Question

Briefly explain to a non-statistician why ANOVA (which is a test of means) is called analysis of variance rather than analysis of means? what variances do we compare in anova?

Answer 1

Because while it compares means, it analyzes variances.

We might be interested in testing the sample means using hypothesis testing, where a specific statement is generated about a population parameter(means) and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The specific test considered here is called analysis of variance (ANOVA) and is a test of the hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials, there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and a standard treatment (i.e., medication currently being used). In a very famous observational study (Framingham Heart Study), the ANOVA technique can be applied to compare mean blood pressure or mean cholesterol levels for different age and sex.

Analysis of Variance (ANOVA) is a statistical technique that is used to test differences between two or more means. It may seem odd that the method is called "Analysis of Variance" rather than "Analysis of Means." As you will see, the name is appropriate because inferences about means are made by analyzing variance. ANOVA is used to test general rather than specific differences among means.

How does it work?

While comparing means of more than two groups ANOVA compares the variation between and the variation within. If the between is large compared to the within, then we have sufficient evidence of a group difference. If the between is small relative to the within, then there is much less evidence.

If we analyze means only then it means we find only difference between the means but that would be quite misleading because it is possible that means could differ a lot but the variation within each group is quite high. In this situation, it is difficult to conclude unless we incorporate the variation both between and within.

Thanks