## MANOVA

- It is an extension to univariate analysis (ANOVA). In ANOVA we have one continuous dependent variable while in MANOVA we have multiple continuous dependent variables.
- In one way MANOVA, one cannot tell that which specific group are significantly different from each other, it only tells that at least two groups are significantly different. As we have many groups in study it is important to know that which groups are significantly different, to do this we use post-hoc test.
- MANOVA is used when there are several correlated dependent variable and the researcher want to use a single test on these set of variable instead of performing multiple individual test.
- How independent variable influence the response of the dependent variable.
- The computation of several ANOVAs instead of one MANOVA increases the Type I error and thus the probability to wrongly reject the null hypothesis increases.
- The general purpose of multivariate analysis of variance (MANOVA) is to determine whether multiple levels of independent variables on their own or in combination with one another have an effect on the dependent variables. (https://uk.sagepub.com/sites/default/files/upm-binaries/9761_036226ch02.pdf)
- Like an ANOVA, MANOVA examines the degree of variance within the independent variables and determines whether it is smaller than the degree of variance between the independent variables. If the within subjects variance is smaller than the between subjects variance it means the independent variable has had a significant effect on the dependent. (https://uk.sagepub.com/sites/default/files/upm-binaries/9761_036226ch02.pdf)

Examples Where MANOVA can be used

- Suppose you want to know that which mall out of two mall (M1 & M2) is better considered for the factors like price, recreational activities, fashion, value for money, etc.
- For example, you might wish to test the hypothesis that sex and ethnicity interact to influence a set of job-related outcomes including attitudes toward co-workers, attitudes toward supervisors, feelings of belonging in the work environment, and identification with the corporate culture
- As another example, you might want to test the hypothesis that three different methods of teaching writing result in significant differences in ratings of student creativity, student acquisition of grammar, and assessments of writing quality by an independent panel of judges

Difference between MANOVA & ANOVA

- There are two main differences between MANOVAs and ANOVAs. The first is that MANOVAs are able to take into account multiple independent and multiple dependent variables within the same model, permitting greater complexity. (https://uk.sagepub.com/sites/default/files/upm-binaries/9761_036226ch02.pdf)
- Secondly, rather than using the F value as the indicator of significance a number of multivariate measures (Wilks’ lambda, Pillai’s trace, Hotelling trace and Roy’s largest root) are used. (https://uk.sagepub.com/sites/default/files/upm-binaries/9761_036226ch02.pdf)
- ANOVA is a special case of MANVOA in which only one dependent variable is analysed.

- There are majorly two types of MANOVA - one way MANOVA, two way MANOVA.
- When we compare two or more continuous response variable (dependent) by a single factor (independent), a one way MANOVA is used.
- In the same way when we compare two or more continuous response variable (dependent) by at least two factors (independent) , a two way MANOVA is used.

- In above one way MANOVA we can see that Independent Variable Education Level is used to find the impact on Monthly Income & Marks Secured and in this way they are compared.
- Similarily, in Two way MANOVA we can see two independent variables "Level of Education" & "No. of children" are used to compare their impact on Monthly Income & Marks Scored.
- A very classy video that I came across to understand MANOVA is linked below https://www.youtube.com/watch?v=jUksjmKvwos

MANOVA analysis using R

- Suppose we have dataset skull in which there are four dependent variables - mb,bh,bl,nh (all in integer format), and one independent variable "epoch" .
- To know MANOVA we use the following command

>summary(manova(cbind(mb,bh,bl,nh)~epoch,data=skull)) # to get the complete details

>summary(manova(cbind(mb,bh,bl,nh)~epoch,data=skull),"Pillai") # other test options are"Wilks", "Hotelling-Lawley","Pillai" and "Roy". The detail of these will be discussed in advanced tutorials.

Independent Variable & Dependent Variable

- Sometimes it become difficult to define Independent and Dependent variable, here is a short and clear video explaining the differentiation between Independent and dependent variable. https://www.youtube.com/watch?v=aeH1FzqdQZ0 https://www.youtube.com/watch?v=-mVHzIDN0A8