It is argued that the ANCOVA of original responses is essentially the ANOVA of the regression-adjusted or statistically controlled measurements obtained from the linear regression of unadjusted responses on the covariates that is common to all treatment groups. This lack of further discussion can partly be attributed to the simple framework and conceptual modification of Cohen ( 1988) on the use of ANOVA method for power evaluation in ANCOVA research. However, relatively little research has attempted to address the corresponding issues for ANCOVA. The corresponding results for multiple regression and correlation, especially the distinct notion of fixed and random regression settings, were given in Gatsonis and Sampson ( 1989), Mendoza and Stafford ( 2001), Sampson ( 1974), and Shieh ( 2006, 2007). Specifically, various algorithms and tables for power and sample size calculations in ANOVA have been presented in the classic sources of Bratcher et al. There are numerous published sources that address statistical theory and applications of power analysis for ANOVA and multiple linear regression. Accordingly, it is of great practical value to develop theoretically sound and numerically accurate power and sample size procedures for detecting treatment differences within the context of ANCOVA.
The importance and implications of statistical power analysis in scientific research are well demonstrated in Cohen ( 1988), Kraemer and Blasey ( 2015), Murphy et al. The extensive literature shows that it is one of the major methods of statistical analysis in applied research across many scientific fields. It is essential to note that ANCOVA provides a useful approach for combining the advantages of two highly acclaimed procedures of analysis of variance (ANOVA) and multiple linear regression. Comprehensive introduction and fundamental principles can be found in the excellent texts of Fleiss ( 2011), Huitema ( 2011), Keppel and Wickens ( 2004), Maxwell and Delaney ( 2004), and Rutherford ( 2011).
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. The analysis of covariance (ANCOVA) has notably proven to be an effective tool in a broad range of scientific applications.
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