SSCP Matrices and Multivariate Tests
This table displays the hypothesis and error sum-of-squares and cross-products (SSCP) matrices for testing model effects. Since there are two dependent variables, each matrix has two columns and two rows. For example, the 2x2 matrix associated with CLOTSOLV in the table is the hypothesis matrix for testing the significance of Clot-dissolving drugs. The matrix associated with PROC in the table is the hypothesis matrix for testing the significance of Surgical treatment, and the matrix associated with PROC*CLOTSOLV is used for testing their interaction effect.
The error matrix is used in testing each effect. In analogy to the test for models with one dependent variable, the "ratio" of the hypothesis SSCP matrix to the error matrix is used to evaluate the effect of interest. More specifically, the eigenvalues of the test matrix defined by the matrix product of the appropriate hypothesis SSCP matrix and the inverse of the error SSCP matrix are used to compute the statistics in the multivariate tests table.
The multivariate tests table displays four tests of signifcance for each model effect.
- Pillai's trace is a positive-valued statistic. Increasing values of the statistic indicate effects that contribute more to the model.
- Wilks' Lambda is a positive-valued statistic that ranges from 0 to 1. Decreasing values of the statistic indicate effects that contribute more to the model.
- Hotelling's trace is the sum of the eigenvalues of the test matrix. It is a positive-valued statistic for which increasing values indicate effects that contribute more to the model. Hotelling's trace is always larger than Pillai's trace, but when the eigenvalues of the test matrix are small, these two statistics will be nearly equal. This indicates that the effect probably does not contribute much to the model.
- Roy's largest root is the largest eigenvalue of the test matrix. Thus, it is a positive-valued statistic for which increasing values indicate effects that contribute more to the model. Roy's largest root is always less than or equal to Hotelling's trace. When these two statistics are equal, the effect is predominantly associated with just one of the dependent variables, there is a strong correlation between the dependent variables, or the effect does not contribute much to the model.
There is evidence that Pillai's trace is more robust than the other statistics to violations of model assumptions1.
Each multivariate statistic is transformed into a test statistic with an approximate or exact F distribution. The hypothesis (numerator) and error (denominator) degrees of freedom for that F distribution are shown.
The significance values of the main effects, CLOTSOLV and PROC, are less than 0.05, indicating that the effects contribute to the model. By contrast, their interaction effect does not contribute to the model. However, though CLOTSOLV does contribute to the model, since the value of Pillai's trace is close to Hotelling's trace, it doesn't contribute very much. A more straightforward way to see this is to look at partial eta squared. The partial eta squared statistic reports the "practical" significance of each term, based upon the "ratio" of the variation accounted for by the effect to the sum of the variation accounted for by the effect and the variation left to error. More specifically, partial eta squared is the matrix product of the hypothesis SSCP matrix and the inverse of the sum of the hypothesis and error SSCP matrices. Larger values of partial eta squared indicate a greater amount of variation accounted for by the model effect, to a maximum of 1. Since partial eta squared is very small for CLOTSOLV, it does not contribute very much to the model. By comparison, partial eta squared for PROC is quite large, which is to be expected. The surgical procedure a patient must undergo for MI treatment is going to have a much greater effect on the length of their hospital stay and final costs than the type of thrombolytic they receive. In this case, it is enough for the multivariate tests to show that CLOTSOLV is significant, which means that the effect of at least one of the drugs is different from the others. The contrast results will show you where the differences are.