variance - • Fit a simple linear regression relating the...

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Variance Stabilizing Transformations
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Variance is Related to Mean Usual Assumption in ANOVA and Regression is that the variance of each observation is the same Problem: In many cases, the variance is not constant, but is related to the mean. Poisson Data (Counts of events): E(Y) = V(Y) = μ Binomial Data (and Percents): E(Y) = n π V(Y) = n π(1-π29 General Case: E(Y) = μ V(Y) = Ω(μ29 Power relationship: V(Y) = σ 2 = α 2 μ * * ) ln( ) ln( ) ln( βμ α μ β α σ αμ σ β + = + = =
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Transformation to Stabilize Variance (Approximately) V(Y) = σ 2 = Ω(μ29 . Then let: [ ] [ ] constant ) ( ) ( 1 ) ( 2 / 1 = Y f V d f μ μ μ This results from a Taylor Series expansion: [ ] [ ] 1 )) ( ( 1 ) ( )) ( ( ) ( ' ) ( ) ( ) ( ) ( ' ) ( ) ( ) ( 2 2 / 1 2 2 2 = - - - + μ μ μ μ μ μ μ μ Y f V f Y f Y f f Y f y f
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Special Case: σ 2 = α 2 μ [ ] [ ] ) ln( 1 1 ) ( 1 ) ( 1 : 2 Case 1 1 1 ) ( 1 ) ( : 1 : 1 Case 2 / 1 1 1 2 / 1 μ α μ αμ μ μ μ β μ β μ α μ αμ μ μ μ β β β β = = = = = + - = = = - + - d d f c d d f
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Estimating β From Sample Data For each group in an ANOVA (or similar X levels
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Unformatted text preview: • Fit a simple linear regression, relating the log of the standard deviation to the log of the mean • The regression coefficient of the log of the mean is an estimate of β Example - Bovine Growth Hormone Bovine Growth Hormone Data 10 20 30 40 50 60 70 50 100 150 200 250 300 350 400 450 500 Mean Std Dev Example - Bovine Growth Hormone ln(mean) ln(sd) 5.7807 3.6687 6.0684 4.0993 5.7900 3.6661 5.7621 3.9703 5.7838 3.8351 5.7930 3.7612 4.9972 3.1946 5.3799 3.4751 4.9416 2.9755 5.0239 3.4340 4.9904 3.0910 5.0239 3.0204 Coefficients Standard Error Intercept-1.0553 0.5373 ln(mean) 0.8396 0.0984 Estimated β = .84 ≈ 1, A logarithmic transformation on data should have approximately constant variance...
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