Anova Tables for Various Statistical Models
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 class '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 class 'aov'
Anova(mod, ...)
## S3 method for class '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 class 'multinom'
Anova(mod, type = c("II","III", 2, 3), ...)
## S3 method for class 'polr'
Anova(mod, type = c("II","III", 2, 3), ...)
## S3 method for class '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 class 'manova'
Anova(mod, ...)
## S3 method for class 'mlm'
Manova(mod, ...)
## S3 method for class 'Anova.mlm'
print(x, ...)
## S3 method for class 'Anova.mlm'
summary(object, test.statistic, univariate=object$repeated,
multivariate=TRUE, p.adjust.method, ...)
## S3 method for class 'summary.Anova.mlm'
print(x, digits = getOption("digits"),
SSP=TRUE, SSPE=SSP, ... )
## S3 method for class 'univaov'
print(x, digits = max(getOption("digits") - 2L, 3L),
style=c("wide", "long"),
by=c("response", "term"),
...)
## S3 method for class 'univaov'
as.data.frame(x, row.names, optional, by=c("response", "term"), ...)
## S3 method for class 'coxph'
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("LR", "Wald"), ...)
## S3 method for class 'coxme'
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Wald", "LR"), ...)
## S3 method for class 'lme'
Anova(mod, type=c("II","III", 2, 3),
vcov.=vcov(mod, complete=FALSE), singular.ok, ...)
## S3 method for class 'mer'
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE), singular.ok, ...)
## S3 method for class 'merMod'
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE), singular.ok, ...)
## S3 method for class 'svyglm'
Anova(mod, ...)
## S3 method for class 'rlm'
Anova(mod, ...)
## Default S3 method:
Anova(mod, type=c("II", "III", 2, 3),
test.statistic=c("Chisq", "F"), vcov.=vcov(mod, complete=FALSE),
singular.ok, ...)mod |
|
error |
for a linear model, an |
type |
type of test, |
singular.ok |
defaults to |
test.statistic |
for a generalized linear model, whether to calculate
|
error.estimate |
for F-tests for a generalized linear model, base the
dispersion estimate on the Pearson residuals ( |
white.adjust |
if not |
SSPE |
For |
SSP |
if |
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 |
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
|
imatrix |
as an alternative to specifying |
x, object |
object of class |
multivariate, univariate |
compute and print multivariate and univariate tests for a repeated-measures
ANOVA or multivariate linear model; the default is |
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 |
digits |
minimum number of significant digits to print. |
style |
for printing univariate tests if requested for a multivariate linear model; one of |
by |
if univariate tests are printed in |
row.names, optional |
not used. |
vcov. |
in the |
... |
do not use. |
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).
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.
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.
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:
## Not run:
library(nlme)
example(lme)
Anova(fm2)
## End(Not run)
## Not run:
library(lme4)
example(glmer)
Anova(gm1)
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.