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INTERPRETING MANOVA PARAMETER ESTIMATES David P. Nichols From SPSS Keywords, February 1993 MANOVA is only one of a number of SPSS procedures in which categorical independent variables (factors) are handled automatically by the procedure via creation of sets of contrast variables. Because what a procedure needs in terms of raw variable codings are not always the same as the resulting contrasts these raw variable codings produce, the design information produced sometimes confuses users when they attempt to interpret the parameter estimates. The purpose of this article is to clarify the default procedure by which MANOVA estimates parameters. Let's consider an example in which we have two factors, A with two levels, and B with three levels. The following commands submitted to MANOVA request a full factorial analysis of the 2 by 3 design:
The PRINT subcommand requests MANOVA to provide us with the ONEWAY and OVERALL DESIGN or basis matrices, as well as the parameter estimates. The OMEANS subcommand requests the grand mean, A and B marginal means, and the mean for each cell. The DESIGN subcommand explicitly requests that the test of significance and parameter estimate for the CONSTANT be included on our printout. While these are not printed by default (except with SPSS/PC+), they are included in the model unless NOCONSTANT is requested on the METHOD subcommand. The omission of any CONTRAST subcommands means that MANOVA will fit the default DEVIATION contrasts for both factors. These contrasts compare all but one of the levels of a factor against the remaining categories. This is equivalent to comparing each level to the mean of all categories. One category is left out (the reference category) because to include it would produce a redundancy in the set of contrasts (recall that with K levels to a factor, there are K-1 degrees of freedom for fitting contrasts). By default, the last category is the reference category. The first thing that MANOVA will print out for us are the tables of "Combined Observed Means" for the CONSTANT, A, B and A BY B, as requested on the OMEANS subcommand. These appear as:
Note that there are two sets of means here, weighted and unweighted. These will always be the same for terms representing individual cell means (the A BY B term here) and will also be the same for marginal and grand means if there are the same number of observations in each cell of the design. If the design has different numbers of observations in the cells, the weighted and unweighted versions of all marginals and the grand mean will generally not be the same. The weighted means weight each cell by the number of observations in it, while the unweighted means are simple means of cell means. Our focus here will be on the unweighted means, since these are what MANOVA uses in parameter estimation. To illustrate how these are calculated, we can see that the first level marginal mean of A is given by:
The grand mean is simply the unweighted average of the six cell means, which is the same as the unweighted average of each set of marginal means:
Now let us see how MANOVA goes about comparing these means. Take the case of the two level factor A. The contrast can be written in two equivalent ways:
While the first is the more intuitive, as it shows that the contrast simply compares the first level of A to the overall A mean, the second expression is easier to use computationally, as it involves only A marginal means. If we express this contrast as a row vector of contrast coefficients, it becomes simply [ 1/2 -1/2 ]. If we add on an expression for a constant term, we have a full contrast matrix for factor A:
If we were to explicitly request DEVIATION contrasts via input of a SPECIAL contrast matrix, this is what we would specify (actually, MANOVA does not allow fractions or factoring out of a term of a contrast matrix, but here we are using the form that is easiest to see in terms of the requested contrasts). Experienced MANOVA users might recall that the CONSTANT row is often specified as all ones, rather than the .5 or 1/2 used here. MANOVA internally converts this to an average, so that the mean of the means is estimated as opposed to the sum of the means. After showing us the means, MANOVA next prints out the ONEWAY design or basis matrix for each factor. For A, this appears as:
Many MANOVA users mistake this matrix for the contrast matrix for factor A, which it is not. It is, however, related to the contrast matrix in the following manner: If the contrast matrix is C, and the basis matrix is X,
This relationship results from the well known expression for least squares estimates of parameters in a linear model:
There is also a way to calculate X from C:
Since users are generally interested in being certain that they know what contrasts are being fitted, interest generally revolves around C. However, MANOVA does not have to form C explicitly in order to produce parameter estimates and significance tests. The optional CHOLESKY method decomposes X ' X and the default QR method decomposes X directly. C is never explicitly formed unless input as a SPECIAL contrast matrix by the user. This is why there is no option to print out the actual contrast matrix. The above equations for C and X can be used in the SPSS MATRIX procedure (not available for SPSS/PC+) to ensure that the desired matrix has been specified. In most cases there are also simpler ways to confirm the specifications. If a contrast matrix is partitioned into a row for the constant and the remaining rows for the factor effect, and the columns of the basis matrix are similarly partitioned, the following features can be observed: The effect row(s) of an orthonormal contrast matrix are the same as the transpose of the effect column(s) of the corresponding basis matrix. Also, two important types of nonorthogonal contrasts, DEVIATION and SIMPLE, are mirror images of each other. That is, the basis matrix for DEVIATION contrasts is the transpose of the contrast matrix for SIMPLE contrasts, and vice versa. This can be seen by comparing the basis matrix for A with a SIMPLE contrast matrix for a two level factor. Recall that SIMPLE contrasts compare each level of a factor except a reference category (or baseline) to that reference category. A SIMPLE contrast matrix for a two level factor would thus look like:
As is easily seen, the nonconstant row here is the transpose of the nonconstant column in the basis matrix for A. The same thing is true of the basis and contrast matrices for factor B. The basis matrix given below
has effect columns that when transposed match SIMPLE contrasts [1 0 -1] and [0 1 -1]. The contrasts specified by the B basis matrix are:
which can also be expressed in a contrast matrix as:
Once the ONEWAY basis matrices have been specified, the overall basis matrix is constructed from Kronecker products of the column partitions of the ONEWAY matrices:
where Ai and Bi represent the ith column partitions of the A and B basis matrices. The result is the overall basis matrix printed by MANOVA:
The resulting overall contrast matrix could be derived from the formula given earlier, or constructed from Kronecker products of the row partitions of the A and B contrast matrices (in a manner similar to that shown for the overall basis construction). The resulting overall contrast matrix is:
Factoring out the divisor of 6 which applies to all terms allows us to see the nature of the contrasts more easily. The CONSTANT term is simply the unweighted grand mean:
The A contrast measures the deviation of level 1 of A from the grand mean:
The B contrasts measure the deviations of levels 1 and 2 of B from the grand mean:
The interaction contrasts are specified by the main effect contrasts. The first contrast tests whether the deviation of level 1 of factor A is the same at level 1 of factor B as at levels 2 and 3 of factor B, which is equivalent to testing whether the deviation of level 1 of factor B is the same at both levels of factor A. The last contrast tests whether the deviation of level 1 of factor A is the same at level 2 of B as at levels 1 and 3, which is equivalent to testing whether the deviation of level 2 of factor B is the same at both levels of factor A. In comparing these formulas with the output given below, note that the parameter numbering in MANOVA is hard coded and cannot be changed by the user.
Now that you have seen how MANOVA handles default contrasts, you might want to apply the formulas listed earlier that relate the contrast and basis matrices to other types of contrasts you use in your own analyses. This is only one of a number of topics on which the SPSS MATRIX procedure can be an invaluable aid in understanding the functioning of MANOVA and other SPSS procedures. Appendix: Data
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