What is ANOVA? Types, Assumptions

What is ANOVA?

Analysis of Variance (ANOVA) is a collection of statistical models and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation. In its simplest form ANOVA provides a statistical test of whether or not the means of several groups are all equal, and therefore generalizes t-test to more than two groups.

Doing multiple two-sample t-tests would result in an increased chance of committing a type I error. For this reason, ANOVAs are useful in comparing two, three or more means.

An important technique for analyzing the effect of categorical factors on a response is to perform an Analysis of Variance. An ANOVA decomposes the variability in the response variable amongst the different factors. Depending upon the type of analysis, it may be important to determine: (a) which factors have a significant effect on the response, and/or (b) how much of the variability in the response variable is attributable to each factor.

Statgraphics Centurion provides several procedures for performing an analysis of variance:

• One-Way ANOVA – used when there is only a single categorical factor. This is equivalent to comparing multiple groups of data.
  
  
  
• Multifactor ANOVA – used when there is more than one categorical factor, arranged in a crossed pattern. When factors are crossed, the levels of one factor appear at more than one level of the other factors.
  
• Variance Components Analysis – used when there are multiple factors, arranged in a hierarchical manner. In such a design, each factor is nested in the factor above it.
  
  
• General Linear Models – used whenever there are both crossed and nested factors, when some factors are fixed and some are random, and when both categorical and quantitative factors are present.

One-Way ANOVA

A one-way analysis of variance is used when the data are divided into groups according to only one factor. The questions of interest are usually:

• Is there a significant difference between the groups and
• If so, which groups are significantly different from which others?

Statistical tests are provided to compare group means, group medians, and group standard deviations. When comparing means, multiple range tests are used, the most popular of which is Tukey’s HSD procedure.

For equal size samples, significant group differences can be determined by examining the means plot and identifying those intervals that do not overlap.

Multifactor ANOVA

When more than one factor is present and the factors are crossed, a multifactor ANOVA is appropriate. Both main effects and interactions between the factors may be estimated. The output includes an ANOVA table and a new graphical ANOVA from the latest edition of Statistics for Experimenters by Box, Hunter and Hunter (Wiley, 2005).

In a graphical ANOVA, the points are scaled so that any levels that differ by more than exhibited in the distribution of the residuals are significantly different.

Variance Components Analysis

A Variance Components Analysis is most commonly used to determine the level at which variability is being introduced into a product. A typical experiment might select several batches, several samples from each batch and then run replicates tests on each sample. The goal is to determine the relative percentages of the overall process variability that is being introduced at each level.

Assumptions of ANOVA

The analysis of variance has been studied from several approaches, the most common of which use a linear model that relates the response to the treatments and blocks. Even when the statistical model is nonlinear, it can be approximated by a linear model for which an analysis of variance may be appropriate.

• The model is correctly specified.
• The ij’s are normally distributed.
• The ij’s have mean zero and a common variance,
• The ij’s are independent across observations.

With multiple populations, detection of violations of these assumptions requires examining the residuals rather than the Y-values themselves.