Calculates type-II or type-III analysis-of-variance tables for
model objects produced by `lm`

, `glm`

, `multinom`

(in the nnet package), `polr`

(in the MASS
package), `coxph`

(in the survival package),
`coxme`

(in the coxme pckage),
`svyglm`

(in the survey package), `rlm`

(in the MASS package),
`lmer`

in the lme4 package,
`lme`

in the nlme package, and (by the default method) for most
models with a linear predictor and asymptotically normal coefficients (see details below). For linear
models, F-tests are calculated; for generalized linear models,
likelihood-ratio chisquare, Wald chisquare, or F-tests are calculated;
for multinomial logit and proportional-odds logit models, likelihood-ratio
tests are calculated. Various test statistics are provided for multivariate
linear models produced by `lm`

or `manova`

. Partial-likelihood-ratio tests
or Wald tests are provided for Cox models. Wald chi-square tests are provided for fixed effects in
linear and generalized linear mixed-effects models. Wald chi-square or F tests are provided
in the default case.

`Anova(mod, ...)`Manova(mod, ...)

# S3 method for lm
Anova(mod, error, type=c("II","III", 2, 3),
white.adjust=c(FALSE, TRUE, "hc3", "hc0", "hc1", "hc2", "hc4"),
vcov.=NULL, singular.ok, ...)

# S3 method for aov
Anova(mod, ...)

# S3 method for glm
Anova(mod, type=c("II","III", 2, 3),
test.statistic=c("LR", "Wald", "F"),
error, error.estimate=c("pearson", "dispersion", "deviance"),
singular.ok, ...)

# S3 method for multinom
Anova(mod, type = c("II","III", 2, 3), ...)

# S3 method for polr
Anova(mod, type = c("II","III", 2, 3), ...)

# S3 method for mlm
Anova(mod, type=c("II","III", 2, 3), SSPE, error.df,
idata, idesign, icontrasts=c("contr.sum", "contr.poly"), imatrix,
test.statistic=c("Pillai", "Wilks", "Hotelling-Lawley", "Roy"),...)

# S3 method for manova
Anova(mod, ...)

# S3 method for mlm
Manova(mod, ...)

# S3 method for Anova.mlm
print(x, ...)

# S3 method for Anova.mlm
summary(object, test.statistic, univariate=object$repeated,
multivariate=TRUE, p.adjust.method, ...)

# S3 method for summary.Anova.mlm
print(x, digits = getOption("digits"),
SSP=TRUE, SSPE=SSP, ... )

# S3 method for univaov
print(x, digits = max(getOption("digits") - 2L, 3L),
style=c("wide", "long"),
by=c("response", "term"),
...)

# S3 method for univaov
as.data.frame(x, row.names, optional, by=c("response", "term"), ...)

# S3 method for coxph
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("LR", "Wald"), ...)

# S3 method for coxme
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Wald", "LR"), ...)

# S3 method for lme
Anova(mod, type=c("II","III", 2, 3),
vcov.=vcov(mod, complete=FALSE), singular.ok, ...)

# S3 method for mer
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE), singular.ok, ...)

# S3 method for merMod
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE), singular.ok, ...)

# S3 method for svyglm
Anova(mod, ...)

# S3 method for rlm
Anova(mod, ...)

# S3 method for default
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE),
singular.ok, ...)

mod

`lm`

, `aov`

, `glm`

, `multinom`

, `polr`

`mlm`

, `coxph`

, `coxme`

, `lme`

, `mer`

, `merMod`

, `svyglm`

,
`rlm`

, or other suitable model object.

error

for a linear model, an `lm`

model object from which the
error sum of squares and degrees of freedom are to be calculated. For
F-tests for a generalized linear model, a `glm`

object from which the
dispersion is to be estimated. If not specified, `mod`

is used.

type

type of test, `"II"`

, `"III"`

, `2`

, or `3`

. Roman numerals are equivalent to
the corresponding Arabic numerals.

singular.ok

defaults to `TRUE`

for type-II tests, and `FALSE`

for type-III tests where the tests for models with aliased coefficients
will not be straightforwardly interpretable;
if `FALSE`

, a model with aliased coefficients produces an error.

test.statistic

for a generalized linear model, whether to calculate
`"LR"`

(likelihood-ratio), `"Wald"`

, or `"F"`

tests; for a Cox
or Cox mixed-effects model, whether to calculate `"LR"`

(partial-likelihood ratio) or
`"Wald"`

tests; in the default case or for linear mixed models fit by
`lmer`

, whether to calculate Wald `"Chisq"`

or Kenward-Roger
`"F"`

tests with Satterthwaite degrees of freedom (*warning:* the KR F-tests
can be very time-consuming).
For a multivariate linear model, the multivariate test statistic to compute --- one of
`"Pillai"`

, `"Wilks"`

, `"Hotelling-Lawley"`

, or `"Roy"`

,
with `"Pillai"`

as the default. The `summary`

method for `Anova.mlm`

objects permits the specification of more than one multivariate
test statistic, and the default is to report all four.

error.estimate

for F-tests for a generalized linear model, base the
dispersion estimate on the Pearson residuals (`"pearson"`

, the default); use the
dispersion estimate in the model object (`"dispersion"`

); or base the dispersion estimate on
the residual deviance (`"deviance"`

). For binomial or Poisson GLMs, where the dispersion
is fixed to 1, setting `error.estimate="dispersion"`

is changed to `"pearson"`

,
with a warning.

white.adjust

if not `FALSE`

, the default,
tests use a heteroscedasticity-corrected coefficient
covariance matrix; the various values of the argument specify different corrections.
See the documentation for `hccm`

for details. If `white.adjust=TRUE`

then the `"hc3"`

correction is selected.

SSPE

For `Anova`

for a multivariate linear model, the
error sum-of-squares-and-products matrix; if missing, will be computed
from the residuals of the model; for the `print`

method for the `summary`

of
an `Anova`

of a multivariate linear model,
whether or not to print the error SSP matrix (defaults to `TRUE`

).

SSP

if `TRUE`

(the default), print the sum-of-squares and
cross-products matrix for the hypothesis and the response-transformation matrix.

error.df

The degrees of freedom for error; if missing, will be taken from the model.

idata

an optional data frame giving a factor or factors defining the
intra-subject model for multivariate repeated-measures data. See
*Details* for an explanation of the intra-subject design and for
further explanation of the other arguments relating to intra-subject factors.

idesign

a one-sided model formula using the ``data'' in `idata`

and
specifying the intra-subject design.

icontrasts

names of contrast-generating functions to be applied by default
to factors and ordered factors, respectively, in the within-subject
``data''; the contrasts must produce an intra-subject model
matrix in which different terms are orthogonal. The default is
`c("contr.sum", "contr.poly")`

.

imatrix

as an alternative to specifying `idata`

, `idesign`

, and
(optionally) `icontrasts`

, the model matrix for the within-subject design
can be given directly in the form of list of named elements. Each element gives
the columns of the within-subject model matrix for a term to be tested, and must
have as many rows as there are responses; the columns of the within-subject model
matrix for different terms must be mutually orthogonal.

x, object

object of class `"Anova.mlm"`

to print or summarize.

multivariate, univariate

compute and print multivariate and univariate tests for a repeated-measures
ANOVA or multivariate linear model; the default is `TRUE`

for both for repeated measures and `TRUE`

for `multivariate`

for a multivariate linear model.

p.adjust.method

if given for a multivariate linear model when univariate tests are requested, the
univariate tests are corrected for simultaneous inference by term; if specified, should be one of the methods
recognized by `p.adjust`

or `TRUE`

, in which case the default (Holm) adjustment is used.

digits

minimum number of significant digits to print.

style

for printing univariate tests if requested for a multivariate linear model; one of `"wide"`

,
the default, or `"long"`

.

by

if univariate tests are printed in `"long"`

`style`

, they can be ordered `by`

`"response"`

, the default, or by `"term"`

.

row.names, optional

not used.

vcov.

in the `default`

method, an optional coefficient-covariance matrix or function
to compute a covariance matrix, computed by default by applying the generic `vcov`

function to the model object.
A similar argument may be supplied to the `lm`

method, and the default (`NULL`

) is to ignore the argument;
if both `vcov.`

and `white.adjust`

are supplied to the `lm`

method, the latter is used.

…

do not use.

An object of class `"anova"`

, or `"Anova.mlm"`

, which usually is printed.
For objects of class `"Anova.mlm"`

, there is also a `summary`

method,
which provides much more detail than the `print`

method about the MANOVA, including
traditional mixed-model univariate F-tests with Greenhouse-Geisser and Huynh-Feldt
corrections.

Be careful of type-III tests: For a traditional multifactor ANOVA model with interactions, for example, these tests will normally only be sensible when using contrasts that, for different terms, are
orthogonal in the row-basis of the model, such as those produced by `contr.sum`

, `contr.poly`

, or `contr.helmert`

, but *not* by the default
`contr.treatment`

. In a model that contains factors, numeric covariates, and interactions, main-effect tests for factors will be for differences over the origin. In contrast (pun intended),
type-II tests are invariant with respect to (full-rank) contrast coding. If you don't understand this issue, then you probably shouldn't use `Anova`

for type-III tests.

The designations "type-II" and "type-III" are borrowed from SAS, but the definitions used here do not correspond precisely to those employed by SAS. Type-II tests are calculated according to the principle of marginality, testing each term after all others, except ignoring the term's higher-order relatives; so-called type-III tests violate marginality, testing each term in the model after all of the others. This definition of Type-II tests corresponds to the tests produced by SAS for analysis-of-variance models, where all of the predictors are factors, but not more generally (i.e., when there are quantitative predictors). Be very careful in formulating the model for type-III tests, or the hypotheses tested will not make sense.

As implemented here, type-II Wald tests are a generalization of the linear hypotheses used to generate these tests in linear models.

For tests for linear models, multivariate linear models, and Wald tests for generalized linear models,
Cox models, mixed-effects models, generalized linear models fit to survey data, and in the default case,
`Anova`

finds the test statistics without refitting the model. The `svyglm`

method simply
calls the `default`

method and therefore can take the same arguments.

The standard R `anova`

function calculates sequential ("type-I") tests.
These rarely test interesting hypotheses in unbalanced designs.

A MANOVA for a multivariate linear model (i.e., an object of
class `"mlm"`

or `"manova"`

) can optionally include an
intra-subject repeated-measures design.
If the intra-subject design is absent (the default), the multivariate
tests concern all of the response variables.
To specify a repeated-measures design, a data frame is provided defining the repeated-measures factor or
factors
via `idata`

, with default contrasts given by the `icontrasts`

argument. An intra-subject model-matrix is generated from the formula
specified by the `idesign`

argument; columns of the model matrix
corresponding to different terms in the intra-subject model must be orthogonal
(as is insured by the default contrasts). Note that the contrasts given in
`icontrasts`

can be overridden by assigning specific contrasts to the
factors in `idata`

. As an alternative, the within-subjects model matrix
can be specified directly via the `imatrix`

argument.
`Manova`

is essentially a synonym for `Anova`

for multivariate linear models.

If univariate tests are requested for the `summary`

of a multivariate linear model, the object returned
contains a `univaov`

component of `"univaov"`

; `print`

and `as.data.frame`

methods are
provided for the `"univaov"`

class.

For the default method to work, the model object must contain a standard
`terms`

element, and must respond to the `vcov`

, `coef`

, and `model.matrix`

functions.
If any of these requirements is missing, then it may be possible to supply it reasonably simply (e.g., by
writing a missing `vcov`

method for the class of the model object).

Fox, J. (2016)
*Applied Regression Analysis and Generalized Linear Models*,
Third Edition. Sage.

Fox, J. and Weisberg, S. (2019)
*An R Companion to Applied Regression*, Third Edition, Sage.

Hand, D. J., and Taylor, C. C. (1987)
*Multivariate Analysis of Variance and Repeated Measures: A Practical
Approach for Behavioural Scientists.* Chapman and Hall.

O'Brien, R. G., and Kaiser, M. K. (1985)
MANOVA method for analyzing repeated measures designs: An extensive primer.
*Psychological Bulletin* **97**, 316--333.

`linearHypothesis`

, `anova`

`anova.lm`

, `anova.glm`

,
`anova.mlm`

, `anova.coxph`

, `svyglm`

.

# NOT RUN { ## Two-Way Anova mod <- lm(conformity ~ fcategory*partner.status, data=Moore, contrasts=list(fcategory=contr.sum, partner.status=contr.sum)) Anova(mod) Anova(mod, type=3) # note use of contr.sum in call to lm() ## One-Way MANOVA ## See ?Pottery for a description of the data set used in this example. summary(Anova(lm(cbind(Al, Fe, Mg, Ca, Na) ~ Site, data=Pottery))) ## MANOVA for a randomized block design (example courtesy of Michael Friendly: ## See ?Soils for description of the data set) soils.mod <- lm(cbind(pH,N,Dens,P,Ca,Mg,K,Na,Conduc) ~ Block + Contour*Depth, data=Soils) Manova(soils.mod) summary(Anova(soils.mod), univariate=TRUE, multivariate=FALSE, p.adjust.method=TRUE) ## a multivariate linear model for repeated-measures data ## See ?OBrienKaiser for a description of the data set used in this example. phase <- factor(rep(c("pretest", "posttest", "followup"), c(5, 5, 5)), levels=c("pretest", "posttest", "followup")) hour <- ordered(rep(1:5, 3)) idata <- data.frame(phase, hour) idata mod.ok <- lm(cbind(pre.1, pre.2, pre.3, pre.4, pre.5, post.1, post.2, post.3, post.4, post.5, fup.1, fup.2, fup.3, fup.4, fup.5) ~ treatment*gender, data=OBrienKaiser) (av.ok <- Anova(mod.ok, idata=idata, idesign=~phase*hour)) summary(av.ok, multivariate=FALSE) ## A "doubly multivariate" design with two distinct repeated-measures variables ## (example courtesy of Michael Friendly) ## See ?WeightLoss for a description of the dataset. imatrix <- matrix(c( 1,0,-1, 1, 0, 0, 1,0, 0,-2, 0, 0, 1,0, 1, 1, 0, 0, 0,1, 0, 0,-1, 1, 0,1, 0, 0, 0,-2, 0,1, 0, 0, 1, 1), 6, 6, byrow=TRUE) colnames(imatrix) <- c("WL", "SE", "WL.L", "WL.Q", "SE.L", "SE.Q") rownames(imatrix) <- colnames(WeightLoss)[-1] (imatrix <- list(measure=imatrix[,1:2], month=imatrix[,3:6])) contrasts(WeightLoss$group) <- matrix(c(-2,1,1, 0,-1,1), ncol=2) (wl.mod<-lm(cbind(wl1, wl2, wl3, se1, se2, se3)~group, data=WeightLoss)) Anova(wl.mod, imatrix=imatrix, test="Roy") ## mixed-effects models examples: # } # NOT RUN { library(nlme) example(lme) Anova(fm2) # } # NOT RUN { # } # NOT RUN { library(lme4) example(glmer) Anova(gm1) # } # NOT RUN { # }