**Multivariate
ANOVA (MANOVA)**

We'll use a simple example adapted from 'Discovering
Statistics with R' by Andy Fields (by the way – it's a pretty good book
explaining statistics).

First, download the data file into your workspace
from OCD and load
it:

> ocdData=read.delim("OCD.dat",
header = TRUE)

If you want, have a look at the data (note one
categorical IV and two numerical DVs)

> ocdData

You should also run some descriptive stats.

Let's code Group levels with shorter names:

> ocdData$Group=factor(ocdData$Group, levels = c("CBT", "BT",
"No Treatment Control"), labels = c("CBT", "BT",
"NT"))

The formula is the same as for ANOVA (i.e. DV ~ IV)
but this time DV is not a vector but a matrix where each column is a separate
DV.

First, we need to create this matrix of DVs:

> outcome=cbind(ocdData$Actions, ocdData$Thoughts)

Then we use *manova**(**)* function to generate the model:

> ocdModel=manova(outcome ~ Group, data = ocdData)

Let's look at the output of the model and the test
statistic:

> ocdModel

> summary(ocdModel,
intercept = TRUE)

Next, you can inspect the differences between
various tests which can be applied to the model:

> summary(ocdModel,
intercept = TRUE, test = "Wilks")

> summary(ocdModel,
intercept = TRUE, test = "Hotelling")

> summary(ocdModel,
intercept = TRUE, test = "Roy")

Watch the difference in p-values!

We can also test for effects on two DVs separately:

> summary.aov(ocdModel)

Alternatively, the same results can be achieved by
explicitly testing two one-way ANOVA models:

> actionModel=aov(Actions ~ Group, data = ocdData)

> thoughtsModel=aov(Thoughts ~ Group, data = ocdData)

> summary(actionModel)

> summary(thoughtsModel)

In our case those two effects are not significant,
so you **cannot** infer anything from planned comparisons. Otherwise, the
procedure for the contrasts is the same as for one-way ANOVA (since actually
you've just run two separate one-way ANOVAs)

> summary.lm(actionModel)

> summary.lm(thoughtsModel)

Finally, there are non-parametric alternatives to
MANOVA *mulrank**(**) *and *cmanova**()*.
They are both part of *WRS* package, if you have spare time try installing
it (instructions are in Tuesday session) and making it work with this dataset.