---
title: "Multivariate Analysis of Variance"
---
## Multivariate analysis of variance
* Standard ANOVA has just one response variable.
* What if you have more than one response?
* Try an ANOVA on each response separately.
* But might miss some kinds of interesting dependence between the responses that distinguish the groups.
## Packages
```{r bManova-1, results='hide', message=FALSE}
library(car) # may need to install first
library(tidyverse)
library(MVTests) # also may need to install
```
## Small example
* Measure yield and seed weight of plants grown under 2 conditions: low and high amounts of fertilizer.
* Data (fertilizer, yield, seed weight):
```{r bManova-2 }
url <- "http://ritsokiguess.site/datafiles/manova1.txt"
hilo <- read_delim(url, " ")
```
* 2 responses, yield and seed weight.
## The data
```{r bManova-3 }
hilo
```
## Boxplot for yield for each fertilizer group
```{r ferto,size="small",fig.height=3}
ggplot(hilo, aes(x = fertilizer, y = yield)) + geom_boxplot()
```
Yields overlap for fertilizer groups.
## Boxplot for weight for each fertilizer group
```{r casteldisangro,size="small",fig.height=3}
ggplot(hilo, aes(x = fertilizer, y = weight)) + geom_boxplot()
```
Weights overlap for fertilizer groups.
## ANOVAs for yield and weight
\small
```{r bManova-4 }
hilo.y <- aov(yield ~ fertilizer, data = hilo)
summary(hilo.y)
hilo.w <- aov(weight ~ fertilizer, data = hilo)
summary(hilo.w)
```
\normalsize
Neither response depends significantly on fertilizer. But...
## Plotting both responses at once
- Have two response variables (not more), so can plot the
response variables against *each other*, labelling points by
which fertilizer group they're from.
\footnotesize
- First, create data frame with points $(31,14)$ and $(38,10)$ (why? Later):
```{r bManova-5, size="footnotesize"}
d <- tribble(
~line_x, ~line_y,
31, 14,
38, 10
)
```
- Then plot data as points, and add line through points in `d`:
```{r bManova-6 }
ggplot(hilo, aes(x = yield, y = weight,
colour = fertilizer)) + geom_point() +
geom_line(data = d,
aes(x = line_x, y = line_y, colour = NULL)) -> g
```
\normalsize
## The plot
```{r charlecombe, echo=F, fig.height=4}
g
```
## Comments
- Graph construction:
- Joining points in `d` by line.
- `geom_line`
inherits `colour` from `aes` in `ggplot`.
- Data frame `d` has
no `fertilizer` (previous `colour`), so have to unset.
- Results:
* High-fertilizer plants have both yield and weight high.
* True even though no sig difference in yield or weight individually.
* Drew line separating highs from lows on plot.
## MANOVA finds multivariate differences
* Is difference found by diagonal line significant? MANOVA finds out.
\footnotesize
```{r bManova-7, echo=FALSE}
options(width = 60)
```
```{r bManova-8}
response <- with(hilo, cbind(yield, weight))
hilo.1 <- manova(response ~ fertilizer, data = hilo)
summary(hilo.1)
```
\normalsize
* Yes! Difference between groups is *diagonally*, not just up/down
(weight) or left-right (yield). The *yield-weight combination* matters.
## Strategy
* Create new response variable by gluing together columns of
responses, using `cbind`.
* Use `manova` with new response, looks like `lm` otherwise.
* With more than 2 responses, cannot draw graph. What then?
* If MANOVA test significant, cannot use Tukey. What then?
* Use *discriminant analysis* (of which more later).
## Another way to do MANOVA
using `Manova` from package `car`:
```{r, include=FALSE}
w <- getOption("width")
options(width = 132)
```
\tiny
```{r bManova-10}
hilo.2.lm <- lm(response ~ fertilizer, data = hilo)
hilo.2 <- Manova(hilo.2.lm)
summary(hilo.2)
```
\normalsize
```{r, include=FALSE}
options(width = w)
```
## Comments
* Same result as small-m `manova`.
* `Manova` will also do *repeated measures*, coming up
later.
## Assumptions
- normality of each response variable within each treatment group
- this is actually *multivariate* normality, with correlations
- equal spreads: each response variable has same variances and correlations (with other response variables) within each treatment group. Here:
- yield has same spread for low and high fertilizer
- weight has same spread for low and high fertilizer
- correlation between yield and weight is same for low and high fertilizer
- test equal spread using Box's $M$ test
- a certain amount of unequalness is OK, so only a concern if P-value from $M$-test is very small (eg. less than 0.001).
## Assumptions for yield-weight data
For normal quantile plots, need "extra-long" with all the data values in one column:
```{r}
hilo %>%
pivot_longer(-fertilizer, names_to = "xname",
values_to = "xvalue") %>%
ggplot(aes(sample = xvalue)) + stat_qq() +
stat_qq_line() +
facet_grid(xname ~ fertilizer, scales = "free") -> g
```
There are only four observations per response variable - treatment group combination, so graphs are not very informative (over):
## The plots
```{r, fig.height=4}
g
```
## Box M test
- Make sure package `MVTests` loaded first.
- inputs:
- the response matrix (or, equivalently, the response-variable columns from your dataframe)
- the column with the grouping variable in it (most easily gotten with `$`).
```{r}
hilo %>% select(yield, weight) -> numeric_values
summary(BoxM(numeric_values, hilo$fertilizer))
```
- No problem at all with unequal spreads.
## Another example: peanuts
* Three different varieties
of peanuts (mysteriously, 5, 6 and 8) planted in two different
locations.
* Three response variables: `y`, `smk` and
`w`.
```{r bManova-11, size="footnotesize"}
u <- "http://ritsokiguess.site/datafiles/peanuts.txt"
peanuts.orig <- read_delim(u, " ")
```
## The data
\small
```{r bManova-12}
peanuts.orig
```
\normalsize
## Setup for analysis
```{r bManova-13 }
peanuts.orig %>%
mutate(
location = factor(location),
variety = factor(variety)
) -> peanuts
response <- with(peanuts, cbind(y, smk, w))
head(response)
```
## Analysis (using `Manova`)
```{r, include=FALSE}
w <- getOption("width")
options(width = 100)
```
\tiny
```{r bManova-14}
peanuts.1 <- lm(response ~ location * variety, data = peanuts)
peanuts.2 <- Manova(peanuts.1)
summary(peanuts.2)
```
\normalsize
```{r, include=FALSE}
options(width = w)
```
## Comments
* Interaction not quite significant, but main effects are.
* Combined response variable `(y,smk,w)` definitely depends
on location and on variety
* Weak dependence of `(y,smk,w)` on the location-variety *combination.*
* Understanding that dependence beyond our scope right now.
## Comments
- this time there are only six observations per location and four per variety, so normality is still difficult to be confident about
- `y` at location 1 seems to be the worst for normality (long tails / outliers), and maybe `y` at location 2 is skewed left, but the others are not bad
- there is some evidence of unequal spread (slopes of lines), but is it bad enough to worry about? (Box M-test, over).
## Box's M tests
- One for location, one for variety:
```{r}
summary(BoxM(response, peanuts$location))
summary(BoxM(response, peanuts$variety))
```
- Neither of these P-values is low enough to worry about. (Remember, the P-value here has to be *really* small to indicate a problem.)