1. Multivariate Analysis of Variance II
BMTRY 726
6/26/18

2. What Happens With > 2 Factors?
So far we’ve only talked about the case where we look at the impact of a single factor/treatment on p outcomes
What happens if we have more than one factor we want to consider…

3. Two-Way MANOVA
What happens if we have 2 treatments we want to compare on p outcomes? Consider the univariate case with treatments 1 and 2 The expected response for kth & lth levels of A & B is
Mean
Response
Overall
level
Factor 1
Effect
Factor 2
Effect
Factor
Interaction
Effect

4. Visualize the Data Suppose we have data with two factors
-treatment A has l = 1,2,3 levels
-treatment B has k = 1,2,3,4 levels

5. Two-Way MANOVA
The vector of measurements taken on the rth subject treated at the lth level of treatment 1 and the kth level of treatment 2 can be written as:

6. Two-way MANOVA So given a multivariate outcome we have: Assumptions:
-These are all p x 1 vectors and all elkr ~N (0,S) are independent random vectors.
-We also constrain the model such that
The response includes p measures replicated n times at each possible combination of treatments 1 and 2.

7. Two-Way MANOVA
We can write this in linear model form:

8. Two-way MANOVA
We can decompose this as follows:

9. From this we can derive the MANOVA table:
Source
VariationI
Matrix Sum of Squares
Degrees of Freedom
Treatment 1
Treatment 2
Interaction
Residual
Total

10. Hypothesis Tests
Generally start by testing for interactions… We can test this hypothesis using Wilk’s lambda NOTE: The LRT requires p < gb(n-1) so SSPres will be positive definite

11. Hypothesis Tests
If we fail to reject the null of an interaction effect, we should then test for our factor effects: We can test this hypothesis using Wilk’s lambda

12. Hypothesis Tests
Note a critical value based on a c2 distribution better for large n For small samples we can compute an F-statistic since this is (sometimes) an exact distribution However, the d.f. are complicated to estimate- the book example works for g = b = 2:

13. Confidence Intervals
As with the one-way MANOVA case, we can estimate Bonferroni confidence intervals

14. Example: Cognitive impairment in Parkinson’s Disease
It is known that lesions in the pre-frontal cortex are responsible for much of the motor dysfunction subject’s with Parkinson’s disease experience. Cognitive impairment is a less well studied adverse outcome in Parkinson’s disease. An investigator hypothesizes that lesions in the locus coeruleus region of the brain are in part responsible for this cognitive deficit. The PI also hypothesizes that this is partly due to decreased expression of BDNF.

15. Experimental Design
The PI wants to investigate the effect of lesion location on cognitive behavior in Parkinson’s model rats.
She also wants to investigate a therapeutic application of BDNF on cognitive performance.
-Outcomes
-Novel Object Recognition (NOR)
-Water Radial Arm Maze (WRAM)
Two experimental factors for six groups:
-Lesion type
-6-OHDA
-DSP-4
-Double
-Therapy
-BDNF microspheres
-No treatment

16. DSP-4 Lesions effect
noradrenergic pathways
in the LC
Two groups of rats receive
these single lesions
-No treatment
-BDNF Treated

17. 6-OHDA Lesions effect
dopaminergic pathways
in the PFC
Two groups of rats receive
these single lesions
-No treatment
-BDNF Treated

18. Double lesion animals
receive both 6-OHDA and
DSP-4 lesions affecting
the LC and PFC
Two groups of rats receive
these single lesions
-No treatment
-BDNF Treated

19. The data The data are arranged in an array. The six matrices represent
the six possible combos
of treatment and lesion type
Column 1 in each matrix are
the NOR task results
Column 2 in each matrix are
the WRAM task results

20. Means for Sums of Squares

21. Means for Sums of Squares

22. Sums of Squares

23. Sums of Squares

24. Hypothesis testing

25. Hypothesis testing

26. A Graphical View of Our Results

27. Bonferroni Simultaneous CIs

28. Bonferroni Simultaneous CIs

29. Conclusions from the Parkinson’s Study
There is not a significant interaction between lesion type and BDNF therapy. Treatment with BDNF in a Parkinson’s rat model significantly increased the mean time rats spent in the NOR task. Treatment with BDNF in a Parkinson’s rat model significantly decreased the time rats needed to complete the WRAM task.

30. Conclusions from the Parkinson’s Study
Rats with double lesions spent significantly short time
with novel objects relative to 6OHDA lesioned rats.
2. Double lesioned rats took a significantly longer amount
of time to complete the WRAM task relative to 6OHDA
rats.

31. Some Things to Note
If interactions are present, interpretation is difficult
One approach is to examine the p variables independently (p univariate ANOVAs) to see which of the p outcomes have interactions
Extension to designs with more than two factors is fairly straight forward
Such models could consider higher order interactions as well

32. 2-Way MANOVA as a Linear Model
As with the one-way MANOVA, SAS thinks about a two-way MANOVA as a linear model Also as in the one-way case, the intercept represents the mean for some reference category However, since we now have two factors, we have to have a reference category for both factors

33. 2-Way MANOVA as a Linear Model
By default in SAS, the reference categories are the highest factor levels of the two treatments
This means that your intercept actually represents the mean for the default levels of factor 1 and factor 2
Let’s look at it in terms of our Parkinson’s rats data…
-Trt: 2 levels (BNDF and no BDNF)
-Lesion type: 3 levels (DSP4, OHDA, and Double)

34. 2-way MANOVA as Linear Model
Assume equal number of observations in each group…
BDNF, DSP4
BDNF, OHDA
BDNF, Double
No BDNF, DSP4
No BDNF, OHDA
No BDNF, Double

35. 2-way MANOVA as Linear Model
Expand the RHS (1st term for vector of 1’s in column 1)

36. 2-way MANOVA as Linear Model
From this we can see:

37. 2-way MANOVA as Linear Model
Our model has parameters m, t1, b1, b2, g11, and g12. What do these parameters represent in terms of our factor levels?

38. 2-way MANOVA as Linear Model
What about the interaction terms?

39. 2-way MANOVA as Linear Model
SAS and R also consider the F distribution (Rao) rather than c2

40. 2-way MANOVA as Linear Model
Consider our test for interaction effects… SAS tests: What are C and M?

41. 2-way MANOVA as Linear Model
Can we find the F-statistic and its d.f. for this contrast?

42. 2-way MANOVA as Linear Model
Can we find the F-statistic and its d.f. for this contrast?

43. 2-way MANOVA as Linear Model
Consider our treatment effect (BDNF vs. no BDNF)… What are C, M, the F-statistics and its d.f.?

44. 2-way MANOVA as Linear Model
Again find the F-statistic and its d.f. for BDNF treatment.

45. Treatment Profiles
Suppose we have data on two (or more) groups treated at with one of p treatments
-Groups: l = 1,2
-Treatment: k = 1,2,3,4 levels
Group 1
Group 2

46. Profile Analysis
Say we are investigating p treatments given to > 2 groups...
There are several hypotheses we can consider.
(1) Are the profiles are parallel?
(2) Are the profiles coincident?
(3) Are the profiles level (i.e. are all treatments equivalent)?

47. Profile Analysis
We can use contrasts again. If we want to test if the profiles are parallel, want is our C?

48. Profile Analysis
In this case, we can use a Hotelling’s T2 statistic that looks like:

49. Profile Analysis
Are the profiles are coincident?

50. Profile Analysis
Are the profiles level?

51. Example
A PI at MUSC hypothesizes that proteins involved in inflammation and immune mediation are elevated in sclera tissue in eyes of patients with age related macular degeneration (AMD)
Study looking as such proteins compared tissue samples from AMD eyes and “healthy” eyes (30/group)
Proteins included:
-C-reactive protein (CRP)
-Complement factor H (CFH)
-Glutamate receptor 2 (glu)

52. Example
Are the profiles are parallel?

53. Example
Are the profiles are parallel?

54. Example
Are the profiles are coincident?

55. Example
Are the profiles are coincident?

56. Example

57. Example
We have been assuming the eyes were independent What if the eyes were from the same individual but one eye had AMD and the other was healthy? Treating the data as we have ignores this dependence (Note this is also true in the book example!) So what can we do if we still want to test for parallel, coincident, or level profiles?

58. Example
We can treat the data as a sample from one population… How would these data look? Recall out original hypothesis for parallel profiles… We can restructure it if we think of these as “paired” samples

59. Example
We now have to re-structure our contrast matrix. Given our newly stated hypothesis this what should our contrast matrix be? What else does this change in our test?

60. Example
What if we want to test coincident profiles What should our hypothesis and contrast matrix be?

61. Strategy for Multivariate Treatment Comparisons
Check assumptions of data
-check univariate/multivariate normality
-are covariance matrices equal?
Perform appropriate multivariate test of hypotheses
-Are you looking at unique groups?
-How many groups are under investigation?
Determine Bonferroni simultaneous CIs
-Ideally, you should know which CIs you want to examine a priori