Analysis of variance (ANOVA) comparing means of more than two groups

74 This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/ by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. Analysis of variance (ANOVA) comparing means of more than two groups Mean values obtained from different groups with different conditions are frequently compared in clinical studies. For example, two mean bond strengths between tooth surface and resin cement may be compared using the parametric Student's t test when independent groups are subjected to the comparison under the assumptions of normal distribution and equal variances (or standard deviation). In a condition of unequal variances we may apply the Welch's t test as an adaptation of the t test. As the nature and specific shape of distributions are predetermined by the assumption, the t test compares only the locations of the distribution represented by means, which is simple and intuitive. The t statistic is the ratio of mean difference and standard errors of the mean difference. Even when more than two groups are compared, some researchers erroneously apply the t test by implementing multiple t tests on multiple pairs of means. It is inappropriate because the repetition of the multiple tests may repeatedly add multiple chances of error, which may result in a larger α error level than the pre-set α level. When we try to compare means of three groups, A, B, and C, using the t test, we need to implement 3 pairwise tests, i.e., A vs B, A vs C, and B vs C. Similarly if comparisons are repeated k times in an experiment and the α level 0.05 was set for each comparison, an unacceptably increased total error rate of 1-(0.95) k may be expected for the total comparison procedure in the experiment. For a comparison of more than two group means the one-way analysis of variance (ANOVA) is the appropriate method instead of the t test. As the ANOVA is based on the same assumption with the t test, the interest of ANOVA is on the locations of the distributions represented by means too. Then why is the method comparing several means the 'analysis of variance', rather than 'analysis of means' themselves? It is because that the relative location of the several group means can be more conveniently identified by variance among the group means than comparing many group …