![]() The value and use of ANCOVA have also received considerable attention in social science, for example, see Elashoff ( 1969), Keselman et al. Its essential nature and principal use were well explicated by Cochran ( 1957) and subsequent articles in the same issue of Biometrics. The analysis of covariance (ANCOVA) was originally developed by Fisher ( 1932) to reduce error variance in experimental studies. In order to facilitate the application of the described power and sample size calculations, accompanying computer programs are also presented. The improved solution is illustrated with an example regarding the comparative effectiveness of interventions. Both theoretical examination and numerical simulation are presented to justify the advantages of the suggested technique over the current formula. An exact approach is proposed for power and sample size calculations in ANCOVA with random assignment and multinormal covariates. This article aims to explicate the conceptual problems and practical limitations of the common method. The frequently recommended procedure is a direct application of the ANOVA formula in combination with a reduced degrees of freedom and a correlation-adjusted variance. Despite the well-documented literature about its principal uses and statistical properties, the corresponding power analysis for the general linear hypothesis tests of treatment differences remains a less discussed issue. Furthermore these differences can be seen in the absence of changes in mean signal.The analysis of covariance (ANCOVA) has notably proven to be an effective tool in a broad range of scientific applications. The most remarkable observation is that these responses can be biphasic and show profound differences in their form depending on the extant task or condition. However, its application to characterizing the temporal aspects of evoked hemodynamic responses reveals some compelling and somewhat unexpected perspectives on transient but stereotyped responses to changes in cognitive or sensorimotor processing. We do not propose that this analysis is a particularly powerful way to make inferences about functional specialization (or more generally functional anatomy) because it only provides statistical inferences about the distributed (whole brain) responses evoked by different conditions. To address this issue we have modeled hemodynamic responses using appropriate temporal basis functions and estimated their exact form within the general linear model using MANCOVA. In particular we do not assume that the neural or hemodynamic response reaches some "steady state" but acknowledge that these physiological changes could show profound task-dependent adaptation and time-dependent changes during the task. We have used these techniques to characterize the form of hemodynamic transients that are evoked during a cognitive or sensorimotor task. This analysis uses standard multivariate statistics (MANCOVA) and the general linear model to make inferences about effects of interest and canonical variates analysis (CVA) to describe the important features of these effects. In this paper we present a multivariate analysis of evoked hemodynamic responses and their spatiotemporal dynamics as measured with fast fMRI.
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