Jackknife Residual

Fig. 6. Repacking of the influenza HA2 hydrophobic core. Left. A ribbon trace of HA2 residues 38 to 127, including the helices that make up the core of the stalk in the native HA structure (see Fig. 3). Middle A hypothetical structure obtained by fusing the base of the coiled coil from the native HA structure with the top of the extended coiled coil from the low pH-converted HA structure. This panel helps distinguish the two major components of the HA conformational change on low pH treatment the existence of such an intermediate structure has not been shovm experimentally for influenza and may exist only transiently if at all. This extended structure, known as a prehairpin intermediate, has been detected indirectly in other virus envelope proteins (reviewed in Chan and Kim, 1998). Right Residues 38 to 127 from low pH-converted HA2 (Bullough et al, 1994). Hydrophobic residues that stabilize the jackknifed structure are indicated in one protomer as gray space-filling atoms. The amino (N) and carboxy (C) termini of a protomer within each trimer structure are indicated.

Under the assumption that the residuals are independent, normally distributed with mean 0 and constant variance, when the sample size is large, studentized residuals greater than 2 are often identified as suspect observations. Like standardized residuals, studentized residuals are not bound by —oo and +oo, but are bounded by - /(n — p) (Gray and Woodall, 1994). An alternative statistic, one that is often erroneously interchanged with standardized residuals, are studentized deleted residuals, which are sometimes called jackknifed... 

As a last comment, caution should be exercised when fitting small sets of data to both structural and residual variance models. It is commonplace in the literature to fit individual data and then apply a residual variance model to the data. Residual variance models based on small samples are not very robust, which can easily be seen if the data are jackknifed or bootstrapped. One way to overcome this is to assume a common residual variance model for all observations, instead of a residual variance model for each subject. This assumption is not such a leap of faith. For GLS, first fit each subject and then pool the residuals. Use the pooled residuals to estimate the residual variance model parameters and then iterate in this manner until convergence. For ELS, things are a bit trickier but are still doable. 

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. 

The method for standardizing the residuals is reasonably straightforward and will not be demonstrated. However, the overall process of residual analysis by smdentizing and jackknifing procedures that use hat matrices will be explored in detail in Chapter 8. 

The reader is directed to Appendix II for a review of matrices and application of matrix algebra. Once that is completed, we will look at examples of Studentized and jackknifed residuals applied to data from simple linear regression models and then discuss rescaling of residuals as it applies to model leveraging due to outliers. 

The fth jackknife residual is computed by deleting the /th residual and, so, is based on 1 observations. The jackknife residual is calculated as... 

The mean of the jackknife residual approximates 0, with a variance of... 

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. 

In practice, Kleimbaum et al. (1998) and this author prefer computing the jackknife residuals over the standardized or Studentized ones, although the same strategy will be relevant to computing those. 

The jackknife procedure reflects an expectation si iV(0, (r ), which is the basis for the Student s f-distribution at a/2 and n k 2 degrees of freedom. The jackknife residual, however, must be adjusted, because there are, in fact, n tests performed, one for each observation. If n = 20, a = 0.05, and a two-tail test is conducted, then the adjustment factor is... 

Table F presents corrected jackknife residual values, which essentially are Bonferroni corrections on the jackknife residuals. For example, let a = 0.05, k = the number of bi values in the model, excluding bo say = 1 and n = 20. In this case. Table F shows that a jackknife residual greater than 3.54 in absolute value, r(, ) > 3.54, would be considered an outlier.

Let us now evaluate the jackknife residuals. The critical jackknife values are found in Table F, where n = 30, k= (representing bi), and a = 0.05. We again need to interpolate. 

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

So, any jackknife residual greater than 3.51, or T(, ) > 3.51, is suspect. Looking down the jackknife residual T(, ) column, we note that the value 4.13288 > 3.51, again at n = 23. Our next question is what happened ... 

Stem-Leaf Display of Jackknife Residuals, Example 8.4... 

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). 

FIGURE 8.16 Boxplot display of jackknife residuals. Example 8.4. 

The jackknife residuals at time 0 are portrayed in the Stem-Leaf display (Figure 8.25). Again, the Subject 4 datum is portrayed as extreme, but not that extreme. 

Figure 8.26 shows the T(, ) jackknife residuals plotted on the Boxplot display and indicates a single outlier. 

FIGURE 8.25 Stem-Leaf display of jackknife residuals r(, -) at time zero. Example 8.4. 

Because we prefer using the jackknife residual, we will look only at the Stem-Leaf and Boxplot displays of these. Note that 5r, = 3.29388 and = 4.13288, both exceed their critical values of 2.61 and 3.65, respectively. 

Figure 8.27 is a Stem-Leaf display of the time 6 h data. We earlier identified the 6.57 value, with a 4.1 jackknife residual, as a spurious data point due to noncompliance by a subject. 

Bonferroni Corrected Jackknife Residual Critical Values... 

TABLE G (continued) Bonferroni Corrected Jackknife Residual Critical Values Level of Significance a = 0.05 ... 

Chapter 8 aids the researcher in determining outlier values of the variables y and x. It also includes residual analysis schemas, such as standardized, Studentized, and jackknife residual analyses. Another important feature of... 

The externally studentized residual is also termed jackknifed residual. 

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