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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)