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Variance Stabilizing Transformations

Variance Stabilizing Transformations . 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) = m

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Variance Stabilizing Transformations

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  1. Variance Stabilizing Transformations

  2. 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) = m • Binomial Data (and Percents): E(Y) = np V(Y) = np(1-p) • General Case: E(Y) = m V(Y) = W(m) • Power relationship: V(Y) = s2 = a2m2b

  3. Transformation to Stabilize Variance (Approximately) • V(Y) = s2 = W(m). Then let: This results from a Taylor Series expansion:

  4. Special Case: s2 = a2m2b

  5. Estimating b From Sample Data • For each group in an ANOVA (or similar X levels in Regression, obtain the sample mean and standard deviation • 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 b • For large n, can fit a regression of squared residuals on predictors expected to be related to variance

  6. Example - Bovine Growth Hormone

  7. Example - Bovine Growth Hormone Estimated b = .84  1, A logarithmic transformation on data should have approximately constant variance

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