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It is indeed gratifying to see SAS users adopting SG Procedures and GTL to create graphs from the simple to the intricate, and presenting their findings and the techniques they have developed to create these graphs. Welcome to Graphically Speaking, a blog by Sanjay Matange focused on the usage of ODS Graphics for data visualization in SAS. The blog content appearing on this site does not necessarily represent the opinions of SAS. Last month’s bestseller list is an eclectic blend of new, tried and true, and titles related to what you do. I’m Maggie Miller and I’ll be providing you the latest updates about SAS books, documentation, tips and industry trends here on the SAS Bookshelf. The name comes from dplyr: originally you created these objects with tbl_df(), which was most easily pronounced as “tibble diff”. Note: Initially in this chapter, the matrix names E and H refer to the error and hypothesis cross products. In multivariate fits, the sums of squares due to hypothesis and error are matrices of squares and cross products instead of single numbers. In the golf balls example, there is only one effect so the Whole Model test and the test for Brand are the same, which show the four multivariate tests with approximate F tests. The red triangle menus on the Whole Model, Intercept, and Brand reports contain options to generate additional information, which includes eigenvalues, canonical correlations, a list of centroid values, a centroid plot, and a Save command that lets you save canonical variates. The effect (Brand in this example) popup menu also includes the option to specify contrasts. The custom test and contrast features are the same as those for regression with a single response. Descriptions of Multivariate Reports describes each Multivariate report (except the Sphericity Test table; see Univariate Tests and the Test for Sphericity). Although the four standard multivariate tests often give similar results, there are situations where they differ, and one might have advantages over another.

When there is a large deviation from the null hypothesis and the eigenvalues differ widely, the order of preference is the reverse (Seber 1984). There are cases, such as a repeated measures model, that allow transformation of a multivariate problem into a univariate problem (Huynh and Feldt 1970). If these conditions hold, the diagonal elements of the E and H test matrices sum to make a univariate sums of squares for the denominator and numerator of an F test.
The sphericity test checks the appropriateness of an unadjusted univariate F test for the within-subject effects using the Mauchly criterion to test the sphericity assumption (Anderson 1958).
If the sphericity Chi-square test is not significant, you can use the unadjusted univariate F-tests. If the sphericity test is significant, use the multivariate or the adjusted univariate tests.
The univariate F statistic has an approximate F-distribution even without sphericity, but the degrees of freedom for numerator and denominator are reduced by some fraction epsilon (Iµ). The epsilon adjusted tests in the multivariate report are labeled G-G (Greenhouse-Geisser) or H-F (Huynh-Feldt), with the epsilon adjustment shown in the value column. Follow along with my talented colleagues, and guest posters as we bring the written word to life and share helpful content with our fellow SAS book enthusiasts. The M-transformed Parameter Estimates report gives the original parameter estimates matrix multiplied by the transpose of the M matrix.
After specification of a response design, E and H refer to those matrices transformed by the response design, which are actually MA?EM and MA?HM. And there are lots of ways to measure how large a value the matrix for the hypothesis sums of squares and cross products (called H or SSCP) is compared to that matrix for the residual (called E). There is only a single intercept with two DF (one for each response), so the F test for it is exact and is labeled Exact F. If the number of response function values (columns specified in the M matrix) is 1 or if an effect has only one degree of freedom per response function, the exact F test is presented.

If the response design yields a single value or if the test is one degree of freedom, this is an exact F test. The sphericity condition means that the M-transformed responses are uncorrelated and have the same variance.
The sphericity test and the univariate tests are always done using an orthonormalized M matrix.
Box (1954), Greenhouse and Geisser (1959), and Huynh-Feldt (1976) offer techniques for estimating the epsilon degrees-of-freedom adjustment.
SAS Press books do cover a wide variety of topics. But these top 9 in November are a particularly striking example of that. Otherwise, the standard four multivariate test statistics are given with approximate F tests: Wilksa€™ Lambda (I›), Pillaia€™s Trace, the Hotelling-Lawley Trace, and Roya€™s Maximum Root. Muller and Barton (1989) recommend the Greenhouse-Geisser version, based on a study of power. We'll feature upcoming posts breaking down books a little more by category in the near future - including SAS documentation.
If you are looking for a test at an exact significance level, you may need to go hunting for tables in reference books. Identity matrices cannot be transformed to a full rank matrix after centralization of column vectors and orthonormalization. If the response design yields a single value, or if the hypothesis is a single degree of freedom, the multivariate tests are equivalent and yield the same exact F test.

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