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`

and `svycoxph`

(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"),
vcov.=vcov(mod, complete=TRUE), 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 svycoxph
Anova(mod, type=c("II", "III", 2, 3),
test.statistic="Wald", ...)

# 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, error.df, ...)

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.

- mod
`lm`

,`aov`

,`glm`

,`multinom`

,`polr`

`mlm`

,`coxph`

,`coxme`

,`lme`

,`mer`

,`merMod`

,`svyglm`

,`svycoxph`

,`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. This argument is available only for some`Anova`

methods.- 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

`error.df`

missing for a multivariate linear model (object of class`"mlm"`

), the error degrees of freedom will be taken from the model.For the

`default`

`Anova`

method, if an F-test is requested and if`error.df`

is missing, the error degrees of freedom will be computed by applying the`df.residual`

function to the model; if`df.residual`

returns`NULL`

or`NA`

, then a chi-square test will be substituted for the F-test (with a message to that effect.- 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.

In the `glm`

method, `vcov.`

is ignored unless `test="Wald"`

; in the `mer`

and `merMod`

methods, `vcov.`

is ignored if `test="F"`

.

- ...
do not use.

John Fox jfox@mcmaster.ca; the code for the Mauchly test and Greenhouse-Geisser and Huynh-Feldt
corrections for non-spericity in repeated-measures ANOVA are adapted from the functions
`stats:::stats:::mauchly.test.SSD`

and `stats:::sphericity`

by R Core; `summary.Anova.mlm`

and `print.summary.Anova.mlm`

incorporates code contributed by Gabriel Baud-Bovy.

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.

```
## 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:
if (FALSE) # loads nlme package
library(nlme)
example(lme)
Anova(fm2)
if (FALSE) # loads lme4 package
library(lme4)
example(glmer)
Anova(gm1)
```

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