MANOVA - Equations

Lecture 11Psy524Andrew Ainsworth

Data design for MANOVAIV1 IV2 DV1 DV2 DVM

S01(11) S01(11) S01(11)

S02(11) S02(11) S02(11) 1

S03(11) S03(11) S03(11)

S04(12) S04(12) S04(12)

S05(12) S05(12) S05(12)

1

2

S06(12) S06(12) S06(12)

S07(21) S07(21) S07(21)

S08(21) S08(21) S08(21) 1

S09(21) S09(21) S09(21)

S10(22) S10(22) S10(22)

S11(22) S11(22) S11(22)

2

2

S12(22) S12(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 –

22 2( ) ( )( ) ( )ij y j y ij j

i j j i j

Y GM n Y GM Y Y

( ) ( ) ( )Total y bg y wg ySS SS SS

Steps to MANOVA

When you have more than one IV the interaction looks something like this:

2 22( ) ( ) ( )

2 2 2

( ) ( ) ( )

( )km km DT k k D m m Tk m k m

km km DT k k D m m Tk m k m

n DT GM n D GM n T GM

n DT GM n D GM n T GM

bg D T DTSS SS SS SS

Steps to MANOVA

The full factorial design is:

2 22( ) ( ) ( )

2 2 2

( ) ( ) ( )

2

( )

( )

ikm ikm k k D m m Ti k m k m

km km DT k k D m m Tk m k m

ikm kmi k m

Y GM n D GM n T GM

n DT GM n D GM n T GM

Y DT

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:

11

115 98 107

108 105 98iY

Matrix Equations You can get the means for disability by

averaging over subjects and treatments

1 2 3

95.83 88.83 82.67

96.50 88.00 77.17D D D

Matrix Equations

Means for treatment by averaging over subjects and disabilities

1 2

99.89 78.33

96.33 78.11T T

Matrix Equations The grand mean is found by averaging

over subjects, disabilities and treatments.

89.11

87.22GM

Matrix Equations Differences are found by subtracting

the matrices, for the first child in the mild/treatment group:

111

115 89.11 25.89( )

108 87.22 20.75Y GM

Matrix Equations Instead of squaring the matrices you

simply multiply the matrix by its transpose, for the first child in the mild/treatment group:

111 111

25.89 670.29 537.99( )( ) ' 25.89 20.75

20.75 537.99 431.81Y GM Y GM

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:

( )( ) ' ( )( ) '

( )( ) ' [ ( )( ) '

( )( ) ' ( )( ) ']

( )( ) '

ikm ikm k k ki k m k

m m m km km kmm km

k k k m m mk m

ikm km ikm kmi k m

Y GM Y GM n D GM D GM

n T GM T GM n DT GM DT GM

n D GM D GM n T GM T GM

Y DT Y DT

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

761.72 1126.78 1767.56 1494.22

2.11 5.28 544.00 31.00

5.28 52.78 31.00 539.33

D T

DT s DT

S S

S S

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

this can be seen as the percent of non-overlap between the effect and the DVs.

error

effect error

S

S S

Test Statistic – Wilk’s Lambda

For the interaction this would be:

/

/

/

292436.52

2.11 5.28 544.00 31.00 546.11 36.28

5.28 52.78 31.00 539.33 36.28 529.11

322040.95

292436.52.908068

322040.95

s DT

DT s DT

DT s DT

S

S S

S S

Test Statistic – Wilk’s Lambda Approximate Multivariate F for Wilk’s

Lambda is

21 2

1

2 21/

2 2

1

2

1( , )

( ) 4 , ,

( ) 5

, ( )

1 ( ) 2( )

2 2

effects

effect

effect

effect effecterror

dfyF df df

y df

p dfwhere y s

p df

p number of DVs df p df

p df p dfdf s df

Test Statistic – Wilk’s Lambda So in the example for the interaction:

2 2

2 2

1/ 2

1

2

(2) (2) 42

(2) (2) 5

.908068 .952926

2(2) 4

( 1) 3(2)(3 1) 12

2 2 1 2(2) 22 12 22

2 2

.047074 22 F(4,22)= 0.2717

.952926 4

error

s

y

df

df dt n

df

Approximate

Eta Squared

2 1

Partial Eta Squared

2 1/1 s