k - Sample t - test Redux

Testing linear combinations of normal group means with unequal variances and possibly unequal samples sizes using a t-statistic is a well known problem for which the effective degrees of freedom are unknown. ? and ? set forth their approximations based on the ratio of combinations of fourth moments, and their methodology remains the de facto standard. We propose a new approach that extends their results by defining what we call Measure of Equivalent Exchange (MEE), which is a function of a tuning parameter, τ , and the eigenvalues of a positive semidefinite matrix. We establish a number of important properties that MEE possesses and demonstrate its efficacy with simulations involving two and three group contrasts. We conclude with some commentary about the potential applications of MEE in other areas such as spatially correlated data and general covariance cross-validation. E-mail addresses: [email protected], [email protected], [email protected]. Date: January 23, 2013.