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heplots

Visualizing Hypothesis Tests in Multivariate Linear Models

Version 1.7.2

Description

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 (H) and for error (E) using ellipses (in two dimensions), ellipsoids (in three dimensions), or by line segments in one dimension. For the theory and applications, see:

If you use this work in teaching or research, please cite it as given by citation("heplots") or see Citation.

Other topics now addressed here include:

  • robust MLMs, using iteratively re-weighted least squared to down-weight observations with large multivariate residuals, robmlm().
  • Mahalanobis() calculates classical and robust Mahalanobis squared distances using MCD and MVE estimators of center and covariance.
  • visualizing tests for equality of covariance matrices in MLMs (Box’s M test), boxM() and plot.boxM().
  • χ2\chi^2 Q-Q plots for MLMs (cqplot()) to detect outliers and assess multivariate normality of residuals.
  • bivariate coefficient plots showing elliptical confidence regions (coefplot()).

In this respect, the heplots package now aims to provide a wide range of tools for analyzing and visualizing multivariate response linear models, together with other packages:

  • The related candisc package provides HE plots in canonical discriminant space, the space of linear combinations of the responses that show the maximum possible effects and for canonical correlation in multivariate regression designs.

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

Installation

CRAN version install.packages("heplots")
Development version remotes::install_github("friendly/heplots")

HE plot functions

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).

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

  • heplot3d() constructs analogous 3D plots for triples of response variables.

  • The pairs method, pairs.mlm() constructs a scatterplot matrix of pairwise HE plots.

  • heplot1d() constructs 1-dimensional analogs of HE plots for model terms and linear hypotheses for single response variables.

Other functions

  • glance.mlm() extends broom::glance.lm() to multivariate response models, giving a one-line statistical summary for each response variable.

  • boxM() Calculates Box’s M test for homogeneity of covariance matrices in a MANOVA design. A plot method displays a visual representation of the components of the test. Associated with this, bartletTests() and levineTests() give the univariate tests of homogeneity of variance for each response measure in a MLM.

  • covEllipses() draw covariance (data) ellipses for one or more group, optionally including the ellipse for the pooled within-group covariance.

Repeated measure designs

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. (At present, this vignette is only available at HE plots for repeated measures designs.)

Datasets

The package also provides a large collection of data sets illustrating a variety of multivariate linear models of the types listed above, together with graphical displays. The table below classifies these with method tags. Their names are linked to their documentation with graphical output on the pkgdown website, [http://friendly.github.io/heplots].

dataset rows cols title tags
AddHealth 4344 3 Adolescent Health Data MANOVA ordered
Adopted 62 6 Adopted Children MMRA repeated
Bees 246 6 Captive and maltreated bees MANOVA
Diabetes 145 6 Diabetes Dataset MANOVA
dogfood 16 3 Dogfood Preferences MANOVA contrasts candisc
FootHead 90 7 Head measurements of football players MANOVA contrasts
Headache 98 6 Treatment of Headache Sufferers for Sensitivity to Noise MANOVA repeated
Hernior 32 9 Recovery from Elective Herniorrhaphy MMRA candisc
Iwasaki_Big_Five 203 7 Personality Traits of Cultural Groups MANOVA
mathscore 12 3 Math scores for basic math and word problems MANOVA
MockJury 114 17 Effects Of Physical Attractiveness Upon Mock Jury Decisions MANOVA candisc
NeuroCog 242 10 Neurocognitive Measures in Psychiatric Groups MANOVA candisc
NLSY 243 6 National Longitudinal Survey of Youth Data MMRA
oral 56 5 Effect of Delay in Oral Practice in Second Language Learning MANOVA
Oslo 332 14 Oslo Transect Subset Data MANOVA candisc
Overdose 17 7 Overdose of Amitriptyline MMRA cancor
Parenting 60 4 Father Parenting Competence MANOVA contrasts
peng 333 8 Size measurements for adult foraging penguins near Palmer Station MANOVA
Plastic 20 5 Plastic Film Data MANOVA
Pottery2 48 12 Chemical Analysis of Romano-British Pottery MANOVA candisc
Probe 11 5 Response Speed in a Probe Experiment MANOVA repeated
RatWeight 27 6 Weight Gain in Rats Exposed to Thiouracil and Thyroxin MANOVA repeated
ReactTime 10 6 Reaction Time Data repeated
Rohwer 69 10 Rohwer Data Set MMRA MANCOVA
RootStock 48 5 Growth of Apple Trees from Different Root Stocks MANOVA contrasts
Sake 30 10 Taste Ratings of Japanese Rice Wine (Sake) MMRA
schooldata 70 8 School Data MMRA robust
Skulls 150 5 Egyptian Skulls MANOVA contrasts
SocGrades 40 10 Grades in a Sociology Course MANOVA candisc
SocialCog 139 5 Social Cognitive Measures in Psychiatric Groups MANOVA candisc
TIPI 1799 16 Data on the Ten Item Personality Inventory MANOVA candisc
VocabGrowth 64 4 Vocabulary growth data repeated
WeightLoss 34 7 Weight Loss Data repeated

Examples

This example illustrates HE plots using the classic iris data set. How do the means of the flower variables differ by Species? This dataset was the impetus for R. A. Fisher (1936) to propose a method of discriminant analysis using data collected by Edgar Anderson (1928). Though some may rightly deprecate Fisher for being a supporter of eugenics, Anderson’s iris dataset should not be blamed.

A basic HE plot shows the H and E ellipses for the first two response variables (here: Sepal.Length and Sepal.Width). The multivariate test is significant (by Roy’s test) iff the H ellipse projects anywhere outside the E ellipse.

The positions of the group means show how they differ on the two response variables shown, and provide an interpretation of the orientation of the H ellipse: it is long in the directions of differences among the means.

iris.mod <- lm(cbind(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) ~ 
                 Species, data=iris)
heplot(iris.mod)

Contrasts

Contrasts or other linear hypotheses can be shown as well, and the ellipses look better if they are filled. We create contrasts to test the differences between versacolor and virginca and also between setosa and the average of the other two. Each 1 df contrast plots as degenerate 1D ellipse– a line.

Because these contrasts are orthogonal, they add to the total 2 df effect of Species. Note how the first contrast, labeled V:V, distinguishes the means of versicolor from virginica; the second contrast, S:VV distinguishes setosa from the other two.

par(mar=c(4,4,1,1)+.1)
contrasts(iris$Species)<-matrix(c(0, -1, 1, 
                                  2, -1, -1), nrow=3, ncol=2)
contrasts(iris$Species)
#>            [,1] [,2]
#> setosa        0    2
#> versicolor   -1   -1
#> virginica     1   -1
iris.mod <- lm(cbind(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) ~ 
                 Species, data=iris)

hyp <- list("V:V"="Species1","S:VV"="Species2")
heplot(iris.mod, hypotheses=hyp, 
       fill=TRUE, fill.alpha=0.1)

All pairwise HE plots

All pairwise HE plots are produced using the pairs() method for MLM objects.In the plot, note how the means of most pairs of variables are very highly correlated, in the order Setosa < Versicolor < Virginica, but this pattern doesn’t hold for relations with Sepal.Width.

pairs(iris.mod, hypotheses=hyp, hyp.labels=FALSE,
      fill=TRUE, fill.alpha=0.1)

Canonical discriminant view

For more than two response variables, a multivariate effect can be viewed more simply by projecting the data into canonical space — the linear combinations of the responses which show the greatest differences among the group means relative to within-group scatter. The computations are performed with the candisc package, which has an heplot.candisc() method.

library(candisc)
iris.can <- candisc(iris.mod) |> print()
#> 
#> Canonical Discriminant Analysis for Species:
#> 
#>    CanRsq Eigenvalue Difference  Percent Cumulative
#> 1 0.96987   32.19193     31.907 99.12126     99.121
#> 2 0.22203    0.28539     31.907  0.87874    100.000
#> 
#> Test of H0: The canonical correlations in the 
#> current row and all that follow are zero
#> 
#>   LR test stat approx F numDF denDF   Pr(> F)    
#> 1      0.02344  199.145     8   288 < 2.2e-16 ***
#> 2      0.77797   13.794     3   145 5.794e-08 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The HE plot in canonical space shows that the differences among species are nearly entirely one-dimensional. The weights for the variables on the first dimension show how Sepal.Width differs from the other size variables.

# HE plot in canonical space
heplot(iris.can)

#> Vector scale factor set to  43.914

Covariance ellipses

MANOVA relies on the assumption that within-group covariance matrices are all equal. It is useful to visualize these in the space of some of the predictors. covEllipses() provides this both for classical and robust (method="mve") estimates. The figure below shows these for the three Iris species and the pooled covariance matrix, which is the same as the E matrix used in MANOVA tests.

par(mar=c(4,4,1,1)+.1)
covEllipses(iris[,1:4], iris$Species)
covEllipses(iris[,1:4], iris$Species, 
            fill=TRUE, method="mve", add=TRUE, labels="")

References

Anderson, E. (1928). The Problem of Species in the Northern Blue Flags, Iris versicolor L. and Iris virginica L. Annals of the Missouri Botanical Garden, 13, 241–313.

Fisher, R. A. (1936). The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics, 8, 379–388.

Friendly, M. (2006). Data Ellipses, HE Plots and Reduced-Rank Displays for Multivariate Linear Models: SAS Software and Examples. Journal of Statistical Software, 17, 1-42.

Friendly, M. (2007). HE plots for Multivariate General Linear Models. Journal of Computational and Graphical Statistics, 16(2) 421-444. DOI.

Fox, J., Friendly, M. & Monette, G. (2009). Visualizing hypothesis tests in multivariate linear models: The heplots package for R Computational Statistics, 24, 233-246.

Friendly, M. (2010). HE plots for repeated measures designs. Journal of Statistical Software, 37, 1–37.

Friendly, M.; Monette, G. & Fox, J. (2013). Elliptical Insights: Understanding Statistical Methods Through Elliptical Geometry Statistical Science, 28, 1-39.

Friendly, M. & Sigal, M. (2017). Graphical Methods for Multivariate Linear Models in Psychological Research: An R Tutorial. The Quantitative Methods for Psychology, 13, 20-45.

Friendly, M. & Sigal, M. (2018): Visualizing Tests for Equality of Covariance Matrices, The American Statistician, DOI