Collections > Electronic Theses and Dissertations > Accounting for Bias and Uncertainty in Power for Multivariate Gaussian Linear Models
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In choosing a sample size for a study with a Gaussian outcome, scientists can nearly always specify, perhaps with some prodding, mean differences of clinical and scientific importance. Any difficulty in providing a believable power analysis revolves around having a believable value for error variance. Multivariate or repeated measures makes the problem far worse. The uncertainty of the result depends not only on the individual variances of the variables, but also on their covariance. Using an estimate of the covariance introduces uncertainty in power. An estimated covariance may also be biased due to distinct populations in the previous and future studies. I show how to overcome both problems in multivariate linear models, uncertainty and bias in power due to estimated covariance. Two different methods help, the confidence interval for power and an internal pilot design. Exact confidence intervals for noncentrality, power and sample size are known for the univariate model only. With an internal pilot design, data from the first stage of the study are used to re-calculate the sample size, based on the estimate of error variance. All data may be used in the final analysis, with no interim data analysis. A wide variety of exact and approximate results for internal pilot designs are known for univariate models, but not for multivariate models. For an important special class of multivariate tests (one "between" degree of freedom), I show how power can be computed from an equivalent univariate linear model. Therefore the theory and application of the univariate results for power confidence intervals can be applied with proper transformation of the problem. A similar approach allows using univariate results for an internal pilot design. Some additional exact results for confidence intervals are provided for another more general collection of models. Finally, approximations which apply to any general multivariate linear model are described and seen to be accurate in simulations.