## MANOVA

### Conceptual Model

### Conceptual model : An Example

### Observation in ANOVA

### MANOVA

### Conceptual Model: Hypothesis

### Conceptual: Model Parameter

### Assumptions of MANOVA

- Population covariance are equal.
- Errors are normally distributed.
- Errors are iid.
- Sensitivity of outliers.

### Test of Equality of Population Covariance's : Box M Test

### What is Spooled variance

### Box M text

### MANOVA table

### Statistical Measures

- Roy’s greatest characteristics root: It is most appropriate when the dependent variables are strongly interrelated. But it is also a measure most likely to be severely affected by violation of the assumption.
- Wilk’s lambda: While Roy is based on first discriminant function, Wilks lambda considers all discriminant function. That is it examines whether the groups are somehow different.
- Pillai criterion and Hoteling's T-square: It is similar to Wilks lambda.

### Factorial Design

- The effects of several independent variables or treatments rather than using only one single treatment. This capability is primary distinction between MANOVA and discriminant Analysis.
- An analysis with two or more treatments is called factorial design.

### MANCOVA

Covariate analysis purpose:

- To eliminate systematic error outside the control of researcher that can bias the result.
### Over all MANOVA

### Practical's

- # Box M test package biotools
- # Correlation matrix : you want various measurement to be higly correlated it will-
- # give you more power if they are correlated and less if they are not. #However they cannot be perfectly correlated.
- corr=cor(iris[,-5],use="pairwise.complete.obs")
- #Manova
- y<-cbind(iris$Sepal.Length,iris$Sepal.Width,iris$Petal.Length,iris$Petal.Width)
- model1<-manova(y~as.factor(iris[,5]),data=iris)
- model1
- summary(model1)
- summary(model1,test = "Wilks")
- summary(model1,test = "Roy")
- summary(model1,test = "Hotelling-Lawley")
- #Post hoc
- anova_sepal_len<-aov(iris$Sepal.Length~iris$Species)
- TukeyHSD(anova_sepal_len)
- anova_sepal_wid<-aov(iris$Sepal.Width~iris$Species)
- TukeyHSD(anova_sepal_wid)
- anova_Petal_len<-aov(iris$Petal.Length~iris$Species)
- TukeyHSD(anova_Petal_len)
- anova_Petal_wid<-aov(iris$Petal.Width~iris$Species)
- TukeyHSD(anova_Petal_wid)

### Quiz :

- Different between ANOVA and MANOVA ?
- What are the assumptions of ANOVA?
- What is the test for covariance?
- Name few statistical test in MANCOVA?
- Difference between Roy’s and Wilks test?
- What is box M test?
- What is Factorial Analysis?
- What is Covariate Analysis?