An Optimum Transformation for Somatic Cell Concentration in Milk

Abstract

Hypothesis testing in analysis of variance requires that the errors be distributed normally and that subclass variances be homogeneous. The purpose of this study was to find the transformation for somatic cell concentration which meets these characteristics of hypothesis testing. Data consisted of 51,800 monthly tests from 52 herds on Dairy Herd Improvement in Wisconsin. Cell concentration by the Filter-DNA method was thousands of cells/ml. Analysis of untransformed data revealed extensive departure from normality and homogeneity. The family of transformations was Y' = (Y+M) L for L~0 or Y' = Log e (Y+M) for L=0, where Y was the untransformed cell concentration. The analysis included L ranging from -.7 to +.6 and M ranging from 0 to 20. The maximum likelihood estimate of L was zero. By log transformation, student's t for skewness and kurtosis and Chi square for heterogeneity of variance were not different from zero. Adding a constant of 10 before taking the log caused a small improvement in tests of normality and heterogeneity of variance.