multivariate analysis of variance, part 2 bmtry 726 2/21/14
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Two-way MANOVAWhat 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
1 1 1 1
1,2,..., ; 1, 2,..., ; 1, 2,...,
0
lkr l k lk lkr
g b g b
l l lk lkl k l k
X
l g k b r n
lkr l k lkrE X
MeanResponse
Overalllevel
Factor 1Effect
Factor 2Effect
Factor Interaction
Effect
Visualize the DataSuppose we have data with two factors -treatment A has l = 1,2,3 levels -treatment B has k = 1,2,3,4 levels
1 2 3 40
1
2
3
4
5
Level Treatment B
Level 1, Trt A
Level 2, Trt A
Level 3, Trt A
Expe
cted
Res
pons
e
2 1
3 3
1 1
1 2
1 3
1 4 3 1
4 2
3 4
2 2
2 3
2 4
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:
1 1 1 1 1 1
2 2 2 2 2 2
where ~ ,
and 1,2,..., (levels o
lkr l k lk lkr
lkr l k lk lkr
lkrp p lp kp lkp lkrp
lkr p
X
X
X
NID
l g
0
f treatment1)
1,2,..., (levels of treatment 2)
1,2,..., (number subjects in each group)
k b
r n
Two-way MANOVAWe can then easily extend to the multivariate case:
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.
lkr l k lk lkr X μ τ β γ ε
1 1 1 1
g b g b
l k lk lkl k l k τ β γ γ 0
Two-Way MANOVA• We can write this in linear model form:
1 2
11 12 1
1,1 1,2 1,1111 1112 111
1121 1122 112 11 12 1
1 2 ,
,
1 1 0 0
1 1 0 0
1 1 0 0
1 0 1 0
1 0 1 0
1 0 0 0
p
p
g g g pp
p
gbn gbn gbnp gbn p
gbn gb
x x x
x x x
x x x
,
1,1 1,2 1,
111 112 11
1, 1,1 1, 1,2 1, 1, ,
p
gbn p
b b b p
p
g b g b g b p gb p
Two-way MANOVAWe can decompose this as follows:
' '
1 1 1 1
'
1
'
1 1
'
1 1 1
. . 1 1 1 1
lkr l k lk l k lkr lk
g b n g
lkr lkr l ll k r l
b
k kk
g b
lk l k lk l kl k
g b n
lkr lk lkr lkl k r
bn
gn
n
d f g b g b g
x x x x x x x x x x x x
x x x x x x x x
x x x x
x x x x x x x x
x x x x
1
1
b n
gbn
From this we can derive the MANOVA table:
SourceVariationI
Matrix Sum of Squares Degrees of Freedom
Treatment 1
Treatment 2
Interaction
Residual
Total
'
1 1
g
l llSSP bn
x x x x
'
interaxn 1 1
g b
k l l k l ll kSSP n
x x x x x x x x
'
2 1
b
k kkSSP gn
x x x x
'
res 1 1 1
g b n
lkr lk lkr lkl k rSSP
x x x x
'
cor 1 1 1
g b n
lkr lkrl k rSSP
x x x x
1g
1b
1 1g b
1gb n
1gbn
Hypothesis TestsGenerally 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
0 11 12: ... . : At least onegb A lkH vs H γ γ γ 0 γ 0
res*
res interaxn
1 1 1 * 22 1 11 lnp g b
g b p
SSP
SSP SSP
where
gb n
Hypothesis TestsIf 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
0 1 2
0 1 2
: ... . : At least one
: ... . : At least one
g A l
g A k
H vs H
H vs H
τ τ τ 0 τ 0
β β β 0 β 0
res*
res 1
1 1 * 22 1
res*
res 2
1 1 * 22 1
Factor1:
1 ln
Factor 2 :
1 ln
p g
g p
p b
b p
SSP
SSP SSP
gb n
SSP
SSP SSP
gb n
Hypothesis TestsNote a critical value based on a c2 distribution better for large nFor small samples we can compute an F-statistic since this is
(sometimes) an exact distributionHowever, the d.f. are complicated to estimate- the book
example works for g = b = 2:
*1 1
* 1 1 1, 1 11 1 1
*1 1
* 1 1, 1 11 1
*1 1
* 1 1, 1 11 1
Interactions:
1~
Factor1:
1~
Factor 2 :
1~
gb n p
g b p gb n pg b p
gb n p
g p gb n pg p
gb n p
b p gb n pb p
F
F
F
Confidence IntervalsAs with the one-way MANOVA case, we can estimate
Bonferroni confidence intervals
21
21
res
Factor1: belongs to
Factor 2 : belongs to
1
diagonalelement of
ii
ii
Eli mi l i m i v v bnpg g
Eki qi ki qi v v gnpb b
thii
x x t
x x t
v gb n
E i SSP
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.
Experimental DesignThe 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
DSP-4 Lesions effect noradrenergic pathwaysin the LC
Two groups of rats receivethese single lesions -No treatment -BDNF Treated
6-OHDA Lesions effect dopaminergic pathwaysin the PFC
Two groups of rats receivethese single lesions -No treatment -BDNF Treated
Double lesion animals receive both 6-OHDA andDSP-4 lesions affecting the LC and PFC
Two groups of rats receivethese single lesions -No treatment -BDNF Treated
The dataBDNF No BDNF
.69 27 .57 28
.72 21 .46 29
DSP4: .72 7 .44 39
.65 27 .51 33
.63 17 .37 24
.85 25 .47 22
.71 6 .49 25
6-OHDA: .63 9 .51 27
.72 32 .50 20
.72 17 .54 28
.70 31
.61 23
Double:
.47 34
.38 37
.64 25 .37 38
.57 27 .37 31
.65 24 .29 42
The data are arranged in anarray.
The six matrices represent the six possible combosof treatment and lesion type
Column 1 in each matrix arethe NOR task results
Column 2 in each matrix arethe WRAM task results
Hypothesis testing
1 2
int
0.4014 16.08 0.0612 5.609
16.08 644.03 5.609 516.07
0.0028 0.0503 0.0825 1.492
0.0503 26.867 1.492 1031.2
How do we test the hypothesis of nointerac
factor factor
eraction residual
SS SS
SS SS
*int
2
tion?
. .d f
Hypothesis testing
1 2
int
0.4014 16.08 0.0612 5.609
16.08 644.03 5.609 516.07
0.0028 0.0503 0.0825 1.492
0.0503 26.867 1.492 1031.2
What about testing for factor effec
factor factor
eraction residual
SS SS
SS SS
*1
2
ts?
. .
fac
d f
Bonferroni Simultaneous CIs
2
1 2 1 2 1 1 1
0.0825 1.492 0.6807 .4493, ,
1.492 1031.2 21.20 30.47
What are the 95% Bonferroni CI's for BDNF versus no BDNF?
NOR...
WRAM...
ii
residual BDNF NoBDNF
Ei i i i bngb n pg g gb n
SS
x x t
x x
Bonferroni Simultaneous CIs
4
1 2 1 2 1
0.0825 1.492 .576 .614 .505, , ,
1.492 1031.2 25.2 21.1 31.2
What are the 95% Bonferroni CI's for comparing different lesion types?
ˆ ˆ
residual DSP OHDA Doub
i i i i gb n p
SS
x x t
x x x
2
1 1
NOR in DSP4 vs 6OHDA...
WRAM in 6OHDA vs Double...
iiEgnb b gb n
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.
Conclusions from the Parkinson’s Study
Rats with double lesions spent significantly short time with novel objects relative to 6OHDA lesioned rats.
Double lesioned rats took a significantly longer amount of time to complete the WRAM task relative to 6OHDA rats.
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
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