MANOVA - Equations
Lecture 11
Psy524
Andrew Ainsworth
Data design for MANOVA
IV1
IV2
1
1
2
1
2
2
DV1
S01(11)
S02(11)
S03(11)
S04(12)
S05(12)
S06(12)
S07(21)
S08(21)
S09(21)
S10(22)
S11(22)
S12(22)
DV2
S01(11)
S02(11)
S03(11)
S04(12)
S05(12)
S06(12)
S07(21)
S08(21)
S09(21)
S10(22)
S11(22)
S12(22)
DVM
S01(11)
S02(11)
S03(11)
S04(12)
S05(12)
S06(12)
S07(21)
S08(21)
S09(21)
S10(22)
S11(22)
S12(22)
Data design for MANOVA
Steps to MANOVA
 MANOVA is a multivariate
generalization of ANOVA, so there are
analogous parts to the simpler
ANOVA equations
Steps to MANOVA
 ANOVA –
 (Y  GM
ij
i
j
)  n Yj  GM ( y )    (Yij  Yj )
2
2
( y)
j
i
j
SSTotal ( y )  SSbg ( y )  SS wg ( y )
2
Steps to MANOVA
 When you have more than one IV the
interaction looks something like this:
nkm  ( DTkm  GM ( DT ) )  nk   Dk  GM ( D )   nm  Tm  GM (T )  
2
2
k
m
k
2
m
2
2
2

 nkm   DTkm  GM ( DT )   nk   Dk  GM ( D )   nm  Tm  GM (T )  
k m
k
m


SSbg  SS D  SST  SS DT
Steps to MANOVA
 The full factorial design is:
 (Y
ikm
i
k
m
 GM (ikm ) )  nk   Dk  GM ( D )   nm  Tm  GM (T )  
2
2
k
2
m
2
2
2

 nkm   DTkm  GM ( DT )   nk   Dk  GM ( D )   nm  Tm  GM (T )  
k m
k
m


 (Yikm  DTkm )2
i
k
m
Steps to MANOVA

MANOVA - You need to think in terms of
matrices





Each subject now has multiple scores, there is a
matrix of responses in each cell
Matrices of difference scores are calculated and the
matrix squared
When the squared differences are summed you get
a sum-of-squares-and-cross-products-matrix (S)
which is the matrix counterpart to the sums of
squares.
The determinants of the various S matrices are
found and ratios between them are used to test
hypotheses about the effects of the IVs on linear
combination(s) of the DVs
In MANCOVA the S matrices are adjusted for by one
or more covariates
Matrix Equations
 If you take the three subjects in the
treatment/mild disability cell:
115  98  107 
Yi11       
108 105  98 
Matrix Equations
 You can get the means for disability by
averaging over subjects and treatments
95.83
88.83
82.67 
D1  
D2  
D3  



96.50
88.00
 77.17 
Matrix Equations
 Means for treatment by averaging
over subjects and disabilities
99.89
78.33
T1  
T

2



96.33
78.11




Matrix Equations
 The grand mean is found by averaging
over subjects, disabilities and treatments.
89.11
GM  

87.22


Matrix Equations
 Differences are found by subtracting
the matrices, for the first child in the
mild/treatment group:
115  89.11  25.89
(Y111  GM )     



108 87.22  20.75
Matrix Equations
 Instead of squaring the matrices you
simply multiply the matrix by its
transpose, for the first child in the
mild/treatment group:
 25.89
670.29 537.99
(Y111  GM )(Y111  GM ) '  
25.89 20.75  



20.75
537.99
431.81




Matrix Equations
 This is done on the whole data set at the
same time, reorganaizing the data to get
the appropriate S matrices. The full break
of sums of squares is:
 (Y
ikm
i
k
 GM )(Yikm  GM ) '  nk  ( Dk  GM )( Dk  GM ) '
m
k
 nm  (Tm  GM )(Tm  GM ) ' [nkm  ( DTkm  GM )( DTkm  GM ) '
m
km
nk  ( Dk  GM )( Dk  GM ) ' nm  (Tm  GM )(Tm  GM ) ']
k
m
  (Yikm  DTkm )(Yikm  DTkm ) '
i
k
m
Matrix Equations
 If you go through this for every
combination in the data you will get
four S matrices (not including the S
total matrix):
570.29 761.72 
 2090.89 1767.56
SD  
ST  


761.72 1126.78
1767.56 1494.22
S DT
 2.11 5.28 
544.00 31.00 

S s / DT  


5.28
52.78
31.00
539.33




Test Statistic – Wilk’s Lambda
 Once the S matrices are found the
diagonals represent the sums of squares
and the off diagonals are the cross
products
 The determinant of each matrix represents
the generalized variance for that matrix
 Using the determinants we can test for
significance, there are many ways to this
and one of the most is Wilk’s Lambda
Test Statistic – Wilk’s Lambda

Serror
Seffect  Serror
 this can be seen as the percent of
non-overlap between the effect and
the DVs.
Test Statistic – Wilk’s Lambda
 For the interaction this would be:
S s / DT  292436.52
 2.11 5.28  544.00 31.00  546.11 36.28 
S DT  S s / DT  





5.28 52.78  31.00 539.33  36.28 529.11
S DT  S s / DT  322040.95
292436.52

 .908068
322040.95
Test Statistic – Wilk’s Lambda
 Approximate Multivariate F for Wilk’s
Lambda is
 1  y   df 2 
F (df1 , df 2 )  


 y   df1 
where y  1/ s , s 
p 2 (df effect ) 2  4
p  (df effect )  5
2
2
,
p  number of DVs, df1  p(df effect )
p  df effect  1   p(df effect )  2 

df 2  s (df error ) 


2
2

 

Test Statistic – Wilk’s Lambda
 So in the example for the interaction:
(2) 2 (2) 2  4
s
2
2
2
(2)  (2)  5
y  .9080681/ 2  .952926
df1  2(2)  4
df error  dt (n  1)  3(2)(3  1)  12
2  2  1   2(2)  2 

df 2  2 12 

 22


2   2 

 .047074   22 
Approximate F(4,22)= 
    0.2717
 .952926   4 
Eta Squared
  1 
2
Partial Eta Squared
  1 
2
1/ s