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# notes11 - Sections 3.9 and 6.8 Transformations Timothy...

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Sections 3.9 and 6.8: Transformations Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1 / 24

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Transformations of variables (Section 3.9 & p. 236) Some violations of our model assumptions may be fixed by transforming one or more predictors x 1 , . . . , x k or Y . If the only problem is a nonlinear relationship between Y and the predictors, i.e. constant variance seems okay, a transformation of one or more of the x 1 , . . . , x k is preferred. If non-constant variance appears in one or more plots of Y versus the predictors, a transformation in Y can help...or make it worse! Data analysis is an art. The best way to learn how to analyze data is to analyze data. A nonlinear relationship could manifest itself the scatterplot matrix of Y i versus x ij for j = 1 , . . . , k , or the residuals e i versus x ij from an initial fit. The chosen transformation should roughly mimic the relationship seen in the plot. 2 / 24
Transformations for x i 1 , . . . , x ik Examples of transformations for predictors are: x * = log( x ) x * = x x * = 1 / x x * = exp( x ) or x * = exp( - x ) See Figure 3.13, page 130. 3 / 24

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Example 1: transforming a predictor ●● ●● ●● ●●● 0 50 100 150 0 2 4 6 x y ●● ●● ●● ●● ●●● 1 2 3 4 5 0 2 4 6 log(x) y ●● ●● ●● ●●● 2 4 6 8 10 0 2 4 6 sqrt(x) y 4 / 24
Example 2: transforming a predictor ●● ●● ●● ●● ●● ●● ●● ●● ●● 0.0 1.0 2.0 2 4 6 8 10 x y ●● ●● ●● ●● ●●● ●● ●● ●● ●● ●●● 0 1 2 3 4 5 2 4 6 8 10 x^2 y ●● ●● ●●

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