Visualizing Hypothesis Tests in Multivariate Linear Models

The heplots package provides functions for visualizing hypothesis tests in multivariate linear models (MANOVA, multivariate multiple regression, MANCOVA, and repeated measures designs). HE plots represent sums-of-squares-and-products matrices for linear hypotheses and for error using ellipses (in two dimensions), ellipsoids (in three dimensions), or by line segments in one dimension.

Details

The basic theory behind HE plots is described by Friendly (2007). See Fox, Friendly and Monette (2007) for a brief introduction; Friendly & Sigal (2016) for a tutorial on these methods; and Friendly, Monette and Fox (2013) for a general discussion of the role of elliptical geometry in statistical understanding.

Other topics now addressed here include robust MLMs, tests for equality of covariance matrices in MLMs, and chi square Q-Q plots for MLMs.

The package also provides a collection of data sets illustrating a variety of multivariate linear models of the types listed above, together with graphical displays.

Several tutorial vignettes are also included. See vignette(package="heplots").

The graphical functions contained here all display multivariate model effects in variable (data) space, for one or more response variables (or contrasts among response variables in repeated measures designs).

list(list("heplot"))

constructs two-dimensional HE plots for model terms and linear hypotheses for pairs of response variables in multivariate linear models.

list(list("heplot3d"))

constructs analogous 3D plots for triples of response variables.

list(list("pairs.mlm"))

constructs a “matrix” of pairwise HE plots.

list(list("heplot1d"))

constructs 1-dimensional analogs of HE plots for model terms and linear hypotheses for single response variables.

For repeated measure designs, between-subject effects and within-subject effects must be plotted separately, because the error terms (E matrices) differ. For terms involving within-subject effects, these functions carry out a linear transformation of the matrix Y of responses to a matrix Y M, where M is the model matrix for a term in the intra-subject design and produce plots of the H and E matrices in this transformed space. The vignette repeated describes these graphical methods for repeated measures designs.

The related car package calculates Type II and Type III tests of multivariate linear hypotheses using the Anova and linearHypothesis functions.

The candisc-package package provides functions for visualizing effects for MLM model terms in a low-dimensional canonical space that shows the largest hypothesis relative to error variation. The candisc package now also includes related methods for canonical correlation analysis.

The heplots package also contains a large number of multivariate data sets with examples of analyses and graphic displays. Use data(package="heplots") to see the current list.

References

Friendly, M. (2006). Data Ellipses, HE Plots and Reduced-Rank Displays for Multivariate Linear Models: SAS Software and Examples. Journal of Statistical Software, 17(6), 1-42. https://www.jstatsoft.org/v17/i06/, doi:10.18637/jss.v017.i06

Friendly, M. (2007). HE plots for Multivariate General Linear Models. Journal of Computational and Graphical Statistics, 16(2) 421-444. http://datavis.ca/papers/jcgs-heplots.pdf, doi:10.1198/106186007X208407

Fox, J., Friendly, M. & Monette, G. (2007). Visual hypothesis tests in multivariate linear models: The heplots package for R. DSC 2007: Directions in Statistical Computing. https://socialsciences.mcmaster.ca/jfox/heplots-dsc-paper.pdf

Friendly, M. (2010). HE Plots for Repeated Measures Designs. Journal of Statistical Software, 37(4), 1-40. doi:10.18637/jss.v037.i04 .

Fox, J., Friendly, M. & Weisberg, S. (2013). Hypothesis Tests for Multivariate Linear Models Using the car Package. The R Journal, 5(1), https://journal.r-project.org/archive/2013-1/fox-friendly-weisberg.pdf.

Friendly, M., Monette, G. & Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry. Statistical Science, 2013, 28 (1), 1-39, http://datavis.ca/papers/ellipses.pdf.

Friendly, M. & Sigal, M. (2014). Recent Advances in Visualizing Multivariate Linear Models. Revista Colombiana de Estadistica, 37, 261-283

Friendly, M. & Sigal, M. (2016). Graphical Methods for Multivariate Linear Models in Psychological Research: An R Tutorial. Submitted for publication.

See also

Anova, linearHypothesis for Anova.mlm computations and tests

candisc-package for reduced-rank views in canonical space

manova for a different approach to testing effects in MANOVA designs

Author

Michael Friendly, John Fox, and Georges Monette

Maintainer: Michael Friendly, friendly@yorku.ca, http://datavis.ca