Analysis of variance.

Analysis of variance (ANOVA) is a statistical technique to analyze variation in a response variable (continuous random variable) measured under conditions defined by discrete factors (classification variables, often with nominal levels). Frequently, we use ANOVA to test equality among several means by comparing variance among groups relative to variance within groups (random error). Sir Ronald Fisher pioneered the development of ANOVA for analyzing results of agricultural experiments.1 Today, ANOVA is included in almost every statistical package, which makes it accessible to investigators in all experimental sciences. It is easy to input a data set and run a simple ANOVA, but it is challenging to choose the appropriate ANOVA for different experimental designs, to examine whether data adhere to the modeling assumptions, and to interpret the results correctly. The purpose of this report, together with the next 2 articles in the Statistical Primer for Cardiovascular Research series, is to enhance understanding of ANVOA and to promote its successful use in experimental cardiovascular research. My colleagues and I attempt to accomplish those goals through examples and explanation, while keeping within reason the burden of notation, technical jargon, and mathematical equations. Here, I introduce the ANOVA concept and provide details for 2 common models. The first model, 1-way fixed-effects ANOVA, is an extension of the Student 2-independentsamples t test that lets us simultaneously compare means among several independent samples. The second model, 2-way fixed-effects ANOVA, has 2 factors, A and B, and each level of factor A appears in combination with each level of factor B. This model lets us compare means among levels of factor A and among levels of factor B; furthermore, we may examine whether combined factors induce interaction effects (synergistic or antagonistic) on the response. In the second ANOVA article, the author reviews several multiple-comparisons procedures for analysis of differences among means, including comparisons between pairs of group means and more general contrasts among group means. Usually, multiple-comparisons procedures are used to control type I error rate across numerous hypothesis tests. In the third ANOVA report, the author introduces repeated-measures ANOVA for use when each experimental unit contributes response data at each level of a fixed factor (eg, different treatment doses). Statistical textbooks2,3 and online documents4,5 provide readers with more technical detail for similar material or with broader coverage of topics beyond the scope of these articles.