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 and svycoxph (in the survey package), rlm (in the MASS package), lmer (in the lme4 package), lme (in the nlme package), clm and clmm (in the ordinal 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"),
   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 clm
Anova(mod, ...)
# S3 method for clmm
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, ...)

Arguments

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.

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


## 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()

## use of vcov.; the following are equivalent:

Anova(mod, white.adjust = TRUE)
Anova(mod, vcov. = hccm) # vcov. is a function, type = "hc3" is the default
Anova(mod, vcov. = hccm(mod, type = "hc3")) # vcov. is a matrix
Anova(mod, vcov. = function(m) hccm(m, type = "hc3")) # passing type as an argument

## 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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