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In this case, the nonzero bit in the XOR difference occurs since y$ = 1 and y3 = 0. However, the same XOR difference in bit 3 would result if y$ = 0 and y3 = 1. For Wang s attack, we need to distinguish between cases such as these. To do so, we use a signed difference that is essentially a signed version of the XOR difference. That is, when there is a 1 in bit i of the XOR difference, we put a + if y = 1 and yi = 0 and we put a - if yi = 0 k and yi = 1. We use a . to indicate those bits that are 0 in the XOR difference, which simply means that y; = yi. We use the notation Vy this signed difference. As in the MD4 attack, we denote the modular difference as Ay. Then for y = 00010101 and y = 00000101 and we have Ay = 24 and Vy is given by . . .+ . . . . I . Now consider

Historical Note Sir Francis Galton rst used the term regression analysis in a study of the heights of fathers (x) and sons ( y) Galton t a least squares line and used it to predict the son s height from the fathers height He found that if a father s height was above average, the son s height would also be above average, but not by as much as the father s height was A similar effect was observed for short heights That is, the son s height regressed toward the average Consequently, Galton referred to the least squares line as a regression line Abuses of Regression Regression is widely used and frequently misused; several common abuses of regression are brie y mentioned here Care should be taken in selecting variables with which to construct regression equations and in determining the form of the model.

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It is possible to develop statistically signi cant relationships among variables that are completely unrelated in a causal sense For example, we might attempt to relate the shear strength of spot welds with the number of empty parking spaces in the visitor parking lot A straight line may even appear to provide a good t to the data, but the relationship is an unreasonable one on which to rely You can t increase the weld strength by blocking off parking spaces A strong observed association between variables does not necessarily imply that a causal relationship exists between those variables This type of effect is encountered fairly often in retrospective data analysis, and even in observational studies Designed experiments are the only way to determine causeand-effect relationships Regression relationships are valid only for values of the regressor variable within the range of the original data.

The linear relationship that we have tentatively assumed may be valid over the original range of x, but it may be unlikely to remain so as we extrapolate that is, if we use values of x beyond that range In other words, as we move beyond the range of values of x for which data were collected, we become less certain about the validity of the assumed model Regression models are not necessarily valid for extrapolation purposes Now this does not mean don t ever extrapolate There are many problem situations in science and engineering where extrapolation of a regression model is the only way to even approach the problem However, there is a strong warning to be careful A modest extrapolation may be perfectly all right in many cases, but a large extrapolation will almost never produce acceptable results..

= 00100101- 00010101= 24.

11-8.3

Lack-of-Fit Test (CD Only)

(5.46)

Regression models are often t to data to provide an empirical model when the true relationship between the variables Y and x is unknown. Naturally, we would like to know whether the order of the model tentatively assumed is correct. This section describes a test for the validity of this assumption. The danger of using a regression model that is a poor approximation of the true functional relationship is illustrated in Fig. S11-1. Obviously, a polynomial of degree two or greater in x should have been used in this situation. We present a test for the goodness of t of the regression model. Speci cally, the hypotheses we wish to test are H0: The simple linear regression model is correct. H1: The simple linear regression model is not correct.

11-1

111 111 113 114 114 115 115 118 119 120 121 124 124 125 125 125 126

z 3 2 = 00100101

11-2

The test involves partitioning the error or residual sum of squares into the following components: SSE SSPE SSLOF (S11-1)

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