Anova: Anova Tables for Various Statistical Models

Description

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.

Usage

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, ...)

Arguments

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".

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.

Value

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.

Warning

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.

Details

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).

References

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.

Examples

Run this code

# 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 {
# }

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