MANOVA

Mechanics

• MANOVA is a multivariate generalization of ANOVA, so there are analogous parts to the simpler ANOVA equations

• First lets revisit Anova• Anova tests the null hypothesis• H0: 1= 2… = k

• How do we determine whether to reject?

• SSTotal =

• SSbg =

• SSwg =

2..( )jn X X

2..( )ijX X

2( )ij jX X

S S treatm ent S S error

S S total

S S Between groups S S with in groups

S S total

Steps to MANOVA

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

• SSbg breaks down into main effects and interaction

2**

2**

2**

( )

( )

( )

A ii

B jj

AB ij i ji j

SS n Y Y

SS n Y Y

SS n Y Y Y Y

• With one-way anova our F statistic is derived from the following formula

)/(

)1/(F ,1 kNSS

kSS

w

bkNk

Steps to MANOVA

• The multivariate test considers not just SSb and SSw for the dependent variables, but also the relationship between the variables

• Our null hypothesis also becomes more complex. Now it is

• With Manova we now are dealing with matrices of response values– Each subject now has multiple scores, there is a matrix, as

opposed to a vector, 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

• We’ll start with this matrix Now consider the matrix product, X'X.                                               

• The result (product) is a square matrix.

• The diagonal values are sums of squares and the off-diagonal values are sums of cross products. The matrix is an SSCP (Sums of Squares and Cross Products) matrix.

• So Anytime you see the matrix notation X'X or D'D or Z'Z, the resulting product will be a SSCP matrix.

Manova

• Now our sums of squares goes something like this

• T = B + W• Total SSCP Matrix = Between SSCP + Within SSCP

• Wilk’s lambda equals |W|/|T| such that smaller values are better (less effect attributable to error)

• We’ll use the following dataset• We’ll start by calculating the W matrix for each

group, then add them together– The mean for Y1 group 1 = 3, Y2 group 2 = 4

• W = W1 + W2 + W3

Group Y1 Y21 2 31 3 41 5 41 2 52 4 82 5 62 6 73 7 63 8 73 9 53 7 6

Means 5.67 5.75

1 121

21 2

ss ss

ss ss

W

• So

• We do the same for the other groups

• Adding all 3 gives us

1

6 0

0 2

W

2

2 1

1 2

W 3

6.8 2.6

2.6 5.2

W

14.8 1.6

1.6 9.2

W

• Now the between groups part• The diagonals of the B matrix are the sums of

squares from the univariate approach for each variable and calculated as:

2..

1

j

..

( )

where n is the number of subjects in group j,

is the mean for the variable i in group j, and

is the grand mean for variable i

k

ii j ij ij

ij

i

b n Y Y

Y

Y

• The off diagonals involve those same mean differences but are the products of the differences found for each DV

.. ..1

( )( )k

mi im j ij i mj jj

b b n Y Y Y Y

• Again T = B + W

• And now we can compute a chi-square statistic* to test for significance

• Same as we did with canonical correlation (now we have p = number of DVs and k = number of groups)