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Variance Stabilizing Transformations: Addressing Non-constant Variance in ANOVA and Regression

Variance is often related to the mean, leading to assumptions in ANOVA and regression that can be violated. When variance is not constant, it poses challenges for analysis. This guide discusses the Poisson and Binomial data relationships and introduces a power transformation to stabilize variance: V(Y) = a²m²b. By estimating parameters using sample data, including the regression of log standard deviation against log mean, we can ensure more robust statistical modeling. Examples, such as the Bovine Growth Hormone study, illustrate effective applications.

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Variance Stabilizing Transformations: Addressing Non-constant Variance in ANOVA and Regression

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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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