ch15 - Chapter 15. Supplemental Text Material S15-1. The...

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Chapter 15. Supplemental Text Material S15-1. The Form of a Transformation In Section 3-4.3 of the textbook we introduce transformations as a way to stabilize the variance of a response and to (hopefully) induce approximate normality when inequality of variance and nonnormality occur jointly (as they often do). In Section 15-1.1 of the book the Box-Cox method is presented as an elegant analytical method for selecting the form of a transformation. However, many experimenters select transformations empirically by trying some of the simple power family transformations in Table 3-9 of Chapter 3 ( yy ,ln( ), / or 1 y , for example) or which appear on the menu of their computer software package. It is possible to give a theoretical justification of the power family transformations presented in Table 3-9. For example, suppose that y is a response variable with mean E y () = µ and variance Vy . That is, the variance of y is a function of the mean. We wish to find a transformation h ( y ) so that the variance of the transformed variable is a constant unrelated to the mean of y . In other words, we want Vh to be a constant that is unrelated to . f == σ 2 R xh y = ( ) y [ ( )] Ehy [() ] Expand x = h ( y ) in a Taylor series about µ , resulting in y hhy = =+ −+ ≅+ ( ) ( ) µµ where R is the remainder in the first-order Taylor series, and we have ignored the remainder. Now the mean of x is Ex Eh h y h ( ) [ ( ) ( )( )] = + = and the variance of x is Vx Ex Ex h y h y h ( ) ( ) =− −− = = 2 2 2 2 2 σµ Since , we have 2 = f ( ) f h () () = 2 We want the variance of x to be a constant, say c 2 . So set cf h 2 2 =
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and solve for , giving hy ( ) = h c f () µ Thus, the form of the transformation that is required is hc dt ft cG k = =+ z where k is a constant. As an example, suppose that for the response variable y we assumed that the mean and variance were equal. This actually happens in the Poisson distribution. Therefore, µσ == 2 implying that t So dt t ct d t k c t k ct k (/) / = = −+ + z z 12 1 12 1 2 This implies that taking the square root of y will stabilize the variance. This agrees with the advice given in the textbook (and elsewhere) that the square root transformation is very useful for stabilizing the variance in Poisson data or in general for count data where the mean and variance are not too different. As a second example, suppose that the square root of the mean is approximately equal to the variance; that is, . Essentially, this says that σ 2 / = 2 2 2 which implies that t Therefore, dt t c dt t k ctkt log( ) , = z z 2 0 if > This implies that for a positive response where the log of the response is an appropriate variance-stabilizing transformation. 2 / =
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S15-2. Selecting λ in the Box-Cox Method In Section 15-1.1 of the Textbook we present the Box-Cox method for analytically selecting a response variable transformation, and observe that its theoretical basis is the method of maximum likelihood. In applying this method, we are essentially maximizing Ln S S E () l n λλ =− 1 2 or equivalently, we are minimizing the error sum of squares with respect to λ . An approximate 100(1- α
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This note was uploaded on 03/20/2011 for the course STATISTIC 101 taught by Professor Fandia during the Spring '10 term at UCLA.

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ch15 - Chapter 15. Supplemental Text Material S15-1. The...

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