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This test procedure may be easily introduced into the analysis of variance conducted for the signi cance of regression. If the null hypothesis of model adequacy is rejected, the model must be abandoned and attempts must be made to nd a more appropriate model. If H0 is not rejected, there is no apparent reason to doubt the adequacy of the model, and MSPE and MSLOF are often combined to estimate 2. EXAMPLE S11-1 Consider the data on two variables y and x shown below. Fit a simple linear regression model and test for lack of t, using 0.05.

(10100101,10010101) (11100000,11010000)

x 1.0 2.0 3.3 4.0 5.0 y 2.3, 1.8 2.8 1.8, 3.7 2.6, 2.6, 2.2 2.0 x 5.6 6.0 6.5 6.9 y 3.5, 2.8, 2.1 3.4, 3.2 3.4 5.0

The regression model is y 1.697 0.259x, and the regression sum of squares is SSR The pure-error sum of squares is computed as follows:

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The analysis of variance is summarized in Table S11-1. Since the lack-of- t F-statistic is f0 1.42, which has a P-value of P 0.3276, we cannot reject the hypothesis that the tentative model adequately describes the data. We will pool lack-of- t and pure-error mean squares to form the residual mean square that is the denominator mean square in the test for signi cance of regression. In addition, since the P-value for the statistic f0 6.66 with 1 and 14 degrees of freedom associated with signi cance of regression is P 0.0216, we conclude that 1 0. In tting a regression model to experimental data, a good practice is to use the lowest degree model that adequately describes the data. The lack-of- t test may be useful in this

( z k , z z ) = (11101111, 11011111)

Table S11-1 Analysis of Variance for Example S11-1 Source of Variation Regression Residual (Lack of t) (Pure error) Total Sum of Squares 3.4930 7.3372 4.3005 3.0366 10.8300 Degrees of Freedom 1 14 7 7 15 Mean Square 3.4930 0.5241 0.6144 0.4338 f0 6.66 1.42 P-value 0.0218 0.3276

11-4

The symbol 0 is usually pronounced as nabla or del , but feel free to call it upside-down triangle if you prefer.

respect. However, it is always possible to t a polynomial of degree n 1 to n data points, and the experimenter should not consider using a model that is saturated that is, that has very nearly as many independent variables as observations on y.

11-10

MORE ABOUT TRANSFORMATIONS (CD ONLY)

and many others. The point here is that while the signed differential is more restrictive than either a modular difference or an XOR difference (or a combination of the two), it still allows for considerable freedom in the choice of values that satisfy a given differential.

128 129 130 133 134 134 136 136 137 137

An Example As noted earlier in Section 11-9, transformations can be very useful in many situations where the true relationship between the response Y and the regressor x is not well approximated by a straight line. The utility of a transformation is illustrated in the following example.

EXAMPLE S11-2

Here, we attempt to provide some motivation for Wang s attack However, it appears that the actual development of the attack was essentially ad hoc Consequently, the motivation is not always clear, but Daum [34] has pulled together enough threads to provide a plausible explanation for much of the reasoning that most likely went into the development of the attack The remainder of this section draws primarily on material found in Daum s dissertation [34] Below, we distinguish between input differences and output differences Input differences are modular differences between input words of messages M and M , whereas output differences are differences between corresponding intermediate values, Q: and Qi The output difference operator is the signed difference discussed above, which is more restrictive than the modular difference or the XOR difference, but offers correspondingly greater control Recall that Dobbertin s MD4 attack discussed in Section 53.

A research engineer is investigating the use of a windmill to generate electricity and has collected data on the DC output from this windmill and the corresponding wind velocity. The data are plotted in Figure S11-2 and listed in Table S11-2. Inspection of the scatter diagram indicates that the relationship between DC output Y and wind velocity (x) may be nonlinear. However, we initially t a straight-line model to the data. The regression model is

0.2411 x

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