## 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 :

1. Different between ANOVA and MANOVA ?
2. What are the assumptions of ANOVA?
3. What is the test for covariance?
4. Name few statistical test in MANCOVA?
5. Difference between Roy’s and Wilks test?
6. What is box M test?
7. What is Factorial Analysis?
8. What is Covariate Analysis?