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) =  –Binomial Data (and Percents): E(Y) = n  V(Y) = n  –General Case: E(Y) =  V(Y) =  –Power relationship: V(Y) =  2 =    

Transformation to Stabilize Variance (Approximately) V(Y) =  2 = . Then let: This results from a Taylor Series expansion:

Special Case:      

Estimating  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  For large n, can fit a regression of squared residuals on predictors expected to be related to variance

Example - Bovine Growth Hormone

Estimated  =.84  1, A logarithmic transformation on data should have approximately constant variance