The MANOVA Test

Thank you for reading this article on the MANOVA test. Please note that this is a preview article. To read the full piece, please visit our website.

Multivariate analysis of variance (MANOVA) is an extension of the commonly used analysis of variance (ANOVA) method, allowing statistical comparisons across three or more groups of data and involving multiple outcome variables at the same time. This article will cover the theory underpinning MANOVA, the types of MANOVA and a worked example of the test.

What is MANOVA?

MANOVA is a statistical test that extends the scope of the more commonly used ANOVA, that allows differences between three or more independent groups of explanatory (independent or predictor) variables across multiple outcome (dependent or response) variables to be tested simultaneously. It is used when the explanatory variables are categorical (arranged into a limited number of values, e.g. gender, blood type) and the outcome variables are quantitative or continuous (able to take any number in a range of values, e.g. age, height). An example might be that we are interested in the effect of three different medications (explanatory variable) on both weight change and cholesterol levels (outcome variables). MANOVA does this by assessing the combined effect of the groups on the outcome variables based on the means in each of the independent groups.

Multivariate ANOVA formula

MANOVA follows similar analytical steps to the ANOVA method. It involves comparing the means of the outcome variables across each group and quantifying the within- and between-group variance. Some key steps in the calculation are the degrees of freedom (df) of the explanatory variable (the degrees to which the values of an analysis can vary), sum of squares (which summarizes the total variation between the group means and overall mean), the mean of the sum of squares (calculated by dividing the sum of squares by the degrees of freedom).

A step in MANOVA that differs from ANOVA is calculation of the test statistic. There are various test statistics that can be used in MANOVA, each suitable for different situations with regards to overall sample size, group size in the explanatory variable and the assumptions of the test being violated. These include Wilks’ Lambda, Pillai’s trace, Hotelling’s trace and Roy’s largest root. Wilks’ Lambda is most commonly used and so will be the focus of this article.

The formula for the Wilks’ Lambda test statistic in MANOVA is: A = E/T

Where E is the within-group covariance matrix, which measures variability of the outcome variables within each group of the explanatory variable. A matrix is a set of numbers arranged into rows and columns, and they are commonly used in statistics. T is the total covariance matrix, which includes E plus the between-group covariance matrix (H) which measures the variability of the outcome variables between each group of the explanatory variable. Wilks’ Lambda can then be converted to an F statistic to conduct a hypothesis test.

MANOVA assumptions

Some key assumptions of MANOVA are as follows:

• Multivariate normality: that the outcome variables follow a multivariate Normal distribution within each group of the explanatory variable.
• Homogeneity of covariance matrices: that the covariance matrices of the outcome variables should be equal across each group of the explanatory variable.
• Independence: that the observations in the data should be independent from one another and collected as a random sample of a population.

Types of multivariate analysis of variance

As with ANOVA, even more complex variations of the MANOVA method can be undertaken (Figure 1):??

• One way MANOVA: is the simplest form, comparing the means of three or more groups of data.
• Two way MANOVA: extends the method allowing for comparison between three or more groups of data across two explanatory (or independent) variables.
• Repeated measures MANOVA: extends the method to allow the same individuals to have multiple measures across different time points, meaning the data points are not independent.

Even more complex types of the test can be conducted, including multi-way MANOVA (more than two explanatory variables), and two way repeated measures MANOVA with an additional covariate. Multivariate analysis of covariance (MANCOVA) is an extension to MANOVA where two or more explanatory variables are compared simultaneously.??

Canonical correlation analysis (CCA) and other multivariate techniques

MANOVA is one of a group of techniques called multivariate statistical methods. “Multivariate” refers to multiple outcome variables assessed at once, this contrasts with “multiple” or “multivariable,” which usually refers to multiple explanatory variables. Other key techniques include canonical correlation analysis (CCA), where the relationship between two sets of variables is explored by finding linear combinations of the variables that are maximally correlated with each other. Another is principal component analysis (PCA), which transforms a dataset to reduce its complexity while minimizing information loss. Others include cluster analysis, multivariate regression and multidimensional scaling (MDS).

Written by Elliot McClenaghan

Read the full article to learn about the differences between MANOVA and ANOVA, when to use MANOVA analysis and see an example.