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We would like to examine the influential points to determine whether they control many model properties. If these influential points are bad points, or erroneous in any way, they should be eliminated. On the other hand, there may be nothing wrong with these points, but at least we would like to determine whether or not they produce results consistent with the rest of the data. In any event, even if an influential point is a valid one, if it controls important model properties, we would like to know this, since it could have an impact on the use of the model. Montgomery, Peck, and Vining (2001) and Myers (1990) describe several methods for detecting influential observations. An excellent diagnostic is the distance measure developed by Dennis R. Cook. This is a measure of the squared distance between the usual least squares estimate of based on all n observations and the estimate obtained when the ith point is removed, say, 1i2. The Cook distance measure is 1 1i2

// Is key K at position t findKey(i,l,j)

Clearly, if the ith point is influential, its removal will result in 1i2 changing considerably from the value . Thus, a large value of Di implies that the ith point is influential. The statistic Di is actually computed using

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A companion to, but somewhat opposite of, Night Scene, Night Snapshot assumes you are not using a tripod when taking pictures at night and reduces camera shake blur by amping up the sensitivity of the image sensor so that a faster, more easily handheld shutter speed can be used Kids & Pets This scene mode optimizes the camera settings for fast focus response and quick shutter speeds to catch all the action Red-eye reduction is turned on but can be overridden in the Shooting menu Kids & Pets mode provides increased sensitivity and fast camera operation but may yield increased noise levels Indoor Higher sensitivity of the image sensor by way of increased ISO Speed settings to avoid camera shake in indoor images is why you d choose Indoor scene mode.

From Equation 12-44 we see that Di consists of the squared studentized residual, which reflects how well the model fits the ith observation yi [recall that ri ei 2 2 11 hii 2 4 and a component that measures how far that point is from the rest of the data 3hii 11 hii 2 is a measure of the distance of the ith point from the centroid of the remaining n 1 points]. A value of Di 1 would indicate that the point is influential. Either component of Di (or both) may contribute to a large value. EXAMPLE 12-10 Table 12-11 lists the values of the hat matrix diagonals hii and Cook s distance measure Di for the wire bond pull strength data in Example 12-1. To illustrate the calculations, consider the first observation: D1 31.57 25.235211 3 0.035

Y = Spa&

(12-44)

h11 2 4 2

0.15732 4 2

Y = Fz(E(P,Y ) )

0.1573 0.15732

The Cook distance measure Di does not identify any potentially influential observations in the data, for no value of Di exceeds unity.

1t o t - j - 1

Table 12-11 Influence Diagnostics for the Wire Bond Pull Strength Data 2 Observations i 1 2 3 4 5 6 7 8 9 10 11 12 13 hii 0.1573 0.1116 0.1419 0.1019 0.0418 0.0749 0.1181 0.1561 0.1280 0.0413 0.0925 0.0526 0.0820 Cook s Distance Measure Di 0.035 0.012 0.060 0.021 0.024 0.007 0.036 0.020 0.160 0.001 0.013 0.001 0.001 Observations i 14 15 16 17 18 19 20 21 22 23 24 25 hii 0.1129 0.0737 0.0879 0.2593 0.2929 0.0962 0.1473 0.1296 0.1358 0.1824 0.1091 0.0729 Cook s Distance Measure Di 0.003 0.187 0.001 0.565 0.155 0.018 0.000 0.052 0.028 0.002 0.040 0.000

EXERCISES FOR SECTION 12-5

return(K) e l s e / / false alarm r e t u r n (not found) end i f end findKey block as necessary. In contrast, the issue of merging and cycling chains is of fundamental importance in this TMTO attack.

12-36 Consider the regression model for the NFL data in Exercise 12-4 (a) What proportion of total variability is explained by this model (b) Construct a normal probability plot of the residuals What conclusion can you draw from this plot (c) Plot the residuals versus y and versus each regressor, and comment on model adequacy (d) Are there any influential points in these data 12-37 Consider the gasoline mileage data in Exercise 12-5 (a) What proportion of total variability is explained by this model (b) Construct a normal probability plot of the residuals and comment on the normality assumption (c) Plot residuals versus y and versus each regressor Discuss these residual plots (d) Calculate Cook s distance for the observations in this data set Are any observations influential 12-38 Consider the electric power consumption data in Exercise 12-6 (a) Calculate R2 for this model Interpret this quantity.

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