# Analysis of Variance and Covariance

Parametric and nonparametric analysis of variance, interactive and noninteractive analysis of covariance, multiple comparisons

Analysis of variance (ANOVA) is a procedure for assigning sample variance to different sources and deciding whether the variation arises within or among different population groups. Samples are described in terms of variation around group means and variation of group means around an overall mean. If variations within groups are small relative to variations between groups, a difference in group means may be inferred. Hypothesis tests are used to quantify decisions. Statistics and Machine Learning Toolbox™ offers several ways to perform ANOVA, including an `anova` object, command line functions, and an interactive app.

## Functions

expand all

 `anova` Analysis of variance (ANOVA) results
 `boxchart` Box chart (box plot) for analysis of variance (ANOVA) `groupmeans` Mean response estimates for analysis of variance (ANOVA) `multcompare` Multiple comparison of means for analysis of variance (ANOVA) `plotComparisons` Interactive plot of multiple comparisons of means for analysis of variance (ANOVA) `stats` Analysis of variance (ANOVA) table `varianceComponent` Variance component estimates for analysis of variance (ANOVA)
 `anova1` One-way analysis of variance `anova2` Two-way analysis of variance `anovan` N-way analysis of variance `canoncorr` Canonical correlation `dummyvar` Create dummy variables `friedman` Friedman’s test `kruskalwallis` Kruskal-Wallis test `multcompare` Multiple comparison test
 `aoctool` Interactive analysis of covariance

## Topics

• One-Way ANOVA

Use one-way ANOVA to determine whether data from several groups (levels) of a single factor have a common mean.

• Two-Way ANOVA

In two-way ANOVA, the effects of two factors on a response variable are of interest.

• N-Way ANOVA

In N-way ANOVA, the effects of N factors on a response variable are of interest.

• ANOVA with Random Effects

ANOVA with random effects is used where a factor's levels represent a random selection from a larger (infinite) set of possible levels.

• Other ANOVA Models

N-way ANOVA can also be used when factors are nested, or when some factors are to be treated as continuous variables.

• Multiple Comparisons

Multiple comparison procedures can accurately determine the significance of differences between multiple group means.

• Analysis of Covariance

Analysis of covariance is a technique for analyzing grouped data having a response (y, the variable to be predicted) and a predictor (x, the variable used to do the prediction).

• Nonparametric Methods

Statistics and Machine Learning Toolbox functions include nonparametric versions of one-way and two-way analysis of variance.