Compute a (generalized) analysis of variance table for one or more multivariate linear models.

```
# S3 method for mlm
anova(object, …,
test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy",
"Spherical"),
Sigma = diag(nrow = p), T = Thin.row(proj(M) - proj(X)),
M = diag(nrow = p), X = ~0,
idata = data.frame(index = seq_len(p)), tol = 1e-7)
```

object

an object of class `"mlm"`

.

…

further objects of class `"mlm"`

.

test

choice of test statistic (see below). Can be abbreviated.

Sigma

(only relevant if `test == "Spherical"`

). Covariance
matrix assumed proportional to `Sigma`

.

T

transformation matrix. By default computed from `M`

and
`X`

.

M

formula or matrix describing the outer projection (see below).

X

formula or matrix describing the inner projection (see below).

idata

data frame describing intra-block design.

tol

tolerance to be used in deciding if the residuals are
rank-deficient: see `qr`

.

An object of class `"anova"`

inheriting from class `"data.frame"`

The `anova.mlm`

method uses either a multivariate test statistic for
the summary table, or a test based on sphericity assumptions (i.e.
that the covariance is proportional to a given matrix).

For the multivariate test, Wilks' statistic is most popular in the
literature, but the default Pillai--Bartlett statistic is
recommended by Hand and Taylor (1987). See
`summary.manova`

for further details.

For the `"Spherical"`

test, proportionality is usually with the
identity matrix but a different matrix can be specified using `Sigma`

.
Corrections for asphericity known as the Greenhouse--Geisser,
respectively Huynh--Feldt, epsilons are given and adjusted \(F\) tests are
performed.

It is common to transform the observations prior to testing. This
typically involves
transformation to intra-block differences, but more complicated
within-block designs can be encountered,
making more elaborate transformations necessary. A
transformation matrix `T`

can be given directly or specified as
the difference between two projections onto the spaces spanned by
`M`

and `X`

, which in turn can be given as matrices or as
model formulas with respect to `idata`

(the tests will be
invariant to parametrization of the quotient space `M/X`

).

As with `anova.lm`

, all test statistics use the SSD matrix from
the largest model considered as the (generalized) denominator.

Contrary to other `anova`

methods, the intercept is not excluded
from the display in the single-model case. When contrast
transformations are involved, it often makes good sense to test for a
zero intercept.

Hand, D. J. and Taylor, C. C. (1987)
*Multivariate Analysis of Variance and Repeated Measures.*
Chapman and Hall.

# NOT RUN { require(graphics) utils::example(SSD) # Brings in the mlmfit and reacttime objects mlmfit0 <- update(mlmfit, ~0) ### Traditional tests of intrasubj. contrasts ## Using MANOVA techniques on contrasts: anova(mlmfit, mlmfit0, X = ~1) ## Assuming sphericity anova(mlmfit, mlmfit0, X = ~1, test = "Spherical") ### tests using intra-subject 3x2 design idata <- data.frame(deg = gl(3, 1, 6, labels = c(0, 4, 8)), noise = gl(2, 3, 6, labels = c("A", "P"))) anova(mlmfit, mlmfit0, X = ~ deg + noise, idata = idata, test = "Spherical") anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ noise, idata = idata, test = "Spherical" ) anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ deg, idata = idata, test = "Spherical" ) f <- factor(rep(1:2, 5)) # bogus, just for illustration mlmfit2 <- update(mlmfit, ~f) anova(mlmfit2, mlmfit, mlmfit0, X = ~1, test = "Spherical") anova(mlmfit2, X = ~1, test = "Spherical") # one-model form, eqiv. to previous ### There seems to be a strong interaction in these data plot(colMeans(reacttime)) # }