manova.gls: Multivariate Analysis of Variance

In mvMORPH: Multivariate Comparative Tools for Fitting Evolutionary Models to Morphometric Data

Multivariate Analysis of Variance

Description

Performs a Multivariate Analysis of Variance (MANOVA) on an object fitted by the mvgls or the mvols function. With the regularized methods by penalized likelihood implemented in mvgls and mvols (ridgeArch penalty), this function can be used to compare model fit on high-dimensional datasets (where the number of variables is larger than the number of observations). When model fit is performed by maximum likelihood (method="LL"), both parametric and permutation tests are possible.

Usage

manova.gls(object, test=c("Pillai", "Wilks", "Hotelling-Lawley", "Roy"),
            type=c("I","II","III"), nperm=1000L, L=NULL, ...)
  

Arguments

object	A model fit obtained by the mvgls or mvols function.
test	The multivariate test statistic to compute - "Wilks", "Pillai", "Hotelling-Lawley", or "Roy"
type	The type of test (sums of squares and cross-products) - "I", "II", or "III"
nperm	The number of permutations used for building the null distribution of the chosen statistic. Permutation is the only available approach for high-dimensional PL models, but either permutations or parametric tests can be used with maximum likelihood (method "LL" in mvgls and mvols)
L	A (contrasts) matrix or a vector giving linear combinations of the coefficients rows.
...	Further arguments to be passed through. (e.g., nbcores=2L to provide the number of cores used for parallel calculus; parametric=FALSE to obtain permutation instead of parametric tests for maximum likelihood fit; verbose=TRUE to display a progress bar during permutations; rhs=0 the "right-hand-side" vector for general linear hypothesis testing; P can be used to specify a matrix of contrasts giving linear combinations of the coefficient columns. See details)

Details

manova.gls allows performing multivariate tests (e.g. Pillai's, Wilks, Hotelling-Lawley and Roy largest root) on generalized least squares (GLS) linear model (objects of class "mvgls", or OLS with objects of class "mvols") fit by either maximum likelihood (method="LL") or penalized likelihood (method="PL-LOO") using the mvgls or mvols function.

General Linear Hypothesis of the form:

\bold{LB=O}

or

\bold{LBP=O}

Where L is a matrix specifying linear combinations of the model coefficients (B) can be provided through the argument L. This type of "contrasts" matrix allows testing specific hypotheses (for instance pairwise differences - see ?pairwise.contrasts and ?pairwise.glh). The right-hand-side matrix O is a constant matrix (of zeros by default) that can be provided through the argument rhs. P is a matrix specifying linear combinations of the model coefficients (B) estimated for each responses (usually used in repeated measures designs or for testing linear, quadratic, etc. relationships between successive responses).

Permutations on high-dimensional datasets is time consuming. You can use the option nbcores to parallelize the calculus over several cores using forking in UNIX platforms (default is nbcores=1L. Estimated time to completion is displayed when verbose=TRUE.

Value

An object of class 'manova.mvgls' which is usually printed. It contains a list including the following components:

test	the multivariate test statistic used
type	the type of tests used to compute the SSCP matrices
stat	the statistic calculated for each terms in the model
pvalue	the pvalues calculated for each terms in the model

Note

For PL methods, only the "RidgeArch" penalty is allowed for now. A tutorial is available from Dryad: https://doi.org/10.5061/dryad.jsxksn052

Author(s)

J. Clavel

References

Clavel, J., Aristide, L., Morlon, H., 2019. A Penalized Likelihood framework for high-dimensional phylogenetic comparative methods and an application to new-world monkeys brain evolution. Systematic Biology 68(1): 93-116.

Clavel, J., Morlon, H. 2020. Reliable phylogenetic regressions for multivariate comparative data: illustration with the MANOVA and application to the effect of diet on mandible morphology in phyllostomid bats. Systematic Biology 69(5): 927-943.

See Also

mvgls, mvols, pairwise.glh, GIC, EIC

Examples

# ---------------------------------------------------- #
# Multivariate regression tests (continuous predictor) #
# ---------------------------------------------------- #

set.seed(1)
n <- 32 # number of species
p <- 30 # number of traits

tree <- pbtree(n=n) # phylogenetic tree
R <- crossprod(matrix(runif(p*p),p))  # a random symmetric matrix (covariance)

# simulate a dataset
Y <- mvSIM(tree, model="BM1", nsim=1, param=list(sigma=R))
X <- rnorm(n) # continuous
grp <- rep(1:2, each=n/2)
dataset <- list(y=Y, x=X, grp=as.factor(grp))

# Model fit
model1 <- mvgls(y~x, data=dataset, tree=tree, model="BM", method="LOO")

# Multivariate test
(multivariate_test <- manova.gls(model1, nperm=999, test="Pillai"))

# ---------------------------------------------------- #
# Multivariate regression tests (discrete predictor)   #
# ---------------------------------------------------- #

# MANOVA on a binary predictor
model2 <- mvgls(y~grp, data=dataset, tree=tree, model="lambda", method="LOO")

# Multivariate test
(multivariate_test <- manova.gls(model2, nperm=999, test="Pillai", verbose=TRUE))

# ---------------------------------------------------- #
# Parametric MANOVA tests                              #
# ---------------------------------------------------- #

# When p<n we can use non-penalized approaches and parametric tests
model2b <- mvgls(y[,1:2]~grp, data=dataset, tree=tree, model="lambda", method="LL")
(multivariate_test2b <- manova.gls(model2b, test="Pillai"))

# ---------------------------------------------------- #
# Multivariate contrasts tests                         #
# ---------------------------------------------------- #

# Multivariate contrasts allow testing specific hypotheses 
# (see also ?pairwise.glh and ?pairwise.contrasts)

# We can replicate the above result by testing if the
# group means are different using the following contrast:
L = matrix(c(0,1), ncol=2)
(manova.gls(model2b, test="Pillai", L=L))

# ---------------------------------------------------- #
# Repeated measures design tests                       #
# ---------------------------------------------------- #

# Contrasts can be used also to test if there's differences
# between repeated measures (responses variables)
# For instance, for comparing y[,1] and y[,2], define the contrast:
P = matrix(c(1,-1), nrow=2)
(manova.gls(model2b, test="Pillai", P=P, L=L))

mvMORPH documentation built on March 31, 2023, 6:25 p.m.