Analysis of Variance in Medical Research.

Abstract

KEY POINT: Analysis of variance (ANOVA) is used to test for the equality of the mean values of a continuous outcome between groups.In this issue of Anesthesia & Analgesia, Sabourdin et al1 report results of a study on the effect of propofol concentrations on pupillary diameter. Patients were randomized to 3 groups based on a targeted propofol concentration. These authors used several analysis techniques, including a comparison of pupillary diameters between the groups with a 1-way analysis of variance (ANOVA). ANOVA is a family of statistical methods to compare the mean values of different groups. The term “analysis of variance” is based on the principle that ANOVA partitions the total observed variability (variance) in the outcome variable into distinct components, as described in more detail below. The most simple ANOVA method is 1-way ANOVA, which involves one categorical independent (predictor) variable—typically, the study group variable—and one continuous dependent (outcome) variable. It extends the 2-sample unpaired t test2 to >2 groups, and tests the null hypothesis that all the population means are equal. In a 1-way ANOVA, the between-group variance is compared to the within-group variance, and the ratio of the 2 is summarized in a so-called F statistic. Assuming the null hypothesis is true, the variability attributable to between-group differences should be relatively low, and most of the observed variability should be attributable to within-group differences. Thus, a large F statistic suggests that the null hypothesis is implausible. If it is larger than a certain threshold—which depends on the α level and the degrees of freedom—the associated P value is lower than α, the null hypothesis is rejected, and the authors can claim an overall between-group difference. While ANOVA tells us whether group mean values are significantly different, it does not tell us which specific groups differ from each other. A variety of post hoc tests are available to address this question, such as the Tukey, Bonferroni, or Dunnett test. These post hoc tests are conceptually similar to performing multiple pairwise t tests, but they adjust for inflation of the type I error risk due to multiple testing.3 The exception is the Fisher least significant difference (LSD) test. The choice depends on which groups are being compared (eg, each group with each other versus with a control group) and on how conservative one wishes to be on the multiplicity adjustment. Valid inferences from a 1-way ANOVA rely on several assumptions being met: The observations are independent of each other (both within and between groups). The dependent variable is approximately normally distributed in each group. The variances in each group are approximately equal. For study designs in which >1 independent variable is of interest, different ANOVA methods are available. A factorial ANOVA allows for >1 categorical independent (predictor) variable. Two-way ANOVA is an example that allows testing for the effects of 2 categorical variables on the outcome (eg, treatment group and patient sex), as well as for the interaction of the 2. Analysis of covariance (ANCOVA) is yet another method that allows adjusting for a continuous covariate (eg, patient age).4 For nonindependent data—quite commonly, data repetitively measured over time in the same subjects—repeated-measures ANOVA methods are available and more appropriate.5Figure.: Figure 1 from Sabourdin et al1 and an excerpt from their results sections. The Figure shows the individual pupillary diameters (circles) and means (solid horizontal line) for each randomized propofol group. The Figure includes a dashed line because authors also tested for a linear relationship between the targeted propofol concentration and pupillary diameter. The 1-way ANOVA F statistic is 15.9, with 2 (numerator) and 37 (denominator) degrees of freedom. The F statistic is a ratio (see text). This F statistic corresponds to a P value of <0.001, allowing the null hypothesis of equal group means to be rejected. ANOVA indicates analysis of variance; Cet, effect site concentration.