# anova.mlm: Comparisons between Multivariate Linear Models

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```## S3 method for class '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) ```

## Arguments

 `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`.

## Details

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.

## Value

An object of class `"anova"` inheriting from class `"data.frame"`

## Note

The Huynh–Feldt epsilon differs from that calculated by SAS (as of v. 8.2) except when the DF is equal to the number of observations minus one. This is believed to be a bug in SAS, not in R.

## References

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

`summary.manova`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```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)) ```