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We are frequently interested in testing hypotheses on the individual regression coefficients. Such tests would be useful in determining the potential value of each of the regressor variables in the regression model. For example, the model might be more effective with the inclusion of additional variables or perhaps with the deletion of one or more of the regressors presently in the model. Adding a variable to a regression model always causes the sum of squares for regression to increase and the error sum of squares to decrease (this is why R2 always increases when a variable is added). We must decide whether the increase in the regression sum of squares is large enough to justify using the additional variable in the model. Furthermore, adding an unimportant variable to the model can actually increase the error mean square, indicating that adding such a variable has actually made the model a poorer fit to the data (this is why R2 is adj a better measure of global model fit then the ordinary R2). The hypotheses for testing the significance of any individual regression coefficient, say , are j

H0: H1:

0 0 (12-23)

where the conditions that determine which half of the equation applies are analogous to those in Problem 21. 25. Suppose that A and B are randomly sclected 32-bit integers. Let

If H0: j 0 is not rejected, this indicates that the regressor xj can be deleted from the model. The test statistic for this hypothesis is 2

(12-24)

x = A + ( w 2 4 ...3 1 ,

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where Cjj is the diagonal element of 1X X2 1 corresponding to j. Notice that the denominator of Equation 12-24 is the standard error of the regression coefficient j. The null hypothesis H0: 0 is rejected if 0 t0 0 t 2,n p. This is called a partial or marginal test because the j regression coefficient j depends on all the other regressor variables xi (i j) that are in the model. More will be said about this in the following example. EXAMPLE 12-4 Consider the wire bond pull strength data, and suppose that we want to test the hypothesis that the regression coefficient for x2 (die height) is zero. The hypotheses are H0: H1:

3.3 To show some motion in the background nightscape for this wedding portrait, I switched to Shutter Priority mode and then set the shutter speed to 1/13 second. Taken at ISO 400, f/8, 1/13 second with a 6.1mm lens setting.

2 2

215.2352210.00000152 0.01253

(x)O7 # (A)0 77 ... ...

0.0000015,

Note that we have used the estimate of 2 reported to four decimal places in Table 12-10. Since t0.025,22 2.074, we reject H0: 2 0 and conclude that the variable x2 (die height) contributes significantly to the model. We could also have used a P-value to draw conclusions. The P-value for t0 4.4767 is P 0.0002, so with = 0.05 we would reject the null hypothesis. Note that this test measures the marginal or partial contribution of x2 given that x1 is in the model. That is, the t-test measures the contribution of adding the variable x2 die height to a model that already contains x1 wire length. Table 12-4 shows the value of the t-test computed by Minitab. The Minitab t-test statistic is reported to two decimal places. Note that the computer produces a t-test for each regression coefficient in the model. These t-tests indicate that both regressors contribute to the model. There is another way to test the contribution of an individual regressor variable to the model. This approach determines the increase in the regression sum of squares obtained by adding a variable xj (say) to the model, given that other variables xi (i j) are already included in the regression equation. The procedure used to do this is called the general regression significance test, or the extra sum of squares method. This procedure can also be used to investigate the contribution of a subset of the regressor variables to the model. Consider the regression model with k regressor variables y X (12-25)

26. In the PKZIP attack, show that if you are given (y2-1)0...7 and (x)0...7, then the riurnber of 32-bit values K-1 (Xi-1)24...31 and J' that satisfy (3.19) is in the range of 65,534 to 65,538, inclusive.

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