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The ith studentized residual is de ned as ri B ei
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(a) Explain why ri has unit standard deviation. (b) Do the standardized residuals have unit standard deviation (c) Discuss the behavior of the studentized residual when the sample value xi is very close to the middle of the range of x. (d) Discuss the behavior of the studentized residual when the sample value xi is very near one end of the range of x.
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We occasionally nd that the straight-line regression model Y is inappropri0 1x ate because the true regression function is nonlinear. Sometimes nonlinearity is visually determined from the scatter diagram, and sometimes, because of prior experience or underlying theory, we know in advance that the model is nonlinear. Occasionally, a scatter diagram will exhibit an apparent nonlinear relationship between Y and x. In some of these situations, a nonlinear function can be expressed as a straight line by using a suitable transformation. Such nonlinear models are called intrinsically linear. As an example of a nonlinear model that is intrinsically linear, consider the exponential function Y
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This function is intrinsically linear, since it can be transformed to a straight line by a logarithmic transformation ln Y ln
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This transformation requires that the transformed error terms ln are normally and independently distributed with mean 0 and variance 2. Another intrinsically linear function is Y By using the reciprocal transformation z Y
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Sometimes several transformations can be employed jointly to linearize a function. For example, consider the function Y letting Y* exp 1 1
1 Y , we have the linearized form ln Y*
0 1x
For examples of tting these models, refer to Montgomery, Peck, and Vining (2001) or Myers (1990).
11-10 11-11
Our development of regression analysis has assumed that x is a mathematical variable, measured with negligible error, and that Y is a random variable. Many applications of regression analysis involve situations in which both X and Y are random variables. In these situations, it
is usually assumed that the observations (Xi, Yi), i 1, 2, p , n are jointly distributed random variables obtained from the distribution f (x, y). For example, suppose we wish to develop a regression model relating the shear strength of spot welds to the weld diameter. In this example, weld diameter cannot be controlled. We would randomly select n spot welds and observe a diameter (Xi) and a shear strength (Yi) for each. Therefore (Xi, Yi) are jointly distributed random variables. We assume that the joint distribution of Xi and Yi is the bivariate normal distribution pre2 2 sented in 5, and Y and Y are the mean and variance of Y, X and X are the mean and variance of X, and is the correlation coef cient between Y and X. Recall that the correlation coef cient is de ned as
where XY is the covariance between Y and X. The conditional distribution of Y for a given value of X fY 0 x 1 y2 where
Y 0 Y Y 1 X X X
x is
0 Y 0x 1x
exp c
1 y a 2
(11-37) (11-38) x is (11-39)
and the variance of the conditional distribution of Y given X
2 Y 0x 2 Y 11 2
That is, the conditional distribution of Y given X E1Y 0 x2
x is normal with mean
and variance 2 0 x . Thus, the mean of the conditional distribution of Y given X x is a Y simple linear regression model. Furthermore, there is a relationship between the correlation 0, then 1 0, which coef cient and the slope 1. From Equation 11-38 we see that if implies that there is no regression of Y on X. That is, knowledge of X does not assist us in predicting Y. The method of maximum likelihood may be used to estimate the parameters 0 and 1. It can be shown that the maximum likelihood estimators of those parameters are
X 1