Analysis of variance is a regression model frequently employed to assess experimental data. Using multivariate analysis of variance, the relationship between multiple category-independent parameters and two or more metric-dependent variables is studied. MANOVA is an ANOVA with two or more continuous response variables, where M represents the multivariate nature of the analysis. MANOVA studies the dependent connection between a collection of variables across a set of groups, whereas ANOVA explores the differences across groups. MANOVA differs from other statistical tests because the independent variables are categorical, and the dependent variable is metric.

## MANOVA typologies #

MANOVA, like ANOVA, may be divided into three types according to the criterion utilized. A one-way MANOVA contrasts test scores and income by a single factor variable, for example, a categorical study period. This technique is identical to the one-way ANOVA (one criterion is used). It studies the link between a multilevel independent variable and many dependent variables.

A two-way MANOVA compares test scores and income by two factored factors, a categorical study term in this case. This technique is identical to the one-way ANOVA (one criterion is used). It studies the link between a multilevel independent variable and many dependent variables.

This technique is analogous to the two-way ANOVA (two criteria are used). It examines the association between a dichotomous independent variable and many dependent variables.

Factorial MANOVA is comparable to factorial ANOVA (when more than two criteria are used). It investigates the relationship between the variance of many nominal independent and dependent variables. Based on the subject design, there are three distinct factorial MANOVA.

- a) Factorial across subject designs: This design compares a single variable across multiple groups.
- b) Factorial within-subject design: Each respondent is evaluated based on multiple variables in this design. Utilized frequently in time-series analysis.
- c) Combined between-subject and within-subject design: Since both between-subject and within-subject designs can be useful in some situations, they are both employed.

## MANOVA: significance testing and model fit #

In the first phase of MANOVA analysis, researchers use the F test to examine the null hypothesis that there is no difference in the means of the dependent variables between groups. Other significant tests for multiple dependent variables that follow the F distribution include Hotelling’s T square, Wilks’ Lambda, and Pillai-Bartlett trace.

Post-Hoc Test: After determining the significance with the F test, researchers can utilize the post-hoc (meaning after this in Latin) F test to conclude the difference between groups. MANOVA investigates model fit by calculating the mean vector equivalents across groups. The post-hoc F test determines if the centroid of means of the dependent variables is the same for all independent variable groups.

## When should one use MANOVA? #

When your dependent variables are correlated, one should consider using multivariate ANOVA. The correlation structure between the dependent variables offers extra information to the model, enhancing MANOVA’s capabilities as follows:

- Greater statistical power: MANOVA can detect smaller effects than standard ANOVA when the dependent variables are correlated.
- Examine relationships between numerous dependent variables: Instead of impacting a single dependent variable, the model’s components might influence the connection between dependent variables.

Further, when you do a series of ANOVA tests because you have several dependent variables, the joint likelihood of rejecting a true null hypothesis grows with each subsequent test. Instead, if only one MANOVA test is performed, the error rate equals the significance threshold.

## MANOVA and ANOVA difference #

MANOVA enhances the possibilities of analysis of variance (ANOVA) by evaluating several dependent variables simultaneously. However, ANOVA has a disadvantage. It can only evaluate one dependent variable at a time. This restriction can be a huge concern in some situations since it can prevent you from recognizing existing impacts. Further, for some research queries, MANOVA gives a solution. This statistical approach simultaneously tests numerous dependent variables.