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The ANOVA Procedure |

**MANOVA***< test-options >< / detail-options >***;**

When a MANOVA statement appears before the first RUN statement, PROC ANOVA enters a multivariate mode with respect to the handling of missing values; in addition to observations with missing independent variables, observations with

**H=***effects*| INTERCEPT | _ALL_-
specifies effects in the preceding
model to use as hypothesis matrices.
for multivariate tests
For each SSCP matrix
**H**associated with an effect, the H= specification computes an analysis based on the characteristic roots of**E**^{-1}**H**, where**E**is the matrix associated with the error effect. The characteristic roots and vectors are displayed, along with the Hotelling-Lawley trace, Pillai's trace, Wilks' criterion, and Roy's maximum root criterion with approximate*F*statistics. Use the keyword INTERCEPT to produce tests for the intercept. To produce tests for all effects listed in the MODEL statement, use the keyword _ALL_ in place of a list of effects. For background and further details, see the "Multivariate Analysis of Variance" section in Chapter 30, "The GLM Procedure." **E=***effect*-
specifies the error effect.
If you omit the E= specification, the ANOVA procedure uses the
error SSCP (residual) matrix from the analysis.
**M=***equation,...,equation*| (*row-of-matrix,...,row-of-matrix*)-
specifies a transformation matrix for the
dependent variables listed in the MODEL statement.
The equations in the M= specification are of the form
*c*_{i}values are coefficients for the various*dependent-variables*. If the value of a given*c*_{i}is 1, it may be omitted; in other words 1 ×*Y*is the same as*Y*. Equations should involve two or more dependent variables. For sample syntax, see the "Examples" section.

Alternatively, you can input the transformation matrix directly by entering the elements of the matrix with commas separating the rows, and parentheses surrounding the matrix. When this alternate form of input is used, the number of elements in each row must equal the number of dependent variables. Although these combinations actually represent the columns of the**M**matrix, they are displayed by rows.

When you include an M= specification, the analysis requested in the MANOVA statement is carried out for the variables defined by the equations in the specification, not the original dependent variables. If you omit the M= option, the analysis is performed for the original dependent variables in the MODEL statement.

If an M= specification is included without either the MNAMES= or the PREFIX= option, the variables are labeled MVAR1, MVAR2, and so forth by default. For further information, see the section "Multivariate Analysis of Variance" in Chapter 30, "The GLM Procedure." **MNAMES=***names*-
provides names for the variables defined
by the equations in the M= specification.
Names in the list correspond to the M= equations or
the rows of the
**M**matrix (as it is entered). **PREFIX=***name*-
is an alternative means of identifying the
transformed variables defined by the M= specification.
For example, if you specify PREFIX=DIFF, the transformed
variables are labeled DIFF1, DIFF2, and so forth.

**CANONICAL**-
produces a canonical analysis of the
**H**and**E**matrices (transformed by the**M**matrix, if specified) instead of the default display of characteristic roots and vectors. **ORTH**-
requests that the transformation matrix in the
M= specification of the MANOVA statement be
orthonormalized by rows before the analysis.
**PRINTE**-
displays the error SSCP matrix
**E**. If the**E**matrix is the error SSCP (residual) matrix from the analysis, the partial correlations of the dependent variables given the independent variables are also produced.

For example, the statementmanova / printe;

displays the error SSCP matrix and the partial correlation matrix computed from the error SSCP matrix. **PRINTH**-
displays the hypothesis SSCP matrix
**H**associated with each effect specified by the H= specification. **SUMMARY**-
produces analysis-of-variance tables for each dependent variable.
When no
**M**matrix is specified, a table is produced for each original dependent variable from the MODEL statement; with an**M**matrix other than the identity, a table is produced for each transformed variable defined by the**M**matrix.

proc anova; class A B; model Y1-Y5=A B(A); manova h=A e=B(A) / printh printe; manova h=B(A) / printe; manova h=A e=B(A) m=Y1-Y2,Y2-Y3,Y3-Y4,Y4-Y5 prefix=diff; manova h=A e=B(A) m=(1 -1 0 0 0, 0 1 -1 0 0, 0 0 1 -1 0, 0 0 0 1 -1) prefix=diff; run;

The first MANOVA statement specifies A as the hypothesis effect and B(A) as the error effect. As a result of the PRINTH option, the procedure displays the hypothesis SSCP matrix associated with the A effect; and, as a result of the PRINTE option, the procedure displays the error SSCP matrix associated with the B(A) effect.

The second MANOVA statement specifies
B(A) as the hypothesis effect.
Since no error effect is specified, PROC ANOVA uses the error
SSCP matrix from the analysis as the **E** matrix.
The PRINTE option displays this **E** matrix.
Since the **E** matrix is the error SSCP matrix
from the analysis, the partial correlation matrix
computed from this matrix is also produced.

The third MANOVA statement requests the same analysis as the first MANOVA statement, but the analysis is carried out for variables transformed to be successive differences between the original dependent variables. The PREFIX=DIFF specification labels the transformed variables as DIFF1, DIFF2, DIFF3, and DIFF4.

Finally, the fourth MANOVA statement has the identical effect as the third, but it uses an alternative form of the M= specification. Instead of specifying a set of equations, the fourth MANOVA statement specifies rows of a matrix of coefficients for the five dependent variables.

As a second example of the use of the M= specification, consider the following:

proc anova; class group; model dose1-dose4=group / nouni; manova h = group m = -3*dose1 - dose2 + dose3 + 3*dose4, dose1 - dose2 - dose3 + dose4, -dose1 + 3*dose2 - 3*dose3 + dose4 mnames = Linear Quadratic Cubic / printe; run;

The M= specification gives a transformation of the dependent variables dose1 through dose4 into orthogonal polynomial components, and the MNAMES= option labels the transformed variables as LINEAR, QUADRATIC, and CUBIC, respectively. Since the PRINTE option is specified and the default residual matrix is used as an error term, the partial correlation matrix of the orthogonal polynomial components is also produced.

For further information, see the "Multivariate Analysis of Variance" section in Chapter 30, "The GLM Procedure."

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