---
title: "HE plot MANOVA Examples"
author: Michael Friendly
date: "`r Sys.Date()`"
package: heplots
output: 
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
    fig_caption: yes
    toc: true
pkgdown:
  as_is: true
bibliography: "HE-examples.bib"
link-citations: yes
csl: apa.csl
vignette: >
  %\VignetteIndexEntry{HE plot MANOVA Examples}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

<!--
output:
    bookdown::html_document2:
      base_format: rmarkdown::html_vignette
    toc_float: true
      
at end:  sessioninfo::package_info()
-->

```{r, include = FALSE}
knitr::opts_chunk$set(
  message = FALSE,
  warning = FALSE,
  fig.height=5,
  fig.width=5,
  # results='hide',
  # fig.keep='none',
  fig.path='fig/manova-',
  echo=TRUE,
  collapse = TRUE,
  comment = "#>"
  )

```

```{r setup, echo=FALSE}
set.seed(1071)
options(width=80, digits=5, continue="  ")
library(heplots)
library(candisc)
library(car)
```

Vignette built using `heplots`, version `r packageDescription("heplots")[["Version"]]`
and `candisc`, version `r packageDescription("candisc")[["Version"]]`.

# Multivariate Analysis of Variance Designs {-}

This vignette provides some worked examples of the analysis of multivariate linear models 
(MLMs) for MANOVA designs where all predictors are factors, and the goal is to determine
how the group means differ on several response variables in relation to the factors
and possible interactions.

Graphical methods for visualizing results using the `heplots` and the `candisc` packages
are illustrated.
The emphasis here is on using these methods in R, and understanding how they help reveal
aspects of these models that might not be apparent from other graphical displays.

No attempt is made here to describe the theory of MLMs or the statistical details behind
HE plots and their reduced-rank canonical cousins.
For that, see @FoxFriendlyMonette:09:compstat; @Friendly:07:manova; @Friendly:06:hesoft.
	
# Plastic film data

An experiment was conducted to determine the optimum conditions for extruding plastic film. 
Three responses, `tear` resistance, film `gloss` and film `opacity`
were measured in relation to two factors, `rate` of extrusion and amount of an `additive`,
both of these being set to two values, High and Low. The data set comes from
@JohnsonWichern:92.

```{r plastic-str}
data(Plastic, package="heplots")
str(Plastic)
```


The design is thus a $2\times 2$ MANOVA, with $n=5$ per cell and 3 numeric response variables.
Because the effects of the factors on the responses are likely correlated, it is useful to
consider a multivariate analysis, rather than 3 separate univariate ones.

This example illustrates:

* 2D and 3D HE plots, 
* the difference between "effect" scaling and "evidence" (significance) scaling, and 
* visualizing composite linear hypotheses.
	
We begin with an overall MANOVA for the two-way MANOVA model. In all these analyses, we use
`car::Anova()` for significance tests rather than `stats::anova()`, which only provides
so-called "Type I" (sequential) tests for terms in linear models.

In this example, because each effect has 1 df, all of the multivariate statistics 
(Roy's maximum root test, Pillai and Hotelling trace criteria, Wilks' Lambda)
are equivalent, in that they give the same $F$ statistics and $p$-values.
We specify `test.statistic="Roy"` to emphasize that Roy's test has
a natural visual interpretation in HE plots. 

```{r, plastic-mod}
plastic.mod <- lm(cbind(tear, gloss, opacity) ~ rate*additive, data=Plastic)
Anova(plastic.mod, test.statistic="Roy")
```

For the three responses jointly, the main effects of  `rate` and `additive`
are significant, while their interaction is not.
In some approaches to testing effects in multivariate linear models (MLMs),
significant multivariate tests are often followed by univariate tests on each
of the responses separately to determine which responses contribute to each
significant effect.  

In R, univariate analyses are conveniently performed
using the `update()` method for the `mlm` object `plastic.mod`, which re-fits the model with
only a single outcome variable.
```{r, plastic-univar}
Anova(update(plastic.mod, tear ~ .))

Anova(update(plastic.mod, gloss ~ .))

Anova(update(plastic.mod, opacity ~ .))
```

The results above show significant main effects for `tear`,
a significant main effect of `rate` for `gloss`,
and no significant effects for `opacity`, but they don't shed light on the
*nature* of these effects.
Traditional univariate plots of the means for each variable separately
are useful, but they don't allow visualization of the
*relations* among the response variables.

We can visualize these
effects for pairs of variables in an HE plot, showing the "size" and
orientation of hypothesis variation ($\mathbf{H}$) in relation to error
variation ($\mathbf{E}$) as ellipsoids.
When, as here, the model terms have 1 degree of freedom, the
$\mathbf{H}$ ellipsoids degenerate to a line.

In HE plots, the $\mathbf{H}$ ellipses can be scaled relative to the 
$\mathbf{E}$ to show **significance** of effects (`size="evidence"`),
or **effect size** (`size="effect"`). In the former case, a model term
is significant (using Roy's maximum root test) _iff_ the 
$\mathbf{H}$ projects anywhere outside the $\mathbf{E}$ ellipse.

This plot overlays those for both scaling, using thicker lines for the
effect scaling.

```{r, plastic1a, echo=-1}
#| echo=-1,
#| fig.cap = "HE plot for effects on `tear` and `gloss` according to the factors `rate`, `additive` and their interaction, `rate:additive`. The thicker lines show effect size scaling; the thinner lines show significance scaling."
par(mar = c(4,4,1,1)+.1)
## Compare evidence and effect scaling 
colors = c("red", "darkblue", "darkgreen", "brown")
heplot(plastic.mod, size="evidence", 
       col=colors, cex=1.25,
       fill=TRUE, fill.alpha=0.1)
heplot(plastic.mod, size="effect", 
       add=TRUE, lwd=5, term.labels=FALSE, col=colors)
```

The interpretation can be easily read from the plot, at least for the two response variables
(`tear` and `gloss`) that are shown in this bivariate view. The effect of `rate` of extrusion is
highly significant: high rate shows greater `tear` compared to low rate. The effect of amount of 
additive is not significant in this view, but high level of additive has greater `tear` and `gloss`.

With effect scaling, both the $\mathbf{H}$ and $\mathbf{E}$ sums of squares and products
matrices are both divided by the error df, giving multivariate analogs of univariate
measures of effect size, e.g., $(\bar{y}_1-\bar{y}_2) / s$.
With significance scaling, the $\mathbf{H}$ ellipse is further divided by
$\lambda_\alpha$, the critical value of Roy's largest root statistic.
This scaling has the property that an $\mathbf{H}$ ellipse will protrude somewhere
outside the $\mathbf{E}$ ellipse *iff* the
multivariate test is significant at level $\alpha$.
Figure \@ref(fig:plastic1) shows both scalings, using a thinner line for significance scaling.
Note that the (degenerate) ellipse for `additive` is significant, but
does not protrude outside the $\mathbf{E}$ ellipse in this view.
All that is guaranteed is that it will protrude somewhere in the 3D space of
the responses.

By design, means for the levels of interaction terms are not shown in the HE plot,
because doing so in general can lead to messy displays.
We can add them here for the term `rate:additive` as follows:

```{r, plastic1}
#| echo=-1,
#| fig.cap = "HE plot for effects on `tear` and `gloss` according to the factors `rate`, `additive` and their interaction, `rate:additive`. Annotations have added means for the combinations of `rate` and `additive`."
par(mar = c(4,4,1,1)+.1)
# Compare evidence and effect scaling 
colors = c("red", "darkblue", "darkgreen", "brown")
heplot(plastic.mod, size="evidence", 
       col=colors, cex=1.25,
       fill=TRUE, fill.alpha=0.05)
heplot(plastic.mod, size="effect", 
       add=TRUE, lwd=5, term.labels=FALSE, col=colors)

## add interaction means
intMeans <- termMeans(plastic.mod, 'rate:additive', abbrev.levels=2)
points(intMeans[,1], intMeans[,2], pch=18, cex=1.2, col="brown")
text(intMeans[,1], intMeans[,2], rownames(intMeans), 
     adj=c(0.5, 1), col="brown")
lines(intMeans[c(1,3),1], intMeans[c(1,3),2], col="brown")
lines(intMeans[c(2,4),1], intMeans[c(2,4),2], col="brown")
```

The factor means in this plot (Figure \@ref(fig:plastic1) have a simple interpretation:
The high `rate` level yields greater `tear` resistance but lower `gloss`
than the low level.
The high `additive` amount produces greater `tear` resistance and greater `gloss`.

The `rate:additive` interaction is not significant overall, though it
approaches significance for `gloss`. 
The cell means for the combinations
of `rate` and `additive` shown in this figure suggest an explanation,
for tutorial purposes:
with the low level of `rate`, there is little difference in `gloss`
for the levels of `additive`. At the high level of `rate`, there is
a larger difference in `gloss`.  The $\mathbf{H}$ ellipse for the interaction
of `rate:additive` therefore "points" in the direction of `gloss`
indicating that this variable contributes to the interaction in the 
multivariate tests.

In some MANOVA models, it is of interest to test sub-hypotheses 
of a given main effect or interaction, or conversely to test composite
hypotheses that pool together certain effects to test them jointly.
All of these tests (and, indeed, the tests of terms in a given model)
are carried out as tests of general linear hypotheses in the MLM.

In this example, it might be useful to test two composite hypotheses:
one corresponding to both main effects jointly, and another corresponding
to no difference among the means of the four groups (equivalent to
a joint test for the overall model). These tests are specified in terms
of subsets or linear combinations of the model parameters.

```{r, plastic-mod1}
plastic.mod
```

Thus, for example, the joint test of both main effects tests the parameters
`rateHigh` and `additiveHigh`.

```{r, plastic-tests}
print(linearHypothesis(plastic.mod, c("rateHigh", "additiveHigh"), title="Main effects"), 
      SSP=FALSE)

print(linearHypothesis(plastic.mod, c("rateHigh", "additiveHigh", "rateHigh:additiveHigh"), 
                       title="Groups"), SSP=FALSE)
```

Correspondingly, we can display these tests in the HE plot by specifying these tests in the
`hypothesis` argument to `heplot()`, as shown in Figure \@ref(fig:plastic2).

```{r, plastic2}
#| echo=-1,
#| fig.cap="HE plot for `tear` and `gloss`, supplemented with ellipses representing
#|     the joint tests of main effects and all group differences"
par(mar = c(4,4,1,1)+.1)
heplot(plastic.mod, 
       hypotheses=list("Group" =  c("rateHigh", "additiveHigh", "rateHigh:additiveHigh ")),
       col=c(colors, "purple"),
       fill = TRUE, fill.alpha = 0.1,
       lwd=c(2, 3, 3, 3, 2), cex=1.25)
heplot(plastic.mod, 
       hypotheses=list("Main effects" = c("rateHigh", "additiveHigh")), 
       add=TRUE,
       col=c(colors, "darkgreen"), cex=1.25)
```


Finally, a 3D HE plot can be produced with `heplot3d()`, giving Figure \@ref(fig:plastic1-HE3D).
This plot was rotated interactively to a view that shows both main effects
protruding outside the error ellipsoid.
```{r, plastic1-HE3D-code, eval=FALSE}
colors = c("pink", "darkblue", "darkgreen", "brown")
heplot3d(plastic.mod, col=colors)
```
```{r, plastic1-HE3D}
#| echo=FALSE,
#| fig.cap="3D HE plot for the plastic MLM"
knitr::include_graphics("fig/plastic-HE3D.png")
```


# Effects of physical attractiveness on mock jury decisions

In a social psychology
study of influences on jury decisions
by @Plaster:89,
male participants (prison inmates)
were shown a picture of one of three young women.  
Pilot  work
had indicated  that one woman  was beautiful,  another of  average physical
attractiveness, and the third  unattractive.  Participants rated the  woman they
saw on each  of twelve attributes on scales of 1--9.  These measures were used to check on the
manipulation of "attractiveness" by the photo.

Then the participants were told that the person in the photo had committed a
Crime, and asked to rate the seriousness of the crime and recommend a
prison sentence, in Years.  The data are contained in the data frame `MockJury`.[^1]

[^1]:The data were made available courtesy of Karl Wuensch, from
	[http://core.ecu.edu/psyc/wuenschk/StatData/PLASTER.dat](http://core.ecu.edu/psyc/wuenschk/StatData/PLASTER.dat)


```{r, MJdata}
data(MockJury, package = "heplots")
str(MockJury)
```

Sample sizes were roughly balanced  for the independent variables
in the three conditions of the attractiveness of the photo,
and the combinations of this with `Crime`:
```{r, MJdata1}
table(MockJury$Attr)
table(MockJury$Attr, MockJury$Crime)
```

The main questions of interest were:

* Does attractiveness of the "defendant" influence the sentence or perceived seriousness of the crime?  
* Does attractiveness interact with the nature of the crime?

## Manipulation check

But first, as a check on the manipulation of attractiveness,
we try to assess the ratings of the photos in relation to the 
presumed categories of the independent variable `Attr`.  The questions here
are:

* do the ratings of the photos on physical attractiveness
(`phyattr`) confirm the original classification?
* how do other ratings differentiate the photos?

To keep things simple, we consider only a few of the other ratings in a one-way MANOVA.

```{r, jury.mod1}
(jury.mod1 <- lm( cbind(phyattr, happy, independent, sophisticated) ~ Attr, data=MockJury))
Anova(jury.mod1, test="Roy")
```

Note that `Beautiful` is the baseline category of `Attr`, so the
intercept term gives the means for this level.
We see that the means are significantly different on all four variables
collectively, by a joint multivariate test.  A traditional analysis might
follow up with univariate ANOVAs for each measure separately. 

As an aid to interpretation of the MANOVA results
We can examine the test of `Attr` in this model with an HE plot for
pairs of variables, e.g., for `phyattr` and  `happy` (Figure \@ref(fig:jury-mod1-HE)).
The means in this plot show that Beautiful is rated higher on
physical attractiveness than the other two photos, while Unattractive
is rated less happy than the other two. Comparing the sizes of the
ellipses, differences among group means on physical attractiveness
contributes more to significance than do ratings on happy.

```{r, jury-mod1-HE}
#| echo=-1,
#| fig.cap="HE plot for ratings of `phyattr` and `happy` according to the classification of photos on `Attr`"
par(mar = c(4,4,1,1)+.1)
heplot(jury.mod1, main="HE plot for manipulation check",
       fill = TRUE, fill.alpha = 0.1)
```

The function `pairs.mlm()` produces all pairwise HE plots.
This plot (Figure \@ref(fig:jury-mod1-pairs)) shows that the means for `happy`
and `independent` are highly correlated, as are the means for `phyattr`
and `sophisticated`.  In most of these pairwise plots, the means form a
triangle rather than a line, suggesting that these attributes are indeed
measuring different aspects of the photos.


```{r, jury-mod1-pairs}
#| fig.cap="HE plots for all pairs of ratings according to the classification of photos on `Attr`"
pairs(jury.mod1)
```

With 3 groups and 4 variables, the $\mathbf{H}$ ellipsoid has only $s=\min(df_h, p)=2$
dimensions.  `candisc()` carries out a canonical discriminant analysis
for the MLM  and returns an object that can be used to show an HE plot in the
space of the canonical dimensions.  This is plotted in Figure \@ref(fig:jury-can1).

```{r, jury-can1a}
jury.can <- candisc(jury.mod1)
jury.can
```

`heplot.candisc()` is the HE plot method for `candisc` objects
```{r, jury-can1}
#| echo=-1,
#| fig.cap="Canonical discriminant HE plot for the MockJury data. Variable vectors show the correlations of the predictors with the canonical dimensions."
par(xpd=TRUE, mar=c(4,4,3,1)+.1)
heplot(jury.can, 
       rev.axes = TRUE,
       fill = c(TRUE,FALSE),
       prefix="Canonical dimension", 
       main="Canonical HE plot")
```

In this plot, 

* the variable vectors are determined by the canonical structure
coefficients and represent the correlations of the predictor variables with
the canonical variables. Thus, an angle near zero with an axis represents a
correlation close to 1.0; an angle near 90$^o$ represent a correlation close to 0.0.
(The axes must be scaled to have equal unit lengths for angles to be interpretable.)

* The lengths of arrows are scaled to roughly fill the plot, but relative length
represents the overall strength of the relation of the variable with the canonical dimensions.

* Points represent the means of the canonical scores on the two dimensions for the three groups of photos.

From this we can see that 91% of the variation among group means
is accounted for by the first dimension, and this is nearly completely
aligned with `phyattr`. 
The second dimension, accounting for the remaining 9%
is determined nearly entirely by ratings on `happy` and `independent`.
This display gives a relatively simple account of the results of the MANOVA
and the relations of each of the ratings to discrimination among the photos.

## Main analysis

Proceeding to the main questions of interest, we carry out a two-way MANOVA of the responses
`Years` and `Serious` in relation to the independent variables
`Attr` and `Crime`.

```{r, jury-mod2}
# influence of Attr of photo and nature of crime on Serious and Years
jury.mod2 <- lm( cbind(Serious, Years) ~ Attr * Crime, data=MockJury)
Anova(jury.mod2, test="Roy")
```
We see that there is a nearly significant  interaction between `Attr` and `Crime`
and a strong effect of `Attr`.


```{r, jury-mod2-HE}
#| echo=-1,
#| fig.cap="HE plot for the two-way MANOVA for `Years` and `Serious`"
par(mar=c(4,4,3,1)+.1)
heplot(jury.mod2)
```


The HE plot shows that the nearly significant
interaction of `Attr:Crime` is mainly in terms of
differences among the groups on the response of `Years` of sentence,
with very little contribution of `Serious`.  We explore this interaction in a bit more detail
below.  The main effect of `Attr` is also dominated by differences among groups
on `Years`. 

If we assume that `Years` of sentence is the main outcome of interest,
it also makes sense to carry out a step-down test of this variable by itself,
controlling for the rating of seriousness (`Serious`) of the crime.
The model `jury.mod3` below is equivalent to an ANCOVA for `Years`.

```{r, jury-mod3-HE}
# stepdown test (ANCOVA), controlling for Serious
jury.mod3 <- lm( Years ~ Serious + Attr * Crime, data=MockJury)
t(coef(jury.mod3))
Anova(jury.mod3)
```
Thus, even when adjusting for `Serious` rating, there is still a 
significant main effect of `Attr` of the photo, but also a hint of
an interaction of `Attr` with `Crime`. The coefficient for
`Serious` indicates that participants awarded 0.84 additional
years of sentence for each 1 unit step on the scale of seriousness of crime.

A particularly useful
method for visualizing the fitted effects in such univariate response
models is provided by the `effects`.  By default `allEffects()`
calculates the predicted values for all high-order terms in a given
model, and the `plot` method produces plots of these values for
each term.  The statements below produce Figure \@ref(fig:jury-mod3-eff).

```{r, jury-mod3-eff}
#| fig.width=9, 
#| fig.height=4,
#| fig.cap="Effect plots for  `Serious` and the `Attr * Crime` interaction in the ANCOVA model `jury.mod3`."
library(effects)
jury.eff <- allEffects(jury.mod3)
plot(jury.eff, ask=FALSE)
```

<!-- \includegraphics[width=\textwidth]{fig/plot-jury-mod3-eff} -->
<!-- \caption{Effect plots for  `Serious` and the `Attr * Crime` -->
<!-- 	 in the ANCOVA model `jury.mod3`.} -->
<!--  {#fig:jury-mod3-eff} -->

The effect plot for `Serious` shows the expected linear relation
between that variable and `Years`. Of greater interest here is the nature
of the possible interaction of `Attr` and `Crime` on `Years`
of sentence, controlling for `Serious`.
The effect plot shows that for the crime of Swindle, there is a much
greater `Years` of sentence awarded to Unattractive defendants.

# Egyptian skulls from five epochs

This example examines physical measurements of size and shape made on
150 Egyptian skulls from five epochs ranging from
4000 BC to 150 AD.
The measures are: maximal breadth (`mb`), basibregmatic height (`bh`),
basialiveolar length (`bl`), and nasal height (`nh`) of each skull.
<!-- See [http://www.redwoods.edu/instruct/agarwin/anth_6_measurements.htm] -->
<!-- for the definitions of these measures, and Figure \@ref(fig:skulls) for a diagram. -->
See Figure \@ref(fig:skulls) for a diagram.
The question of interest is whether and how these measurements change over time.
Systematic changes over time is of interest because this
would indicate interbreeding with immigrant populations.


```{r skulls}
#| echo=FALSE,
#| out.width="60%",
#| fig.cap='Diagram of the skull measurements. Maximal breadth and basibregmatic height are the basic measures of "size" of a skull.  Basialveolar length and nasal height are important anthropometric measures of "shape".'
knitr::include_graphics("fig/skulls.jpg")
```


```{r, skulls1}
data(Skulls)
str(Skulls)
table(Skulls$epoch)
```

Note that `epoch` is an ordered factor, so the default contrasts
will be orthogonal polynomials.  This assumes that `epoch`
values are equally spaced, which they are not.  However, examining
the linear and quadratic trends is useful to a first approximation.

For ease of labeling various outputs, it is useful to trim the
`epoch` values and assign more meaningful variable labels.
```{r, skulls2}
# make shorter labels for epochs
Skulls$epoch <- factor(Skulls$epoch, labels=sub("c","",levels(Skulls$epoch)))
# assign better variable labels
vlab <- c("maxBreadth", "basibHeight", "basialLength", "nasalHeight")
```

We start with some simple displays of the means by epoch.  From the numbers,
the means don't seem to vary much.  
A `pairs` plot, Figure \@ref(fig:skulls4), joining points
by `epoch` is somewhat more revealing for the bivariate relations among means.

```{r, skulls3}
means <- aggregate(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls, FUN=mean)[,-1]
rownames(means) <- levels(Skulls$epoch)
means
```

```{r, skulls4}
#| fig.width=7,
#| fig.height=7,
#| fig.cap="Pairs plot of means of Skulls data, by epoch."
pairs(means, vlab,
      panel = function(x, y) {
          text(x, y, levels(Skulls$epoch))
          lines(x,y)
      })
```


Perhaps better for visualizing the trends over time is a set of boxplots,
joining means over `epoch`. Using `bwplot()` from the `lattice`
package requires reshaping the data from wide to long format.  The following
code produces Figure \@ref(fig:skulls-bwplot).

<!-- TODO: do this with ggplot2 -->

```{r, skulls-bwplot}
#| fig.height=7,
#| fig.width=7,
#| fig.cap="Boxplots of Skulls data, by epoch, for each variable."
library(lattice)
library(reshape2)
sklong <- melt(Skulls, id="epoch")

bwplot(value ~ epoch | variable, data=sklong, scales="free", 
	ylab="Variable value", xlab="Epoch",
	strip=strip.custom(factor.levels=paste(vlab,
	                                       " (", levels(sklong$variable), ")",
	                                       sep="")),
	panel = function(x,y, ...) {
		panel.bwplot(x, y, ...)
		panel.linejoin(x,y, col="red", ...)
	}) 
```

The trend lines aren't linear, but neither are they random, so something systematic has been going
on!

Now, fit the MANOVA model, and test the effect of `epoch` with `car::Anova()`.
We see that the multivariate means differ substantially.
```{r, skulls5}
# fit manova model
sk.mod <- lm(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls)
Anova(sk.mod)
```

Perhaps of greater interest are the more focused tests of trends over time.
These are based on tests of the coefficients in the model `sk.mod`
being jointly equal to zero, for subsets of the 
(polynomial) contrasts in `epoch`.
```{r, skulls5a}
coef(sk.mod)
```

We use `linearHypothesis()` for a multivariate test of the
`epoch.L` linear effect.
The linear trend is highly significant.  It is not obvious from 
Figure \@ref(fig:skulls4) that maximal breadth and nasal are increasing
over time, while the other two measurements have negative slopes.

```{r, skulls6}
coef(sk.mod)["epoch.L",]
print(linearHypothesis(sk.mod, "epoch.L"), SSP=FALSE) # linear component
```

`linearHypothesis()` can also be used to test composite hypotheses.
Here we test all non-linear coefficients jointly.  The result indicates
that, collectively, all non-linear terms are not significantly different
from zero.
```{r, skulls6a}
print(linearHypothesis(sk.mod, c("epoch.Q", "epoch.C", "epoch^4")), SSP=FALSE)
```

Again, HE plots can show the patterns of these tests of multivariate hypotheses.
With four response variables, it is easiest to look at all pairwise
HE plots with the `pairs.mlm()` function. 
The statement below produces Figure \@ref(fig:skulls-HE-pairs).
In this plot, we show the hypothesis ellipsoids for the overall
effect of `epoch`, as well as those for the tests just shown
for the linear trend component `epoch.L`
as well as the joint test of all non-linear terms.

<!-- TODO: shorten labels -->

```{r, skulls-HE-pairs}
#| out.width="100%",
#| fig.cap="Pairs HE plot of Skulls data, showing multivariate tests of `epoch`, as well as tests of linear and nonlinear trends."
pairs(sk.mod, variables=c(1,4,2,3),
	hypotheses=list(Lin="epoch.L", 
	                NonLin=c("epoch.Q", "epoch.C", "epoch^4")), 
	var.labels=vlab[c(1,4,2,3)])
```


<!-- 	\includegraphics[width=.8\textwidth]{fig/plot-skulls-HE-pairs} -->
<!-- \caption{Pairs HE plot of Skulls data, showing multivariate tests of -->
<!-- 	`epoch`, as well as tests of linear and nonlinear trends.} -->
<!--  {#fig:skulls-HE-pairs} -->


These plots have an interesting geometric interpretation:
the $\mathbf{H}$ ellipses for the overall effect of `epoch`
are representations of the additive decomposition of this effect into  
$\mathbf{H}$ ellipses for the linear and nonlinear linear
hypothesis tests according to

$$\mathbf{H}_{\textrm{epoch}} = \mathbf{H}_{\textrm{linear}} + \mathbf{H}_{\textrm{nonlinear}}$$

where the linear term has rank 1 (and so plots as a line), while the nonlinear
term has rank 3.  In each panel, it can be seen that the large direction of
the $\mathbf{H}_{\textrm{epoch}}$ leading to significance of this effect corresponds
essentially to the linear contrast.  $\mathbf{H}_{\textrm{nonlinear}}$ is the orthogonal
complement of $\mathbf{H}_{\textrm{linear}}$ in the space of $\mathbf{H}_{\textrm{epoch}}$,
but nowhere does it protrude beyond the boundary of the $\mathbf{E}$ ellipsoid.




## References