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), lmer in the lme4 package, lme in the nlme package, and for any model with a linear predictor and asymptotically normal coefficients that responds to the vcov and coef functions. 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 class 'lm':
Anova(mod, error, type=c("II","III", 2, 3), 
	white.adjust=c(FALSE, TRUE, "hc3", "hc0", "hc1", "hc2", "hc4"), 
	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, multivariate=TRUE, 
    univariate=TRUE, digits=getOption("digits"), ...)
    
## S3 method for class 'coxph':
Anova(mod, type=c("II","III", 2, 3), 
	test.statistic=c("LR", "Wald"), ...)
	
## S3 method for class 'lme':
Anova(mod, type=c("II","III", 2, 3),
		vcov.=vcov(mod), singular.ok, ...)

## S3 method for class 'mer':
Anova(mod, type=c("II","III", 2, 3), 
	test.statistic=c("chisq", "F"), vcov.=vcov(mod), singular.ok, ...)
		
## S3 method for class 'svyglm':
Anova(mod, ...)
	
## S3 method for class 'default':
Anova(mod, type=c("II","III", 2, 3), 
	test.statistic=c("Chisq", "F"), vcov.=vcov(mod), 
	singular.ok, ...)

Arguments

mod

lm, aov, glm, multinom, polr mlm, coxph, lme, mer, svyglm 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 e

type

type of test, "II", "III", 2, or 3.

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 coef

test.statistic

for a generalized linear model, whether to calculate "LR" (likelihood-ratio), "Wald", or "F" tests; for a Cox model, whether to calculate "LR" (partial-likelihood ratio) or "Wald"

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"), which, e.g., is fixed to 1

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 h

SSPE

The error sum-of-squares-and-products matrix; if missing, will be computed from the residuals of the model.

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 argume

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.

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

x, object

object of class "Anova.mlm" to print or summarize.

multivariate, univariate

print multivariate and univariate tests for a repeated-measures ANOVA; the default is TRUE for both.

digits

minimum number of significant digits to print.

vcov.

an optional coefficient-covariance matrix, computed by default by applying the generic vcov function to the model object.

...

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.

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.

References

Fox, J. (2008) Applied Regression Analysis and Generalized Linear Models, Second Edition. Sage. Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second 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)

## 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)

## 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:

library(nlme)
	example(lme)
	Anova(fm2)

library(lme4)
	example(lmer)
	Anova(gm1)

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