1. Multivariate Analysis of Variance
MANOVA
Multivariate Analysis of Variance

2. One way Multivariate Analysis of Variance (MANOVA)
Comparing k p-variate Normal Populations

3. Comparing k mean vectors
Situation
We have k normal populations
Let denote the mean vector and covariance matrix of population i.
i = 1, 2, 3, … k.
Note: we assume that the covariance matrix for each population is the same.

4. We want to test
against

5. The data Assume we have collected data from each of k populations
Let denote the n observations from population i.
i = 1, 2, 3, … k.

6. The summary statistics
Sample mean vectors
Sample covariance matrices
S1, S2, etc.

7. Computing Formulae:
Compute 1) 2) 3)

8. 4) 5)

9. Let
= the Between SS and SP matrix

10. Let
= the Within SS and SP matrix

11. The Manova Table
Source
SS and SP matrix
Between
Within

12. There are several test statistics for testing
against

13. 1. Roy’s largest root 2. Wilk’s lambda (L)
This test statistic is derived using Roy’s union intersection principle
2. Wilk’s lambda (L)
This test statistic is derived using the generalized Likelihood ratio principle

14. 3. Lawley-Hotelling trace statistic
4. Pillai trace statistic (V)

15. Example
In the following study, n = 15 first year university students from three different School regions (A, B and C) who were each taking the following four courses (Math, biology, English and Sociology) were observed: The marks on these courses is tabulated on the following slide:

16. The data

17. Summary Statistics

18. Computations : 1) 2) 3)

19. 4) =

20. 5) =

21. Now
= the Between SS and SP matrix

22. Let
= the Within SS and SP matrix

23. Using SPSS to perform MANOVA

24. Selecting the variables and the Factors

25. The output

26. Univariate Tests

27. Profile Analysis

28. Repeated Measures Designs

29. In a Repeated Measures Design
We have experimental units that
may be grouped according to one or several factors (the grouping factors)
Then on each experimental unit we have
not a single measurement but a group of measurements (the repeated measures)
The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors)

30. Example
In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
The enzyme was measured
immediately after surgery (Day 0),
one day (Day 1),
two days (Day 2) and
one week (Day 7) after surgery
for n = 15 cardiac surgical patients.

31. The data is given in the table below.
Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

32. The subjects are not grouped (single group).
There is one repeated measures factor -Time – with levels
Day 0,
Day 1,
Day 2,
Day 7
This design is the same as a randomized block design with
Blocks = subjects

33. The Anova Table for Enzyme Experiment
The Subject Source of variability is modelling the variability between subjects
The ERROR Source of variability is modelling the variability within subjects

34. Example : (Repeated Measures Design - Grouping Factor)
In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery.
In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

35. The 24 patients were randomly divided into three groups of n= 8 patients.
The first group of patients were left untreated as a control group while
the second and third group were given drug treatments A and B respectively.
Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

36. Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery for three treatment groups (control, Drug A, Drug B)

37. The subjects are grouped by treatment
control,
Drug A,
Drug B
There is one repeated measures factor -Time – with levels
Day 0,
Day 1,
Day 2,
Day 7

38. The Anova Table
There are two sources of Error in a repeated measures design:
The between subject error – Error1 and
the within subject error – Error2

39. Tables of means Drug Day 0 Day 1 Day 2 Day 7 Overall
Control A B
Overall

41. Example : Repeated Measures Design - Two Grouping Factors
In the following example , the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task.
In addition the researcher was interested in how the error rates also changed over time.
Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.

42. The number of errors committed at each instance is tabulated below.

43. The Anova Table