 1 = 1,. .., n, it is defined as... 

Therefore, the / are masked whenever the MIXx,) are. In particular, Cook s distance in Equation 6.1 can fail, as it can be rewritten as... 

The initial regression is examined for robustness from the standpoint of both influential chemicals and poorly behaved parameters. Ridge regression, Cook s distance, partial correlations, and principal components are used to evaluate the regression. After the poorly behaved parameters are removed,... 

One problem with DFBETAS is that there will be n x p DFBETAS for the analyst to examine, which can be tedious to examine. Cook s distance, Dj, is a composite score that assesses the influence an observation has on the set of regression parameters and is computed by... 

Similar to the linear regression case, influence diagnostics for linear mixed models can be developed using case deletion. These extensions include Cook s distance, DFFITs, COVRATIO, and PRESS residuals. For example, a linear mixed effects extension to DFFITS is... 

Two chemicals, acenaphthene and cyclopenta(cof)pyrene (Nos. 106 and 117 of Table 4, respectively) had rather large residuals (> 60° Q.The Cook s distance for these two chemicals indicated they were influential cases. Given these circumstances, an outlier test was performed and both chemicals had a significant result. After the removal of these chemicals, the following model was developed ... 

The Cook s distance (0,) measures the influence of any one observation relative to the others. That is, it measures the change in bj, the Unear regression coefficient, when that observation, or an observation set, is removed from the aggregate set of observations. The calculation of C, is ... 

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... 

Because MiniTab and other software packages also can provide values for leverage (/ ,) and Cook s distance, let us look at them relative to the previous analysis with the two data points ( 4 and 23) in the analysis. 

Previously in this chapter, we discussed Cook s distance for simple linear regression, a regression diagnostic that is used to detect an extreme value and its influence by removing it from the analysis and then observing the results. In multiple linear regression, the same approach is used, except that a data set is removed. Cook s distance lets the researcher determine just how influential the ith value set is. 

Let us again look at Example 8.5. The Cook s distance values are pro-... 

A large DFBETASy( j) > 2 /n means that the fth observation needs to be checked. The mily problem is that, with small samples, the may not be useful. For large samples, n > 30, it works fine. In smaller samples, use Cook s distance in preference to DFBETASy(, ). 

Instead of removing a specific value, an analogy to a trim mean might be employed. Say 10% of the most extreme absolute residual values. Cook s distance values, or deleted Studentized values are simply removed this is 5% of the extreme positive residuals and 5% of the negative ones. This sort of determination helps prevent distorting the data for one s gain. 

TABLE K Cook s Distance Table Critical Values for the Maximum of n Values of Cook s d(i) x (n - k - 1) (Bonferroni Correction Used) Observations and k Predictors Level of Significance o = 0.1 n... 

To measure the change of the influential observation, the model has to be built by including or deleting it. From the two models, we obtain different estimations for the y values that can be used to compute a measure, the so-called Cook s distance, D ... 

Figure 6.10 Regression diagnostics Residual plot in dependence on for influential observations and out- the calculated y values, (c) Jack-liers. (a) Cook s distance for recogni- knifed residuals according to tion of influential observations, (b) Eq. (6.102).

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