| New methods that improve upon current techniques related to power for UNIREP tests are introduced. The research is motivated by imaging applications, which often generate the type of data that can be handled with UNIREP techniques. The UNIREP Huynh-Feldt test is based on the Huynh-Feldt sphericity estimator. Claiming their estimator was a ratio of unbiased estimators, Huynh and Feldt developed it as an alternative to the sometimes biased Geisser-Greenhouse estimator. The Huynh-Feldt estimator is examined and shown to be a ratio of unbiased estimators only for the special case of rank of the design matrix, X, equal to 1. This realization results in a biased Huynh-Feldt test and power calculation when rank of X is greater than 1. A proper, adjusted Huynh-Feldt estimator for any rank of X is presented and shown to better estimate the population sphericity when rank of X is greater than 1. A power approximation for the rank-adjusted Huynh-Feldt test is also presented. For practical research situations, the rank-adjusted Huynh-Feldt power approximation is shown to perform as well as and, in most cases, better than the most accurate Huynh-Feldt power approximation in use. Furthermore, the Huynh-Feldt power approximation is shown to yield artificially inflated power values at a cost of inflated test size when rank of X is greater than 1. The rank-adjusted test is shown to control test size adequately. Approximate confidence intervals for UNIREP power in the case of an estimated covariance and fixed means are introduced and shown to provide reasonably accurate coverage probabilities for all four UNIREP tests. The approximate confidence intervals perform well in most cases considered, even for small sample sizes. The approximate confidence intervals are shown to perform better for higher power values than for lower power values, making them more useful in practical research conditions. Factors affecting UNIREP power confidence interval coverage probabilities are examined. These factors include sample size, rank of and the degrees of freedom for both X the estimating and target studies, as well as estimated sphericity multipliers. To provide tighter, more informative confidence bounds, one-sided confidence intervals are recommended. |
