MANOVA (Multivariate Analysis of Variance) by R-Bloggers
What is MANOVA (Multivariate Analysis of Variance)?
▪ MANOVA is an extension to univariate ANOVA that includes at least two dependent variables to analyze differences between multiple groups in the independent variable.
▪ In contrast to ANOVA, where we compare individual group means, MANOVA compares the vectors containing the group mean of each dependent variable.
▪ MANOVA uses omnibus Wilk’s Lambda, Pillai’s Trace (most robust to departures from assumptions), Roy’s Largest Root, or Hotelling-Lawley’s test, which are later converted to F statistics for assessing the significance of the group differences. Pillai’s Trace has the highest statistical power.
▪ MANOVA maximizes the discrimination in between-groups than within-groups based on best linear combinations of the multiple response variables.
Why MANOVA is useful?
▪ MANOVA analyzes the differences among groups in an independent variable by considering multiple dependent variables. This reduces the type I error which can be inflated by performing separate univariate ANOVA for each dependent variable.
▪ MANOVA also controls for inter-correlations among the multiple dependent variables in the dataset.
▪ As compared to univariate ANOVA, MANOVA uses more information from the dependent variables i.e., MANOVA may find differences between groups based on combined information from the multiple dependent variables.
If there is only one dependent variable is of interest for quantifying the differences between groups, then MANOVA is not necessary.
Assumptions of MANOVA
MANOVA follows similar assumptions as in ANOVA for the independence of observations and homogeneity of variances
In addition, MANOVA needs to meet the following assumption,
▪ Multivariate normality: data or residuals should have a multivariate normal distribution for each combination of independent and dependent variables (checked by Shapiro-Wilk test for univariate normality and Mardia’s skewness and kurtosis for multivariate normality)
▪ Homogeneity of the variance-covariance matrices: data should have equal variance-covariance matrices for each combination formed by each group in the independent variable. This is a multivariate version of the Homogeneity of variances that is checked in univariate ANOVA. It can be tested using Box’s M test. Box’s M-test has little power and uses a lower alpha level such as 0.001 to assess the p value for significance.
▪ Multicollinearity: There should be no multicollinearity (very high correlations i.e., > 0.9) among dependent variables
▪ Linearity: dependent variables should be linearly related for each group of the independent variable. If there are more than two dependent variables, the pair of dependent variables should be linearly related
▪ There should be no outliers in the dependent variables (multivariate outliers) (checked by Mahalanobis distances)
▪ dependent variables should be continuous
▪ Null hypothesis: group mean vectors are same for all groups
▪ Alternative hypothesis: group mean vectors are not same for all groups