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 jY 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 cellMatrices of difference scores are calculated and the matrix squaredWhen 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 DVsIn 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 107108 105 98iY

=

Matrix Equations

You can get the means for disability by averaging over subjects and treatments

1 2 3

95.83 88.83 82.6796.50 88.00 77.17

D D D

= = =

Matrix Equations

Means for treatment by averaging over subjects and disabilities

1 2

99.89 78.3396.33 78.11

T T

= =

Matrix Equations

The grand mean is found by averaging over subjects, disabilities and treatments.

89.1187.22

GM

=

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 1767.56761.72 1126.78 1767.56 1494.22

2.11 5.28 544.00 31.005.28 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 productsThe determinant of each matrix represents the generalized variance for that matrixUsing 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

SS S

Λ =+

Test Statistic – Wilk’s Lambda

For the interaction this would be:

/

/

/

292436.52

2.11 5.28 544.00 31.00 546.11 36.285.28 52.78 31.00 539.33 36.28 529.11

322040.95292436.52 .908068322040.95

s DT

DT s DT

DT s DT

S

S S

S S

=

+ = + =

+ =

Λ= =

Test Statistic – Wilk’s LambdaApproximate Multivariate F for Wilk’sLambda is

21 2

1

2 21/

2 2

1

2

1( , )

( ) 4 , ,

( ) 5

, ( )

1 ( ) 2( )

2 2

effects

effect

effect

effect effecterror

dfyF df dfy df

p dfwhere y s

p df

p number of DVs df p df

p df p dfdf s df

−=

−= Λ =

+ −

= =

− + − = − −

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

2 2

2 2

1/ 2

1

2

(2) (2) 4 2(2) (2) 5

.908068 .9529262(2) 4

( 1) 3(2)(3 1) 122 2 1 2(2) 22 12 22

2 2.047074 22F(4,22)= 0.2717.952926 4

error

s

ydfdf dt n

df

Approximate

−= =

+ −

= == == − = − =

− + − = − − = =

Eta Squared

2 1η = −Λ

Partial Eta Squared

2 1/ sη = −Λ